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Journal articles on the topic 'Quasi-convex Functions'

Author: Grafiati

Published: 9 March 2023

Last updated: 31 July 2025

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1

Liu, Zheng. "ON INEQUALITIES RELATED TO SOME QUASI-CONVEX FUNCTIONS." Issues of Analysis 22, no. 2 (2015): 45–64. http://dx.doi.org/10.15393/j3.art.2015.2869.

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2

Zhang, Kewei. "Quasi-convex functions on subspaces and boundaries of quasi-convex sets." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 4 (2004): 783–99. http://dx.doi.org/10.1017/s0308210500003486.

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We embed truncations of the epi-graph of quasi-convex functions defined on linear subspaces E ⊂ MN × n of real matrices into MN × n to bound quasi-convex sets by the graph of the functions. We also characterize subspaces E on which all quasi-convex functions are convex and show, by using the Tarski–Seidenberg theorem in real algebraic geometry, that if dim (E) > N + n − 1, then there exist non-trivial quasi-convex functions on E.

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3

Hazim, Revan I., and Saba N. Majeed. "Quasi Semi and Pseudo Semi (p,E)-Convexity in Non-Linear Optimization Programming." Ibn AL-Haitham Journal For Pure and Applied Sciences 36, no. 1 (2023): 355–66. http://dx.doi.org/10.30526/36.1.2928.

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The class of quasi semi -convex functions and pseudo semi -convex functions are presented in this paper by combining the class of -convex functions with the class of quasi semi -convex functions and pseudo semi -convex functions, respectively. Various non-trivial examples are introduced to illustrate the new functions and show their relationships with -convex functions recently introduced in the literature. Different general properties and characteristics of this class of functions are established. In addition, some optimality properties of generalized non-linear optimization problems are disc

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4

Tariq, Muhammad, Sotiris K. Ntouyas, and Asif Ali Shaikh. "A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Quantum Calculus." Foundations 3, no. 2 (2023): 340–79. http://dx.doi.org/10.3390/foundations3020026.

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A review of results on Hermite–Hadamard (H-H) type inequalities in quantum calculus, associated with a variety of classes of convexities, is presented. In the various classes of convexities this includes classical convex functions, quasi-convex functions, p-convex functions, (p,s)-convex functions, modified (p,s)-convex functions, (p,h)-convex functions, tgs-convex functions, η-quasi-convex functions, ϕ-convex functions, (α,m)-convex functions, ϕ-quasi-convex functions, and coordinated convex functions. Quantum H-H type inequalities via preinvex functions and Green functions are also presented

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5

Beer, G., and R. Lucchetti. "Minima of quasi-convex functions." Optimization 20, no. 5 (1989): 581–96. http://dx.doi.org/10.1080/02331938908843480.

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6

Ubhaya, Vasant A. "Uniform approximation by quasi-convex and convex functions." Journal of Approximation Theory 55, no. 3 (1988): 326–36. http://dx.doi.org/10.1016/0021-9045(88)90099-8.

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7

Ubhaya, Vasant A. "Lp approximation by quasi-convex and convex functions." Journal of Mathematical Analysis and Applications 139, no. 2 (1989): 574–85. http://dx.doi.org/10.1016/0022-247x(89)90130-3.

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8

Meftah, B., and A. Souahi. "Cebyšev inequalities for co-ordinated \(QC\)-convex and \((s,QC)\)-convex." Engineering and Applied Science Letters 4, no. 1 (2021): 14–20. http://dx.doi.org/10.30538/psrp-easl2021.0057.

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In this paper, we establish some new Cebyšev type inequalities for functions whose modulus of the mixed derivatives are co-ordinated quasi-convex and \(\alpha \)-quasi-convex and \(s\)-quasi-convex functions.

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9

Tariq, Muhammad, Sotiris K. Ntouyas, and Asif Ali Shaikh. "A Comprehensive Review on the Fejér-Type Inequality Pertaining to Fractional Integral Operators." Axioms 12, no. 7 (2023): 719. http://dx.doi.org/10.3390/axioms12070719.

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A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, (θ,h−m)−p-convex functions, and h-preinvex functions. Include

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10

Hinderer, A., and M. Stieglitz. "Minimization of quasi-convex symmetric and of discretely quasi-convex symmetric functions." Optimization 36, no. 4 (1996): 321–32. http://dx.doi.org/10.1080/02331939608844187.

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11

Tariq, Muhammad, Sotiris K. Ntouyas, and Bashir Ahmad. "Ostrowski-Type Fractional Integral Inequalities: A Survey." Foundations 3, no. 4 (2023): 660–723. http://dx.doi.org/10.3390/foundations3040040.

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This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, (ζ,m)-convex functions, s-convex functions, (s,r)-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, MT-convex functions, P-convex functions, m-convex functions, (s,m)-convex functions, exponentially s-convex functions, (β,m)-convex functions, expo

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12

Syau, Yu-Ru. "A note on convex functions." International Journal of Mathematics and Mathematical Sciences 22, no. 3 (1999): 525–34. http://dx.doi.org/10.1155/s0161171299225252.

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In this paper, we give two weak conditions for a lower semi-continuous function on then-dimensional Euclidean spaceRnto be a convex function. We also present some results for convex functions, strictly convex functions, and quasi-convex functions.

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13

Youness, Ebrahim A. "Quasi and strictly quasiE-convex functions." Journal of Statistics and Management Systems 4, no. 2 (2001): 201–10. http://dx.doi.org/10.1080/09720510.2001.10701038.

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14

Ivanenko, Y., M. Nedic, M. Gustafsson, B. L. G. Jonsson, A. Luger, and S. Nordebo. "Quasi-Herglotz functions and convex optimization." Royal Society Open Science 7, no. 1 (2020): 191541. http://dx.doi.org/10.1098/rsos.191541.

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We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modelling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions and we show that several of the important properties and modelling perspectives are inherited by the new set of quasi-Herglotz functions. In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their b

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15

Dragomir, S. S., and C. E. M. Pearce. "Quasi-convex functions and Hadamard's inequality." Bulletin of the Australian Mathematical Society 57, no. 3 (1998): 377–85. http://dx.doi.org/10.1017/s0004972700031786.

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Some extensions of quasi-convexity appearing in the literature are explored and relations found between them. Hadamard's inequality is connected tenaciously with convexity and versions of it are shown to hold in our setting. Our theorems extend and unify a number of known results. In particular, we derive a generalised Kenyon-Klee theorem.

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16

Mrowiec, Jacek, and Teresa Rajba. "Quasi-convex functions of higher order." Mathematical Inequalities & Applications, no. 4 (2019): 1335–54. http://dx.doi.org/10.7153/mia-2019-22-92.

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17

Özdemir, M. Emin, Ahmet Ocak Akdemir, and Çetin Yıldız. "On co-ordinated quasi-convex functions." Czechoslovak Mathematical Journal 62, no. 4 (2012): 889–900. http://dx.doi.org/10.1007/s10587-012-0072-z.

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18

Sun, Mingbao, and Xiaoping Yang. "Quasi-convex Functions in Carnot Groups*." Chinese Annals of Mathematics, Series B 28, no. 2 (2007): 235–42. http://dx.doi.org/10.1007/s11401-005-0052-9.

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19

Moradi, Hamid Reza, Shigeru Furuichi, and Mohammad Sababheh. "Quasi-convex and Q-class functions." Journal of Mathematical Inequalities, no. 3 (2023): 893–908. http://dx.doi.org/10.7153/jmi-2023-17-57.

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20

Kadakal, Huriye. "Better Approximations for Quasi-Convex Functions." Studia Universitatis Babes-Bolyai Matematica 69, no. 2 (2024): 267–81. http://dx.doi.org/10.24193/subbmath.2024.2.02.

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In this paper, by using Hölder-İşcan, Hölder integral inequality and a general identity for differentiable functions we can get new estimates on generalization of Hadamard, Ostrowski and Simpson type integral inequalities for functions whose derivatives in absolute value at certain power are quasi-convex functions. It is proved that the result obtained Hölder-İşcan integral inequality is better than the result obtained Hölder inequality. Keywords: Hölder-İşcan scan inequality, Hermite-Hadamard inequality, Simpson and Ostrowski type inequality, midpoint and trapezoid type inequality, quasi-conv

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21

Khan, Shahid, Şahsene Altınkaya, Qin Xin, Fairouz Tchier, Sarfraz Nawaz Malik, and Nazar Khan. "Faber Polynomial Coefficient Estimates for Janowski Type bi-Close-to-Convex and bi-Quasi-Convex Functions." Symmetry 15, no. 3 (2023): 604. http://dx.doi.org/10.3390/sym15030604.

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Motivated by the recent work on symmetric analytic functions by using the concept of Faber polynomials, this article introduces and studies two new subclasses of bi-close-to-convex and quasi-close-to-convex functions associated with Janowski functions. By using the Faber polynomial expansion method, it determines the general coefficient bounds for the functions belonging to these classes. It also finds initial coefficients of bi-close-to-convex and bi-quasi-convex functions by using Janowski functions. Some known consequences of the main results are also highlighted.

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22

Choudhary, Masood Ahmed, and ToseefAhmed Malik. "Some properties of harmonic convex and harmonic quasi-convex functions." International Journal of Mathematics Trends and Technology 56, no. 4 (2018): 252–57. http://dx.doi.org/10.14445/22315373/ijmtt-v56p536.

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23

Kim, Hoonjoo. "Some properties of quasi-convex functions on abstract convex spaces." International Journal of Mathematical Analysis 9 (2015): 2431–40. http://dx.doi.org/10.12988/ijma.2015.59238.

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24

Mahmood, Shahid, Sarfraz Nawaz Malik, Sumbal Farman, S. M. Jawwad Riaz, and Shabieh Farwa. "Uniformly Alpha-Quasi-Convex Functions Defined by Janowski Functions." Journal of Function Spaces 2018 (2018): 1–7. http://dx.doi.org/10.1155/2018/6049512.

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In this work, we aim to introduce and study a new subclass of analytic functions related to the oval and petal type domain. This includes various interesting properties such as integral representation, sufficiency criteria, inclusion results, and the convolution properties for newly introduced class.

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25

Pearce, C. E. M., and A. M. Rubinov. "P-functions, Quasi-convex Functions, and Hadamard-type Inequalities." Journal of Mathematical Analysis and Applications 240, no. 1 (1999): 92–104. http://dx.doi.org/10.1006/jmaa.1999.6593.

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26

Noor, Khalida Inayat. "On quasi-convex functions and related topics." International Journal of Mathematics and Mathematical Sciences 10, no. 2 (1987): 241–58. http://dx.doi.org/10.1155/s0161171287000310.

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LetSbe the class of functionsfwhich are analytic and univalent in the unit discEwithf(0)=0,f′(0)=1. LetC,S*andKbe the classes of convex, starlike and close-to-convex functions respectively. The classC*of quasi-convex functions is defined as follows:Letfbe analytic inEandf(0),f′(0)=1. Thenf ϵ C*if and only if there exists ag ϵ Csuch that, forz ϵ ERe(zf′(z))′g′(z)>0.In this paper, an up-to-date complete study of the classC*is given. Its basic properties, its relationship with other subclasses ofS, coefficient problems, arc length problem and many other results are included in this study. Some

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27

Rashid, Saima, Saima Parveen, Hijaz Ahmad та Yu-Ming Chu. "New quantum integral inequalities for some new classes of generalized ψ-convex functions and their scope in physical systems". Open Physics 19, № 1 (2021): 35–50. http://dx.doi.org/10.1515/phys-2021-0001.

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Abstract In the present study, two new classes of convex functions are established with the aid of Raina’s function, which is known as the ψ-s-convex and ψ-quasi-convex functions. As a result, some refinements of the Hermite–Hadamard ( {\mathcal{ {\mathcal H} {\mathcal H} }} )-type inequalities regarding our proposed technique are derived via generalized ψ-quasi-convex and generalized ψ-s-convex functions. Considering an identity, several new inequalities connected to the {\mathcal{ {\mathcal H} {\mathcal H} }} type for twice differentiable functions for the aforesaid classes are derived. The

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28

Altıntaş, Osman, and Melike Aydoğan. "On the Coefficients of Quasi-Convex Functions." Journal of Physics: Conference Series 1562 (June 2020): 012002. http://dx.doi.org/10.1088/1742-6596/1562/1/012002.

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29

Noor, Khalida Inayat. "Some classes of alpha-quasi-convex functions." International Journal of Mathematics and Mathematical Sciences 11, no. 3 (1988): 497–501. http://dx.doi.org/10.1155/s0161171288000584.

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LetC[C,D],−1≤D<C≤1denote the class of functionsg,g(0)=0,g′(0)=1, analytic in the unit diskEsuch that(zg′(z))′g′(z)is subordinate to1+CZ1+DZ,z∈E. We investigate some classes of Alpha-Quasi-Convex Functionsf, withf(0)=f′(0)−1=0for which there exists ag∈C[C,D]such that(1−α)f′(z)g′(z)+α(zf′(z))′g′(z)is subordinate to1+AZ1+BZ′,−1≤B<A≤1. Integral representation, coefficient bounds are obtained. It is shown that some of these classes are preserved under certain integral operators.

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30

Sezer, Sevda, and Zeynep Eken. "The Hermite-Hadamard type inequalities for quasi $ p $-convex functions." AIMS Mathematics 8, no. 5 (2023): 10435–52. http://dx.doi.org/10.3934/math.2023529.

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<abstract><p>In this paper, the Hermite-Hadamard inequality and its generalization for quasi $ p $-convex functions are provided. Also several new inequalities are established for the functions whose first derivative in absolute value is quasi $ p $-convex, which states some bounds for sides of the Hermite-Hadamard inequalities. In the context of the applications of results, we presented some relations involving special means and some inequalities for special functions including digamma function and Fresnel integral for sinus. In addiditon, an upper bound for error in numerical int

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31

H. M. Srivastava, K. A. Selvakumaran, and S. D. Purohit. "Inclusion properties for certain subclasses of analytic functions defined by using the generalized Bessel functions." Malaya Journal of Matematik 3, no. 03 (2015): 360–67. http://dx.doi.org/10.26637/mjm303/015.

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By making use of the operator $B_\nu^c$ defined by the generalized Bessel functions of the first kind, the authors introduce and investigate several new subclasses of starlike, convex, close-to-convex and quasi-convex functions. The authors establish inclusion relationships associated with the aforementioned operator. Some interesting corollaries and consequences of the main inclusion relationships are also considered.

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32

Ferreira, Orizon Pereira, Sándor Zoltán Németh, and Lianghai Xiao. "On the Spherical Quasi-convexity of Quadratic Functions on Spherically Subdual Convex Sets." Journal of Optimization Theory and Applications 187, no. 1 (2020): 1–21. http://dx.doi.org/10.1007/s10957-020-01741-7.

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Abstract In this paper, the spherical quasi-convexity of quadratic functions on spherically subdual convex sets is studied. Sufficient conditions for spherical quasi-convexity on spherically subdual convex sets are presented. A partial characterization of spherical quasi-convexity on spherical Lorentz sets is given, and some examples are provided.

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33

Aslan, Sinan, and Ahmet Ocak Akdemir. "New Estimations for Quasi-Convex Functions and \((h,m)\)-Convex Functions with the Help of Caputo-Fabrizio Fractional Integral Operators." Electronic Journal of Applied Mathematics 1, no. 3 (2023): 38–46. http://dx.doi.org/10.61383/ejam.20231353.

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The main motivation of the paper is to provide new integral inequalities for different kinds of convex functions by using a fractional integral operator with a non-singular kernel. The findings involve several new integral inequalities for quasi-convex functions and \((h,m)\)-convex functions. We have used the algebraic properties of Caputo-Fabrizio fractional operator, definitions of convex functions, and elementary analysis methods for the

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34

Rangel-Oliveros, Yenny, and Eze R. Nwaeze. "Simpson’s type inequalities for exponentially convex functions with applications." Open Journal of Mathematical Analysis 5, no. 2 (2021): 84–94. http://dx.doi.org/10.30538/psrp-oma2021.0096.

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The Simpson's inequality cannot be applied to a function that is twice differentiable but not four times differentiable or have a bounded fourth derivative in the interval under consideration. Loads of articles are bound for twice differentiable convex functions but nothing, to the best of our knowledge, is known yet for twice differentiable exponentially convex and quasi-convex functions. In this paper, we aim to do justice to this query. For this, we prove several Simpson's type inequalities for exponentially convex and exponentially quasi-convex functions. Our findings refine, generalize an

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35

Murota, Kazuo, and Akiyoshi Shioura. "Quasi M-convex and L-convex functions—quasiconvexity in discrete optimization." Discrete Applied Mathematics 131, no. 2 (2003): 467–94. http://dx.doi.org/10.1016/s0166-218x(02)00468-7.

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36

Mahmood, Shahid, Janusz Sokół, Hari Srivastava, and Sarfraz Malik. "Some Reciprocal Classes of Close-to-Convex and Quasi-Convex Analytic Functions." Mathematics 7, no. 4 (2019): 309. http://dx.doi.org/10.3390/math7040309.

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The present paper comprises the study of certain functions which are analytic and defined in terms of reciprocal function. The reciprocal classes of close-to-convex functions and quasi-convex functions are defined and studied. Various interesting properties, such as sufficiency criteria, coefficient estimates, distortion results, and a few others, are investigated for these newly defined sub-classes.

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37

Beer, Gerald. "Quasi-concave functions and convex convergence to infinity." Bulletin of the Australian Mathematical Society 60, no. 1 (1999): 81–94. http://dx.doi.org/10.1017/s0004972700033359.

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By a convex mode of convergence to infinity 〈Ck〉, we mean a sequence of nonempty closed convex subsets of a normed linear space X such that for each k, Ck+1 ⊆ int Ck and and a sequence 〈xn〉 is X is declared convergent to infinity with respect to 〈Ck〉 provided each Ck contains xn eventually. Positive convergence to infinity with respect to a pointed cone with nonempty interior as well as convergence to infinity in a fixed direction fit within this framework. In this paper we study the representation of convex modes of convergence to infinity by quasi-concave functions and associated remetrizati

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38

Sim, Young, Oh Kwon, and Nak Cho. "Geometric Properties of Lommel Functions of the First Kind." Symmetry 10, no. 10 (2018): 455. http://dx.doi.org/10.3390/sym10100455.

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In the present paper, we find sufficient conditions for starlikeness and convexity of normalized Lommel functions of the first kind using the admissible function methods. Additionally, we investigate some inclusion relationships for various classes associated with the Lommel functions. The functions belonging to these classes are related to the starlike functions, convex functions, close-to-convex functions and quasi-convex functions.

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39

Li, Gang, Minghua Li, and Yaohua Hu. "Stochastic quasi-subgradient method for stochastic quasi-convex feasibility problems." Discrete & Continuous Dynamical Systems - S 15, no. 4 (2022): 713. http://dx.doi.org/10.3934/dcdss.2021127.

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<p style='text-indent:20px;'>The feasibility problem is at the core of the modeling of many problems in various disciplines of mathematics and physical sciences, and the quasi-convex function is widely applied in many fields such as economics, finance, and management science. In this paper, we consider the stochastic quasi-convex feasibility problem (SQFP), which is to find a common point of infinitely many sublevel sets of quasi-convex functions. Inspired by the idea of a stochastic index scheme, we propose a stochastic quasi-subgradient method to solve the SQFP, in which the quasi-subg

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40

Ramachandran, C., S. Annamalai, and Basem Frasin. "The q-difference operator associated with the multivalent function bounded by conical sections." Boletim da Sociedade Paranaense de Matemática 39, no. 1 (2021): 133–46. http://dx.doi.org/10.5269/bspm.32913.

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In this paper we obtain some inclusion relations of k - starlike functions, k - uniformly convex functions and quasi-convex functions. Furthermore, we obtain coe¢ cient bounds for some subclasses of fractional q-derivative multivalent functions together with generalized Ruscheweyh derivative.

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41

Sidky, Fawzan Ismail, Doaa Shokry Mohamed, and Amina Ahmed Awad. "Some Inclusion Properties of Certain Subclasses of Analytic Functions Defined by Using the Tremblay Fractional Derivative Operator." WSEAS TRANSACTIONS ON SYSTEMS 20 (July 28, 2021): 209–16. http://dx.doi.org/10.37394/23202.2021.20.23.

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In this paper, we introduce new subclasses of analytic and p-valent functions related to starlike, convex, close-to-convex, and quasi-convex functions by using a p-valent analog of the Tremblay fractional derivative operator. Inclusion relationships for these subclasses are established.

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42

Tariq, Muhammad, Sotiris K. Ntouyas, and Asif Ali Shaikh. "A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Fractional Integral Operators." Mathematics 11, no. 8 (2023): 1953. http://dx.doi.org/10.3390/math11081953.

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In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions, (α,m)-convex functions, (α,m)-geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically (θ

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43

Emam, Tarek. "Logarithmic E-convex multi-objective programming." Control and Cybernetics 53, no. 2 (2024): 335–54. https://doi.org/10.2478/candc-2024-0014.

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Abstract A significant generalization of convexity is constituted by the concept of E-convexity, and our aim in this paper is to define a new class of functions depending on E-convexity, called logarithmic E -convex functions and to discuss their properties. Logarithmic E -convex functions are extended to lo-garithmic quasi E-convex and logarithmic pseudo E-convex functions and some results concerning them are presented. Logarithmic E -convex multi-objective programming problem is formulated. Finally, the sufficient and necessary conditions for a feasible solution to be an efficient solution f

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44

Adamek, Miroslaw. "Quasi-arithmetic F-convex functions and their characterization." Journal of Nonlinear Sciences and Applications 12, no. 11 (2019): 740–44. http://dx.doi.org/10.22436/jnsa.012.11.05.

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45

Lahrech, S., A. Jaddar, J. Hlal, A. Ouahab, and A. Mbarki. "A note on the weakly quasi-convex functions." International Mathematical Forum 2 (2007): 1755–61. http://dx.doi.org/10.12988/imf.2007.07155.

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46

Wu, Min-Chun, and Vladimir Itskov. "A topological approach to inferring the intrinsic dimension of convex sensing data." Journal of Applied and Computational Topology 6, no. 1 (2021): 127–76. http://dx.doi.org/10.1007/s41468-021-00081-3.

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AbstractWe consider a common measurement paradigm, where an unknown subset of an affine space is measured by unknown continuous quasi-convex functions. Given the measurement data, can one determine the dimension of this space? In this paper, we develop a method for inferring the intrinsic dimension of the data from measurements by quasi-convex functions, under natural assumptions. The dimension inference problem depends only on discrete data of the ordering of the measured points of space, induced by the sensor functions. We construct a filtration of Dowker complexes, associated to measurement

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47

Mehmet Zeki SARIKAYA, Hakan Bozkurt, and Mehmet Eyüp KIRIS. "On Hermite-Hadamard type integral inequalities for functions whose second derivative are nonconvex." Malaya Journal of Matematik 2, no. 03 (2014): 293–300. http://dx.doi.org/10.26637/mjm203/016.

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In this paper, we extend some estimates of the right hand side of a Hermite- Hadamard type inequality for nonconvex functions whose second derivatives absolute values are $\varphi$-convex, $\log -\varphi$-convex, and quasi- $\varphi$ convex.

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48

Yildiz, Çetin, and M. Emin Özdemir. "On new Fejér type inequalities for $m-$convex and quasi convex functions." Tbilisi Mathematical Journal 8, no. 2 (2015): 325–33. http://dx.doi.org/10.1515/tmj-2015-0030.

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Özdemir, M. E., Mustafa Gürbüz, and Çetin Yildiz. "Inequalities for mappings whose second derivativesare quasi-convex or $h$-convex functions." Miskolc Mathematical Notes 15, no. 2 (2014): 635. http://dx.doi.org/10.18514/mmn.2014.643.

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Boonmee, Prakassawat, and Santi Tasena. "Quadratic transformation of multivariate aggregation functions." Dependence Modeling 8, no. 1 (2020): 254–61. http://dx.doi.org/10.1515/demo-2020-0015.

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Abstract:

AbstractIn this work, we prove that quadratic transformations of aggregation functions must come from quadratic aggregation functions. We also show that this is different from quadratic transformations of (multivariate) semi-copulas and quasi-copulas. In the latter case, those two classes are actually the same and consists of convex combinations of the identity map and another fixed quadratic transformation. In other words, it is a convex set with two extreme points. This result is different from the bivariate case in which the two classes are different and both are convex with four extreme po

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