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Journal articles on the topic '$\varphi$ functions'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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1

Peleshenko, B. I. "On boundedness of operators of weak type $(\varphi_0, \psi_0, \varphi_1, \psi_1)$ in Lorentz spaces in limit cases." Researches in Mathematics 15 (February 15, 2021): 107. http://dx.doi.org/10.15421/240716.

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We prove theorems on boundedness of operators of weak type $(\varphi_0, \psi_0, \varphi_1, \psi_1)$ from Lorentz space $\Lambda_{\varphi,a}(\mathbb{R}^n)$ to $\Lambda_{\varphi,b}(\mathbb{R}^n)$ in “limit” cases, when one of functions $\varphi(t) / \varphi_0(t)$, $\varphi(t) / \varphi_1(t)$ slowly changes at zero and at infinity.

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2

Bayer, Tomáš. "Plotting the map projection graticule involving discontinuities based on combined sampling." Geoinformatics FCE CTU 17, no. 2 (August 23, 2018): 31–64. http://dx.doi.org/10.14311/gi.17.2.3.

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This article presents new algorithm for interval plotting the projection graticule on the interval $\varOmega=\varOmega_{\varphi}\times\varOmega_{\lambda}$ based on the combined sampling technique. The proposed method synthesizes uniform and adaptive sampling approaches and treats discontinuities of the coordinate functions $F,G$. A full set of the projection constant values represented by the projection pole $K=[\varphi_{k},\lambda_{k}]$, two standard parallels $\varphi_{1},\varphi_{2}$ and the central meridian shift $\lambda_{0}^{\prime}$ are supported. In accordance with the discontinuity direction it utilizes a subdivision of the given latitude/longitude intervals $\varOmega_{\varphi}=[\underline{\varphi},\overline{\varphi}]$, $\varOmega_{\lambda}=[\underline{\lambda},\overline{\lambda}]$ to the set of disjoint subintervals $\varOmega_{k,\varphi}^{g},$$\varOmega_{k,\lambda}^{g}$ forming tiles without internal singularities, containing only "good" data; their parameters can be easily adjusted. Each graticule tile borders generated over $\varOmega_{k}^{g}=\varOmega_{k,\varphi}^{g}\times\varOmega_{k,\lambda}^{g}$ run along singularities. For combined sampling with the given threshold $\overline{\alpha}$ between adjacent segments of the polygonal approximation the recursive approach has been used; meridian/parallel offsets are $\Delta\varphi,\Delta\lambda$. Finally, several tests of the proposed algorithms are involved.

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3

Rooin, Jamal, Akram Alikhani, and Mohammad Sal Moslehian. "Operator m-convex functions." Georgian Mathematical Journal 25, no. 1 (March 1, 2018): 93–107. http://dx.doi.org/10.1515/gmj-2017-0045.

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AbstractThe aim of this paper is to present a comprehensive study of operatorm-convex functions. Let{m\in[0,1]}, and{J=[0,b]}for some{b\in\mathbb{R}}or{J=[0,\infty)}. A continuous function{\varphi\colon J\to\mathbb{R}}is called operatorm-convex if for any{t\in[0,1]}and any self-adjoint operators{A,B\in\mathbb{B}({\mathscr{H}})}, whose spectra are contained inJ, we have{\varphi(tA+m(1-t)B)\leq t\varphi(A)+m(1-t)\varphi(B)}. We first generalize the celebrated Jensen inequality for continuousm-convex functions and Hilbert space operators and then use suitable weight functions to give some weighted refinements. Introducing the notion of operatorm-convexity, we extend the Choi–Davis–Jensen inequality for operatorm-convex functions. We also present an operator version of the Jensen–Mercer inequality form-convex functions and generalize this inequality for operatorm-convex functions involving continuous fields of operators and unital fields of positive linear mappings. Employing the Jensen–Mercer operator inequality for operatorm-convex functions, we construct them-Jensen operator functional and obtain an upper bound for it.

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4

Castillo, René E., Julio C. Ramos-Fernández, and Harold Vacca-González. "Properties of multiplication operators on the space of functions of bounded φ-variation." Open Mathematics 19, no. 1 (January 1, 2021): 492–504. http://dx.doi.org/10.1515/math-2021-0050.

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Abstract In this paper, the functions u ∈ B V φ [ 0 , 1 ] u\in B{V}_{\varphi }\left[0,1] which define compact and Fredholm multiplication operators M u {M}_{u} acting on the space of functions of bounded φ \varphi -variation are studied. All the functions u ∈ B V φ [ 0 , 1 ] u\hspace{-0.08em}\in \hspace{-0.08em}B{V}_{\varphi }\left[0,\hspace{-0.08em}1] which define multiplication operators M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] {M}_{u}:B{V}_{\varphi }\left[0,1]\to B{V}_{\varphi }\left[0,1] with closed range are characterized.

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5

Kofanov, V. A., and I. V. Popovich. "Sharp Nagy type inequalities for the classes of functions with given quotient of the uniform norms of positive and negative parts of a function." Researches in Mathematics 28, no. 1 (August 19, 2020): 3. http://dx.doi.org/10.15421/242001.

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For any $p\in (0, \infty],$ $\omega > 0,$ $d \ge 2 \omega,$ we obtain the sharp inequality of Nagy type$$\|x_{\pm}\|_\infty \le\frac{\|(\varphi+c)_{\pm}\|_\infty}{\|\varphi+c\|_{L_p(I_{2\omega})}} \left\|x \right\|_{L_{p} \left(I_d \right)}$$on the set $S_{\varphi}(\omega)$ of $d$-periodic functions $x$ having zeros with given the sine-shaped $2\omega$-periodiccomparison function $\varphi$, where $c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$ is such that$$ \|x_{+}\|_\infty \cdot\|x_{-}\|^{-1}_\infty = \|(\varphi+c)_{+}\|_\infty \cdot\|(\varphi+c)_{-}\|^{-1}_\infty .$$In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms $\|x_{+}\|_\infty / \|x_-\|_\infty$.

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6

Alexandrova, T. V., and V. A. Kofanov. "Sharp inequalities of various metrics on the classes of functions with given comparison function." Researches in Mathematics 29, no. 1 (July 5, 2021): 11. http://dx.doi.org/10.15421/242102.

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For any $q > p > 0$, $\omega > 0,$ $d \ge 2 \omega,$ we obtain the following sharp inequality of various metrics$$\|x\|_{L_q(I_{d})} \le \frac{\|\varphi +c\|_{L_q(I_{2\omega})}}{\|\varphi + c \|_{L_p(I_{2\omega})}}\|x\|_{L_p(I_{d})}$$on the set $S_{\varphi}(\omega)$ of $d$-periodic functions $x$ having zeros with given the sine-shaped $2\omega$-periodic comparison function $\varphi$, where $c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$ is such that$$\|x_{\pm}\|_{L_p(I_{d})} = \|(\varphi +c)_{\pm}\|_{L_p(I_{2\omega})}\,.$$In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms $\|x_{+}\|_{L_p(I_{d})} / \|x_-\|_{L_p(I_{d})}$.

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7

Zivari-Kazempour, Abbas, and Mohammad Reza Hadadi. "Some properties of $varphi$-convex functions." Mathematical Analysis and Convex Optimization 1, no. 2 (December 1, 2020): 121–27. http://dx.doi.org/10.29252/maco.1.2.12.

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8

Yao, Jen-Chih, Xi Yin Zheng, and Jiangxing Zhu. "Stable Minimizers of $\varphi$-Regular Functions." SIAM Journal on Optimization 27, no. 2 (January 2017): 1150–70. http://dx.doi.org/10.1137/16m1086741.

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9

Ciemnoczołowski, J., and W. Orlicz. "Composing functions of bounded $\varphi$-variation." Proceedings of the American Mathematical Society 96, no. 3 (March 1, 1986): 431. http://dx.doi.org/10.1090/s0002-9939-1986-0822434-6.

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10

Zivari-Kazempour, Abbas, and Mohammad Reza Hadadi. "Some properties of $varphi$-convex functions." Mathematical Analysis and Convex Optimization 1, no. 2 (December 1, 2020): 121–27. http://dx.doi.org/10.29252/maco.1.2.12.

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11

Hong, Qing, Guozhen Lu, and Lu Zhang. "L p boundedness of rough bi-parameter Fourier integral operators." Forum Mathematicum 30, no. 1 (January 1, 2018): 87–107. http://dx.doi.org/10.1515/forum-2016-0221.

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Abstract In this paper, we will investigate the boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form: T(f\/)(x)=\frac{1}{(2\pi)^{2n}}\int_{\mathbb{R}^{2n}}e^{i\varphi(x,\xi,\eta)}% \cdot a(x,\xi,\eta)\cdot\widehat{f}(\xi,\eta)\,d\xi\,d\eta, where {x=(x_{1},x_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{n}} and {\xi,\eta\in\mathbb{R}^{n}\setminus\{0\}} , {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} is the amplitude, and the phase function is of the form \varphi(x,\xi,\eta)=\varphi_{1}(x_{1},\xi\/)+\varphi_{2}(x_{2},\eta) , with \varphi_{1},\varphi_{2}\in L^{\infty}\Phi^{2}(\mathbb{R}^{n}\times\mathbb{R}^{% n}\setminus\{0\}) , and satisfies a certain rough non-degeneracy condition (see (2.2)). The study of these operators are motivated by the {L^{p}} estimates for one-parameter FIOs and bi-parameter Fourier multipliers and pseudo-differential operators. We will first define the bi-parameter FIOs and then study the {L^{p}} boundedness of such operators when their phase functions have compact support in frequency variables with certain necessary non-degeneracy conditions. We will then establish the {L^{p}} boundedness of the more general FIOs with amplitude {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} and non-smooth phase function {\varphi(x,\xi,\eta)} on x satisfying a rough non-degeneracy condition.

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12

Lusky, Wolfgang. "Composition operators on weighted spaces of holomorphic functions on the upper half plane." MATHEMATICA SCANDINAVICA 122, no. 1 (February 20, 2018): 141. http://dx.doi.org/10.7146/math.scand.a-97126.

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We consider moderately growing weight functions $v$ on the upper half plane $\mathbb G$ called normal weights which include the examples $(\mathrm{Im} w)^a$, $w \in \mathbb G$, for fixed $a > 0$. In contrast to the comparable, well-studied situation of normal weights on the unit disc here there are always unbounded composition operators $C_{\varphi }$ on the weighted spaces $Hv(\mathbb G)$. We characterize those holomorphic functions $\varphi \colon \mathbb G \rightarrow \mathbb G$ where the composition operator $C_{\varphi } $ is a bounded operator $Hv(\mathbb G) \rightarrow Hv(\mathbb G)$ by a simple property which depends only on $\varphi $ but not on $v$. Moreover we show that there are no compact composition operators $C_{\varphi }$ on $Hv(\mathbb G)$.

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13

Hue, Nguyen Ngoc, and Duong Quoc Huy. "Monotonicity of sequences involving generalized convexity function and sequences." Tamkang Journal of Mathematics 46, no. 2 (June 30, 2015): 121–27. http://dx.doi.org/10.5556/j.tkjm.46.2015.1626.

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In this paper, by using the theory of generalized convexity functions we introduce and prove monotonicity of sequences of the forms $$ \left\{\left(\prod\limits_{k=1}^nf\left({a_k\over a_n}\right)\right)^{1/n}\right\},\quad \left\{\left(\prod\limits_{k=1}^nf\left({\varphi(k)\over\varphi(n)}\right)\right)^{1/\varphi(n)}\right\}, $$ $$ \left\{{1\over n}\sum_{k=1}^nf\left({a_n\over a_k}\right)\right\}\quad\text{or}\quad \left\{{1\over\varphi(n)}\sum_{k=1}^nf\left({\varphi(n)\over\varphi(k)}\right)\right\}, $$ where $f$ belongs to the classes of $AG$-convex (concave), $HA$-convex (concave), or $HG$-convex (concave) functions defined on suitable intervals, $\{a_n\}$ is a given sequence and $\varphi$ is a given function that satisfy some preset conditions. As a consequence, we obtain some generalizations of Alzer type inequalities.

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14

Lo, Ching-on, and Anthony Wai-keung Loh. "Hilbert-Schmidtness of weighted composition operators and their differences on Hardy spaces." Opuscula Mathematica 40, no. 4 (2020): 495–507. http://dx.doi.org/10.7494/opmath.2020.40.4.495.

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Let \(u\) and \(\varphi\) be two analytic functions on the unit disk \(\mathbb{D}\) such that \(\varphi(\mathbb{D}) \subset \mathbb{D}\). A weighted composition operator \(uC_{\varphi}\) induced by \(u\) and \(\varphi\) is defined on \(H^2\), the Hardy space of \(\mathbb{D}\), by \(uC_{\varphi}f := u \cdot f \circ \varphi\) for every \(f\) in \(H^2\). We obtain sufficient conditions for Hilbert-Schmidtness of \(uC_{\varphi}\) on \(H^2\) in terms of function-theoretic properties of \(u\) and \(\varphi\). Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on \(H^2\).

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15

Logvinenko, A. D. "A Method of Identification of Analyser Characteristics That is Free of Probability Summation Effect." Perception 26, no. 1_suppl (August 1997): 245. http://dx.doi.org/10.1068/v970191.

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A detection model (originally proposed by Quick) comprising, in a sequence of linear analysers, varphi1, …, varphi n, nonlinear transducer functions, and the Minkowski decision rule, is widely used, especially when it is necessary to take into account the effect of probability summation. However, there is a general belief that the analyser characteristics cannot be determined in detection experiments since there is a trade-off between these characteristics and the decision rule. Here we show how to overcome this problem, ie how to identify the analysers varphi1, …, varphi n despite the probability summation between them. The observer's performance is assumed to be quantitatively defined in terms of an equidetection surface (EDS). Each analyser varphi i is expressed as a weighted sum of linear (coordinate) analysers {phi j}: varphi i=sum j=1 n a ijphi j, so that an identification of the analysers {phi i} is then reduced to evaluating the weight matrix A={ a ij}. It has been proven that A can be uniquely recovered from an ellipsoidal approximation of EDS in the neighbourhood of at least two points. More specifically, the following equation holds true: A−1 DA= H1−1 H2, where D is a diagonal matrix, H1 and H2 are the matrices of the quadratic forms determining the n-dimensional ellipsoids approximating EDS. Thus, the matrix H1−1 H2 known from experiment is a similarity transform of the diagonal matrix, the columns of A being the eigenvectors of H1−1 H2. Hence, any eigensystem routine can be used to derive A from H1−1 H2.

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16

Khats', R. V. "Sufficient conditions for the improved regular growth of entire functions in terms of their averaging." Carpathian Mathematical Publications 12, no. 1 (June 12, 2020): 46–54. http://dx.doi.org/10.15330/cmp.12.1.46-54.

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Let $f$ be an entire function of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$ and $h(\varphi)$ be its indicator. In 2011, the author of the article has been proved that if $f$ is of improved regular growth (an entire function $f$ is called a function of improved regular growth if for some $\rho\in (0,+\infty)$ and $\rho_1\in (0,\rho)$, and a $2\pi$-periodic $\rho$-trigonometrically convex function $h(\varphi)\not\equiv -\infty$ there exists a set $U\subset\mathbb C$ contained in the union of disks with finite sum of radii and such that $\log |{f(z)}|=|z|^\rho h(\varphi)+o(|z|^{\rho_1})$, $U\not\ni z=re^{i\varphi}\to\infty$), then for some $\rho_3\in (0,\rho)$ the relation \begin{equation*} \int_1^r {\frac{\log |{f(te^{i\varphi})}|}{t}}\, dt=\frac{r^\rho}{\rho}h(\varphi)+o(r^{\rho_3}),\quad r\to +\infty, \end{equation*} holds uniformly in $\varphi\in [0,2\pi]$. In the present paper, using the Fourier coefficients method, we establish the converse statement, that is, if for some $\rho_3\in (0,\rho)$ the last asymptotic relation holds uniformly in $\varphi\in [0,2\pi]$, then $f$ is a function of improved regular growth. It complements similar results on functions of completely regular growth due to B. Levin, A. Grishin, A. Kondratyuk, Ya. Vasyl'kiv and Yu. Lapenko.

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17

ARENA, ORAZIO, and CRISTINA GIANNOTTI. "ELLIPTIC EXTENSIONS IN THE DISK WITH OPERATORS IN DIVERGENCE FORM." Bulletin of the Australian Mathematical Society 88, no. 1 (August 20, 2012): 51–55. http://dx.doi.org/10.1017/s000497271200069x.

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AbstractLet $\varphi _0$ and $\varphi _1$ be regular functions on the boundary $\partial D$ of the unit disk $D$ in $\mathbb {R}^2$, such that $\int _{0}^{2\pi }\varphi _1\,d\theta =0$ and $\int _{0}^{2\pi }\sin \theta (\varphi _1-\varphi _0)\,d\theta =0$. It is proved that there exist a linear second-order uniformly elliptic operator $L$ in divergence form with bounded measurable coefficients and a function $u$ in $W^{1,p}(D)$, $1 \lt p \lt 2$, such that $Lu=0$ in $D$ and with $u|_{\partial D}= \varphi _0$ and the conormal derivative $\partial u/\partial N|_{\partial D}=\varphi _1$.

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18

Kumar, Manoj, and N. Shravan Kumar. "Convolution structures for an Orlicz space with respect to vector measures on a compact group." Proceedings of the Edinburgh Mathematical Society 64, no. 1 (February 2021): 87–98. http://dx.doi.org/10.1017/s0013091521000018.

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The aim of this paper is to present some results about the space $L^{\varPhi }(\nu ),$ where $\nu$ is a vector measure on a compact (not necessarily abelian) group and $\varPhi$ is a Young function. We show that under natural conditions, the space $L^{\varPhi }(\nu )$ becomes an $L^{1}(G)$-module with respect to the usual convolution of functions. We also define one more convolution structure on $L^{\varPhi }(\nu ).$

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19

Dilnyi, V. M., and T. I. Hishchak. "On the intersection of weighted Hardy spaces." Carpathian Mathematical Publications 8, no. 2 (December 30, 2016): 224–29. http://dx.doi.org/10.15330/cmp.8.2.224-229.

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Let $H^p_\sigma( \mathbb{C}_+),$ $1\leq p <+\infty,$ $0\leq \sigma < +\infty,$ be the space of all functions $f$ analytic in the half plane $ \mathbb{C}_{+}= \{ z: \text {Re} z>0 \}$ and such that $$\|f\|:=\sup\limits_{\varphi\in (-\frac{\pi}{2};\frac{\pi}{2})}\left\{\int\limits_0^{+\infty} |f(re^{i\varphi})|^pe^{-p\sigma r|\sin \varphi|}dr\right\}^{1/p}<+\infty.$$ We obtain some properties and description of zeros for functions from the space $\bigcap\limits_{\sigma>0} H^{p}_{\sigma}(\mathbb C_{+}).$

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20

Chern, Peter Tien-Yu. "Borel direction relative to function-values of meromorphic functions with finite logarithmic order." Tamkang Journal of Mathematics 38, no. 3 (September 30, 2007): 217–24. http://dx.doi.org/10.5556/j.tkjm.38.2007.74.

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It is shown that if $ f(z )$ is meromorphic in the complex plane $ \mathbb C $ with finite positive logarithmic order $ \lambda $ and its characteristic function $ T(r,f) $ satisfies the growth condition $$ \ls \ T(r,f)/(\log r)^2 = + \infty,$$ then there is a number $ \theta $ with $ 0 \le \theta < 2\pi $ such that for each positive number $ \epsilon $, the expression $$ \ls \ \dfrac{\log \bigg\{\displaystyle{\sum^3_{i=1}} \ n(r,\theta, \epsilon, f = a_i(z)) \bigg\}}{\log \log r} = \lambda - 1, $$ holds for any three distinct meromorphic functions $ a_i(z) (i = 1,2,3) $ with $ T(r,a_i) = o(U(r,f)/ $ $ (\log r)^2), $ as $ r \to + \infty $, where $ n(r,\varphi ,\epsilon ,f = a(z)) $ denotes the number of roots counting multiplicitie s of the equation $ f(z) = a(z) $ for $ z$ in the angular domain $ \Omega (r,\varphi ,\epsilon ) = \{z: |\arg z - \varphi | < \epsilon $, $ |z| < r \} $ where $ 0 \leq \varphi < 2\pi $, $ \epsilon > 0$, $U(r,f) = (\log r)^{\lambda (r)} $, and $ \displaystyle{\ls \ \lambda (r) = \lambda} $.

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21

Zabolotskyj, M. V., and Yu V. Basiuk. "Asymptotics of the entire functions with $\upsilon$-density of zeros along the logarithmic spirals." Carpathian Mathematical Publications 11, no. 1 (June 30, 2019): 26–32. http://dx.doi.org/10.15330/cmp.11.1.26-32.

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Let $\upsilon$ be the growth function such that $r\upsilon'(r)/\upsilon (r) \to 0$ as $r \to +\infty$, $l_\varphi^c = \{z=te^{i(\varphi+c \ln t)}, 1 \leqslant t < +\infty\}$ be the logarithmic spiral, $f$ be the entire function of zero order. The asymptotics of $\ln f(re^{i(\theta +c \ln r)})$ along ordinary logarithmic spirals $l_\theta^c$ of the function $f$ with $\upsilon$-density of zeros along $l_\varphi^c$ outside the $C_0$-set is found. The inverse statement is true just in case zeros of $f$ are placed on the finite logarithmic spirals system $\Gamma_m = \bigcup_{j=0}^m l_{\theta_j}^c$.

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22

Andrusyak, I. V., and P. V. Filevych. "The minimal growth of entire functions with given zeros along unbounded sets." Matematychni Studii 54, no. 2 (December 25, 2020): 146–53. http://dx.doi.org/10.30970/ms.54.2.146-153.

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Let $l$ be a continuous function on $\mathbb{R}$ increasing to $+\infty$, and $\varphi$ be a positive function on $\mathbb{R}$. We proved that the condition$$\varliminf_{x\to+\infty}\frac{\varphi(\ln[x])}{\ln x}>0$$is necessary and sufficient in order that for any complex sequence $(\zeta_n)$ with $n(r)\ge l(r)$, $r\ge r_0$, and every set $E\subset\mathbb{R}$ which is unbounded from above there exists an entire function $f$ having zeros only at the points $\zeta_n$ such that$$\varliminf_{r\in E,\ r\to+\infty}\frac{\ln\ln M_f(r)}{\varphi(\ln n_\zeta(r))\ln l^{-1}(n_\zeta(r))}=0.$$Here $n(r)$ is the counting function of $(\zeta_n)$, and $M_f(r)$ is the maximum modulus of $f$.

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23

Peleshenko, B. I. "On interpolation of operator, which is the sum of weighted Hardy-Littlewood and Cesaro mean operators." Researches in Mathematics 27, no. 1 (July 23, 2019): 45. http://dx.doi.org/10.15421/241905.

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It is proved that operators, which are the sum of weighted Hardy-Littlewood $\int\limits_0^1 f(xt) \psi(t) dt$ and Cesaro $\int\limits_0^1 f(\frac{x}{t}) t^{-n} \psi(t) dt$ mean operators, are limited on Lorentz spaces $\Lambda_{\varphi, a} (\mathbb{R})$, if the functions $f(x) \in \Lambda_{\varphi, a}(\mathbb{R})$ satisfy the condition $|f(-x)| = |f(x)|$, $x > 0$, for such non-increasing semi-multiplicative functions $\psi$, for which the next conditions are satisfied: $\frac{M_1}{\psi(t)} \leqslant \psi(\frac{1}{t}) \leqslant \frac{M_2}{\psi(t)}$, for all $0 < t \leqslant 1$; at some $0 < \varepsilon < \frac{1}{2}$, $0 < \delta < \frac{1}{2}$ functions $\psi(t) t^{1-\varepsilon}$, $\psi(\frac{1}{t}) t^{-\delta}$ do not decrease monotonically and functions $\psi(t) t$, $\psi (\frac{1}{t})$ are absolutely continuous. Also, there are proved sufficient conditions that the operators, which are the sum of weighted Hardy-Littlewood and Cesaro mean operators, when $\psi(t) = t^{-\alpha}$, where $\alpha \in (0, \frac{1}{2})$, on Lorentz spaces $\Lambda_{\varphi, a}(\mathbb{R})$, if the functions $f(x) \in \Lambda_{\varphi, a}(\mathbb{R})$ satisfy the condition $|f(-x)| = |f(x)|$, $x > 0$.

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24

Taguchi, Y., and D. Wan. "$L$-functions of $\varphi $-sheaves and Drinfeld modules." Journal of the American Mathematical Society 9, no. 3 (1996): 755–81. http://dx.doi.org/10.1090/s0894-0347-96-00199-3.

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25

Kanas, Stanisława, Vali Soltani Masih, and Ali Ebadian. "Coefficients problems for families of holomorphic functions related to hyperbola." Mathematica Slovaca 70, no. 3 (June 25, 2020): 605–16. http://dx.doi.org/10.1515/ms-2017-0375.

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AbstractWe consider a family of analytic and normalized functions that are related to the domains ℍ(s), with a right branch of a hyperbolas H(s) as a boundary. The hyperbola H(s) is given by the relation $\begin{array}{} \frac{1}{\rho}=\left( 2\cos\frac{\varphi}{s}\right)^s\quad (0 \lt s\le 1,\, |\varphi| \lt (\pi s)/2). \end{array}$ We mainly study a coefficient problem of the families of functions for which zf′/f or 1 + zf″/f′ map the unit disk onto a subset of ℍ(s) . We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.

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26

HU, ZHANGJIAN. "ESSENTIAL NORM OF EXTENDED CESÀRO OPERATORS FROM ONE BERGMAN SPACE TO ANOTHER." Bulletin of the Australian Mathematical Society 85, no. 2 (December 12, 2011): 307–14. http://dx.doi.org/10.1017/s0004972711003108.

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AbstractLet Ap(φ) be the pth Bergman space consisting of all holomorphic functions f on the unit ball B of ℂn for which $\|f\|^p_{p,\varphi }= \int _B |f(z)|^p \varphi (z) \,dA(z)\lt +\infty $, where φ is a given normal weight. Let Tg be the extended Cesàro operator with holomorphic symbol g. The essential norm of Tg as an operator from Ap (φ) to Aq (φ) is denoted by $\|T_g\|_{e, A^p (\varphi )\to A^q (\varphi )} $. In this paper it is proved that, for p≤q, with 1/k=(1/p)−(1/q) , where ℜg(z) is the radial derivative of g; and for p>q, with 1/s=(1/q)−(1/p) .

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27

Mastyło, Mieczysław. "An operator ideal generated by Orlicz spaces." Mathematische Annalen 376, no. 3-4 (October 4, 2019): 1675–703. http://dx.doi.org/10.1007/s00208-019-01904-6.

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Abstract Absolutely $$\varphi $$φ-summing operators between Banach spaces generated by Orlicz spaces are investigated. A variant of Pietsch’s domination theorem is proved for these operators and applied to prove vector-valued inequalities. These results are used to prove asymptotic estimates of $$\pi _\varphi $$πφ-summing norms of finite-dimensional operators and also diagonal operators between Banach sequence lattices for a wide class of Orlicz spaces based on exponential convex functions $$\varphi $$φ. The key here is the description of a space of coefficients of the Rademacher series in this class of Orlicz spaces, proved via interpolation methods. As by-product, some absolutely $$\varphi $$φ-summing operators on the Hilbert space $$\ell _2$$ℓ2 are characterized in terms of its approximation numbers.

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Jiang, Wei-Dong, Da-Wei Niu, and Feng Qi. "Some inequalities of Hermite-Hadamard type for $r$-$\varphi$-preinvex functions." Tamkang Journal of Mathematics 45, no. 1 (March 30, 2014): 31–38. http://dx.doi.org/10.5556/j.tkjm.45.2014.1261.

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Baron, Karol. "Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations." Annales Mathematicae Silesianae 34, no. 1 (July 1, 2020): 36–44. http://dx.doi.org/10.2478/amsil-2019-0015.

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AbstractThe paper consists of two parts. At first, assuming that (Ω, A, P) is a probability space and (X, ϱ) is a complete and separable metric space with the σ-algebra 𝒝 of all its Borel subsets we consider the set 𝒭c of all 𝒝 ⊗ 𝒜-measurable and contractive in mean functions f : X × Ω → X with finite integral ∫ Ωϱ (f(x, ω), x) P (dω) for x ∈ X, the weak limit π f of the sequence of iterates of f ∈ 𝒭c, and investigate continuity-like property of the function f ↦ πf, f ∈ 𝒭c, and Lipschitz solutions φ that take values in a separable Banach space of the equation\varphi \left( x \right) = \int_\Omega {\varphi \left( {f\left( {x,\omega } \right)} \right)P\left( {d\omega } \right)} + F\left( x \right).Next, assuming that X is a real separable Hilbert space, Λ: X → X is linear and continuous with ||Λ || < 1, and µ is a probability Borel measure on X with finite first moment we examine continuous at zero solutions φ : X → 𝔺 of the equation\varphi \left( x \right) = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \mu } \left( x \right)\varphi \left( {\Lambda x} \right)which characterizes the limit distribution π f for some special f ∈ 𝒭c.

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30

de Hierro, A. F. Roldán López, and Bessem Samet. "$$\varvec{\varphi }$$ φ -admissibility results via extended simulation functions." Journal of Fixed Point Theory and Applications 19, no. 3 (November 25, 2016): 1997–2015. http://dx.doi.org/10.1007/s11784-016-0385-x.

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31

Vyas, Rajendra G. "Absolute convergence of multiple Fourier series of a function of p(n)-Λ-BV." Georgian Mathematical Journal 25, no. 3 (September 1, 2018): 481–91. http://dx.doi.org/10.1515/gmj-2017-0008.

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AbstractIn this paper, we obtain sufficiency conditions for generalized β-absolute convergence ({0<\beta\leq 2}) of single and multiple Fourier series of functions of the class {\Lambda\text{-}\mathrm{BV}(p(n)\uparrow\infty,\varphi,[-\pi,\pi])} and the class {(\Lambda^{1},\Lambda^{2},\dots,\Lambda^{N})\text{-}\mathrm{BV}(p(n)\uparrow% \infty,\varphi,[-\pi,\pi]^{N})}, respectively.

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32

Rao, N. V., and A. K. Roy. "MULTIPLICATIVELY SPECTRUM-PRESERVING MAPS OF FUNCTION ALGEBRAS. II." Proceedings of the Edinburgh Mathematical Society 48, no. 1 (February 2005): 219–29. http://dx.doi.org/10.1017/s0013091504000719.

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AbstractLet $\mathcal{A}$ be a closed, point-separating sub-algebra of $C_0(X)$, where $X$ is a locally compact Hausdorff space. Assume that $X$ is the maximal ideal space of $\mathcal{A}$. If $f\in\mathcal{A}$, the set $f(X)\cup\{0\}$ is denoted by $\sigma(f)$. After characterizing the points of the Choquet boundary as strong boundary points, we use this equivalence to provide a natural extension of the theorem in [10], which, in turn, was inspired by the main result in [6], by proving the ‘Main Theorem’: if $\varPhi:\mathcal{A}\rightarrow\mathcal{A}$ is a surjective map with the property that $\sigma(fg)=\sigma(\varPhi(f)\varPhi(g))$ for every pair of functions $f,g\in\mathcal{A}$, then there is an onto homeomorphism $\varLambda:X\rightarrow X$ and a signum function $\epsilon(x)$ on $X$ such that$$ \varPhi(f)(\varLambda(x))=\epsilon(x)f(x) $$for all $x\in X$ and $f\in\mathcal{A}$.AMS 2000 Mathematics subject classification: Primary 46J10; 46J20

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33

Hung, Ha Duy, Luong Dang Ky, and Thai Thuan Quang. "Hausdorff operators on holomorphic Hardy spaces and applications." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 3 (January 30, 2019): 1095–112. http://dx.doi.org/10.1017/prm.2018.74.

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AbstractThe aim of this paper is to characterize the non-negative functions φ defined on (0,∞) for which the Hausdorff operator $${\rm {\cal H}}_\varphi f(z) = \int_0^\infty f \left( {\displaystyle{z \over t}} \right)\displaystyle{{\varphi (t)} \over t}{\rm d}t$$is bounded on the Hardy spaces of the upper half-plane ${\rm {\cal H}}_a^p ({\open C}_ + )$, $p\in [1,\infty ]$. The corresponding operator norms and their applications are also given.

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34

LU, JIN, and XIAOFEN LV. "TOEPLITZ OPERATORS BETWEEN FOCK SPACES." Bulletin of the Australian Mathematical Society 92, no. 2 (June 2, 2015): 316–24. http://dx.doi.org/10.1017/s0004972715000477.

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Given a positive Borel measure ${\it\mu}$ on the $n$-dimensional Euclidean space $\mathbb{C}^{n}$, we characterise the boundedness (and compactness) of Toeplitz operators $T_{{\it\mu}}$ between Fock spaces $F^{\infty }({\it\varphi})$ and $F^{p}({\it\varphi})$ with $0<p\leq \infty$ in terms of $t$-Berezin transforms and averaging functions of ${\it\mu}$. Our result extends recent work of Mengestie [‘On Toeplitz operators between Fock spaces’, Integral Equations Operator Theory78 (2014), 213–224] and others.

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35

MONTALBÁN, ANTONIO, and JAMES WALSH. "ON THE INEVITABILITY OF THE CONSISTENCY OPERATOR." Journal of Symbolic Logic 84, no. 1 (March 2019): 205–25. http://dx.doi.org/10.1017/jsl.2018.65.

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AbstractWe examine recursive monotonic functions on the Lindenbaum algebra of $EA$. We prove that no such function sends every consistent φ to a sentence with deductive strength strictly between φ and $\left( {\varphi \wedge Con\left( \varphi \right)} \right)$. We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function f, if there is an iterate of $Con$ that bounds f everywhere, then f must be somewhere equal to an iterate of $Con$.

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36

Gaarde, Anders, and Gerd Grubb. "Logarithms and sectorial projections for elliptic boundary problems." MATHEMATICA SCANDINAVICA 103, no. 2 (December 1, 2008): 243. http://dx.doi.org/10.7146/math.scand.a-15079.

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On a compact manifold with boundary, consider the realization $B$ of an elliptic, possibly pseudodifferential, boundary value problem having a spectral cut (a ray free of eigenvalues), say $\mathsf{R}_{-}$. In the first part of the paper we define and discuss in detail the operator $\log B$; its residue (generalizing the Wodzicki residue) is essentially proportional to the zeta function value at zero, $\zeta (B,0)$, and it enters in an important way in studies of composed zeta functions $\zeta (A,B,s)= {\operatorname {Tr}}(AB^{-s})$ (pursued elsewhere). There is a similar definition of the operator $\log_{\theta}B$, when the spectral cut is at a general angle $\theta$. When $B$ has spectral cuts at two angles $\theta <\varphi$, one can define the sectorial projection $\Pi_{\theta,\varphi} (B)$ whose range contains the generalized eigenspaces for eigenvalues with argument in $\left]\theta,\varphi \right[$; this is studied in the last part of the paper. The operator $\Pi_{\theta ,\varphi}(B)$ is shown to be proportional to the difference between $\log_{\theta}B$ and $\log_{\varphi} B$, having slightly better symbol properties than they have. We show by examples that it belongs to the Boutet de Monvel calculus in many special cases, but lies outside the calculus in general.

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37

BALADI, V., S. MARMI, and D. SAUZIN. "Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps." Ergodic Theory and Dynamical Systems 34, no. 3 (January 25, 2013): 777–800. http://dx.doi.org/10.1017/etds.2012.161.

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AbstractFor a piecewise expanding unimodal interval map$f$with unique absolutely continuous invariant probability measure$\mu $, a perturbation$X$, and an observable$\varphi $, the susceptibility function is$\Psi _\varphi (z)= \sum _{k=0}^\infty z^k \int X(x) \varphi '( f^k)(x) (f^k)'(x) \, d\mu $. Combining previous results [V. Baladi, On the susceptibility function of piecewise expanding interval maps.Comm. Math. Phys.275(2007), 839–859; V. Baladi and D. Smania, Linear response for piecewise expanding unimodal maps.Nonlinearity21(2008), 677–711] (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer–Simon [Natural boundaries and spectral theory.Adv. Math.226(2011), 4902–4920] (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon [Sur les séries de Dirichlet.Ann. Sci. Éc. Norm. Supér.(3)66(1949), 263–310]), we show that density of the postcritical orbit (a generic condition) implies that$\Psi _\varphi (z)$has a strong natural boundary on the unit circle. The Breuer–Simon method provides uncountably many candidates for the outer functions of$\Psi _\varphi (z)$, associated with precritical orbits. If the perturbation$X$is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the non-tangential limit of$\Psi _\varphi (z)$as$z\to 1$exists and coincides with the derivative of the absolutely continuous invariant probability measure with respect to the map (‘linear response formula’). Applying the Wiener–Wintner theorem, we study the singularity type of non-tangential limits of$\Psi _\varphi (z)$as$z\to e^{i\omega }$for real$\omega $. An additional ‘law of the iterated logarithm’ typicality assumption on the postcritical orbit gives stronger results.

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38

NARKIEWICZ, WŁADYSŁAW. "WEAK PROPER DISTRIBUTION OF VALUES OF MULTIPLICATIVE FUNCTIONS IN RESIDUE CLASSES." Journal of the Australian Mathematical Society 93, no. 1-2 (October 2012): 173–88. http://dx.doi.org/10.1017/s144678871200064x.

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AbstractFor a class of multiplicative integer-valued functions $f$ the distribution of the sequence $f(n)$ in restricted residue classes modulo $N$ is studied. We consider a property weaker than weak uniform distribution and study it for polynomial-like multiplicative functions, in particular for $\varphi (n)$ and $\sigma (n)$.

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39

Quan, L. T., and T. Van An. "On the solutions of a class of nonlinear integral equations in cone $b$-metric spaces over Banach algebras." Carpathian Mathematical Publications 11, no. 1 (June 30, 2019): 163–78. http://dx.doi.org/10.15330/cmp.11.1.163-178.

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In this paper, we study the existence of the solutions of a class of functional integral equations by using some fixed point results in cone $b$-metric spaces over Banach algebras. In order to obtain these results we introduced and proved some properties of generalized weak $\varphi$-contractions, in which the $\varphi$ are nonlinear weak comparison functions. The obtained results are generalizations of results of Van Dung N., Le Hang V. T., Huang H., Radenovic S. and Deng G. Also, some suitable examples are given to illustrate obtained results.

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40

Demkiv, I. I., M. I. Kopach, A. F. Obshta, and B. A. Shuvar. "Unconventional analogs of single-parametric method of iterational aggregation." Carpathian Mathematical Publications 10, no. 2 (December 31, 2018): 296–302. http://dx.doi.org/10.15330/cmp.10.2.296-302.

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When we solve practical problems that arise, for example, in mathematical economics, in the theory of Markov processes, it is often necessary to use the decomposition of operator equations using methods of iterative aggregation. In the studies of these methods for the linear equation $x=Ax+b$ the most frequent are the conditions of positiveness of the operator $A$, constant $b$ and the aggregation functions, and also the implementation of the inequality $\rho(A) <1$ for the spectral radius $\rho(A)$ of the operator $A$. In this article for an approximate solution of a system composed of the equation $x = Ax + b$ represented in the form $ x = A_1x + A_2x + b, $ where $ b \in E, $ $ E $ is a Banach space, $ A_1, A_2 $ are linear continuous operators that act from $ E $ to $ E $ and the auxiliary equation $ y = \lambda y - (\varphi, A_2x) - (\varphi, b) $ with a real variable $ y $, where $ (\varphi, x) $ is the value of the linear functional $ \varphi \in E ^ * $ on the elements $ x \in E $, $ E^* $ is conjugation with space $ E $, an iterative process is constructed and investigated \begin{equation*} \begin{split} x^{(n+1)}&=Ax^{(n)}+b+\frac{\sum\limits_{i=1}^{m}A^i_1x^{(n)}}{(\varphi, x^{(n)})\sum\limits_{i=0}^{m}\lambda^i}(y^{(n)}-y^{(n+1)}) \quad (m<\infty),\\ y^{(n+1)}&=\lambda y^{(n+1)}-(\varphi,A_2x^{(n)})-(\varphi,b). \end{split} \end{equation*} The conditions are established under which the sequences $ {x ^ {(n)}}, {y ^ {(n)}}$, constructed with the help of these formulas, converge to $ x ^ *, y ^ * $ as a component of solving the system constructed from equations $ x = A_1 x + A_2 x + b $ and the equation $ y = \lambda y - (\varphi, A_2 x) - (\varphi, b) $ not slower than the rate of convergence of the geometric progression with the denominator less than $1$. In this case, it is required that the operator $ A $ be a compressive and constant by sign, and that the space $ E $ is semi-ordered. The application of the proposed algorithm to systems of linear algebraic equations is also shown.

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41

Karaman, Özkan. "Spectral Properties of Nonhomogenous Differential Equations with Spectral Parameter in the Boundary Condition." Analele Universitatii "Ovidius" Constanta - Seria Matematica 22, no. 2 (June 1, 2014): 109–20. http://dx.doi.org/10.2478/auom-2014-0036.

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AbstractIn this paper, using the boundary properties of the analytic functions we investigate the structure of the discrete spectrum of the boundary value problem (0.1)$$\matrix{\hfill {iy_1^\prime + q_1 \left(x \right)y_2 - \lambda y_1 = \varphi _1 \left(x \right)\;\;} & \hfill {} \cr \hfill {- iy_2^\prime + q_2 \left(x \right)y_1 - \lambda y_2 = \varphi _2 \left(x \right),} & \hfill {x \in R_ + } \cr }$$ and the condition (0.2)$$\left({a_1 \lambda + b_1 } \right)y_2 \left({0,\lambda } \right) - \left({a_2 \lambda + b_2 } \right)y_1 \left({0,\lambda } \right) = 0$$ where q1,q2, φ1, φ2 are complex valued functions, ak ≠ 0, bk ≠ 0, k = 1, 2 are complex constants and λ is a spectral parameter. In this article, we investigate the spectral singularities and eigenvalues of (0.1), (0.2) using the boundary uniqueness theorems of analytic functions. In particular, we prove that the boundary value problem (0.1), (0.2) has a finite number of spectral singularities and eigenvalues with finite multiplicities under the conditions, $$\matrix{{\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {\varphi _k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] < \infty ,\;\;\;k = 1.2} \hfill \cr {\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {q_k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] < \infty ,\;\;\;k = 1.2} \hfill \cr }$$ for some ε > 0, ${1 \over 2} < \delta < 1$

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42

Elmassoudi, Mhamed, Ahmed Aberqi, and Jaouad Bennouna. "Existence of entropy solutions in Musielak Orlicz spaces via a sequence of penalized equations." Boletim da Sociedade Paranaense de Matemática 38, no. 6 (May 25, 2019): 203–38. http://dx.doi.org/10.5269/bspm.v38i6.37269.

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This paper, is devoted to an existence result of entropy unilateral solutions for the nonlinear parabolic problems with obstacle in Musielak- Orlicz--spaces:$$ \partial_{t}u + A(u) + H(x,t,u,\nabla u) =f + div(\Phi(x,t,u))$$and $$ u\geq \zeta \,\,\mbox{a.e. in }\,\,Q_T.$$Where $A$ is a pseudomonotone operator of Leray-Lions type defined in the inhomogeneous Musielak-Orlicz space $W_{0}^{1,x}L_{\varphi}(Q_{T})$,$H(x,t,s,\xi)$ and $\phi(x,t,s)$ are only assumed to be Crath\'eodory's functions satisfying only the growth conditions prescribed by Musielak-Orlicz functions $\varphi$ and $\psi$ which inhomogeneous and does not satisfies $\Delta_2$-condition. The data $f$ and $u_{0}$ are still taken in $L^{1}(Q_T)$ and $L^{1}(\Omega)$.

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43

Set, Erhan, Mehmet Zeki Sarιkaya, and Ahmet Ocak Akdemir. "Hadamard type inequalities for $\varphi -$convex functions on the co-ordinates." Tbilisi Mathematical Journal 7, no. 2 (2014): 51–60. http://dx.doi.org/10.2478/tmj-2014-0016.

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44

Weingartner, Andreas. "The distribution functions of $\sigma(n)/n$ and $n/\varphi(n)$." Proceedings of the American Mathematical Society 135, no. 09 (February 6, 2007): 2677–82. http://dx.doi.org/10.1090/s0002-9939-07-08771-0.

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45

Koskela, Antti, and Alexander Ostermann. "A Moment-Matching Arnoldi Iteration for Linear Combinations of $\varphi$ Functions." SIAM Journal on Matrix Analysis and Applications 35, no. 4 (January 2014): 1344–63. http://dx.doi.org/10.1137/130945156.

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46

Vasylyshyn, T. V. "Topology on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$." Carpathian Mathematical Publications 9, no. 1 (June 19, 2017): 22–27. http://dx.doi.org/10.15330/cmp.9.1.22-27.

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It is known that the so-called elementary symmetric polynomials $R_n(x) = \int_{[0,1]}(x(t))^n\,dt$ form an algebraic basis in the algebra of all symmetric continuous polynomials on the complex Banach space $L_\infty,$ which is dense in the Fr\'{e}chet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded type on $L_\infty.$ Consequently, every continuous homomorphism $\varphi: H_{bs}(L_\infty) \to \mathbb{C}$ is uniquely determined by the sequence $\{\varphi(R_n)\}_{n=1}^\infty.$ By the continuity of the homomorphism $\varphi,$ the sequence $\{\sqrt[n]{|\varphi(R_n)|}\}_{n=1}^\infty$ is bounded. On the other hand, for every sequence $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C},$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded, there exists $x_\xi \in L_\infty$ such that $R_n(x_\xi) = \xi_n$ for every $n \in \mathbb{N}.$ Therefore, for the point-evaluation functional $\delta_{x_\xi}$ we have $\delta_{x_\xi}(R_n) = \xi_n$ for every $n \in \mathbb{N}.$ Thus, every continuous complex-valued homomorphism of $H_{bs}(L_\infty)$ is a point-evaluation functional at some point of $L_\infty.$ Note that such a point is not unique. We can consider an equivalence relation on $L_\infty,$ defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ The spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, it is naturally to identify $M_{bs}$ with the set of all sequences $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C}$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded.We show that the quotient topology is Hausdorffand that $M_{bs}$ with the operation of coordinate-wise addition of sequences forms an abelian topological group.

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47

Kashuri, Artion, and Rozana Liko. "Hermite-Hadamard type fractional integral inequalities for MT$_{(m,\varphi)}$-preinvex functions." Studia Universitatis Babes-Bolyai Matematica 62, no. 4 (December 3, 2017): 439–50. http://dx.doi.org/10.24193/subbmath.2017.4.03.

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48

Noor, Muhammad Aslam, Khalida Inayat Noor, Muhammad Uzair Awan, and Sundas Khan. "Hermite–Hadamard type inequalities for differentiable $${h_{\varphi}}$$ h φ -preinvex functions." Arabian Journal of Mathematics 4, no. 1 (January 8, 2015): 63–76. http://dx.doi.org/10.1007/s40065-014-0124-3.

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49

Gong, Pan, and Hong Yan Xu. "Oscillation of arbitrary-order derivatives of solutions to the higher order non-homogeneous linear differential equations taking small functions in the unit disc." AIMS Mathematics 6, no. 12 (2021): 13746–57. http://dx.doi.org/10.3934/math.2021798.

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<abstract><p>In this article, we study the relationship between solutions and their arbitrary-order derivatives of the higher order non-homogeneous linear differential equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = F(z) \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>in the unit disc $ \bigtriangleup $ with analytic or meromorphic coefficients of finite $ [p, q] $-order. We obtain some oscillation theorems for $ f^{(j)}(z)-\varphi(z) $, where $ f $ is a solution and $ \varphi(z) $ is a small function.</p></abstract>

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50

Knizhnerman, L. A. "Gauss-Arnoldi quadrature for $ \bigl\langle(zI-A)^{-1}\varphi,\,\varphi\bigr\rangle$ and rational Padé-type approximation for Markov-type functions." Sbornik: Mathematics 199, no. 2 (February 28, 2008): 185–206. http://dx.doi.org/10.1070/sm2008v199n02abeh003915.

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