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Journal articles on the topic 'Opial inequality'

Author: Grafiati

Published: 4 June 2025

Last updated: 1 August 2025

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1

Gudelj, Ana, Kristina Krulić Himmelreich, and Josip Pečarić. "General Opial Type Inequality and New Green Functions." Axioms 11, no. 6 (2022): 252. http://dx.doi.org/10.3390/axioms11060252.

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In this paper we provide many new results involving Opial type inequalities. We consider two functions—one is convex and the other is concave—and prove a new general inequality on a measure space (Ω,Σ,μ). We give an new result involving four new Green functions. Our results include Grüss and Ostrowski type inequalities related to the generalized Opial type inequality. The obtained inequalities are of Opial type because the integrals contain the function and its integral representation. They are not a direct generalization of the Opial inequality.

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2

Barbir, Ana, Kristina Krulić Himmelreich, and Josip Pečarić. "General Opial type inequality." Aequationes mathematicae 89, no. 3 (2014): 641–55. http://dx.doi.org/10.1007/s00010-013-0252-4.

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3

Alp, Necmettin, Candan Bilişik, and Mehmet Sarıkaya. "On q-Opial type inequality for quantum integral." Filomat 33, no. 13 (2019): 4175–84. http://dx.doi.org/10.2298/fil1913175a.

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4

Mirković, Tatjana, and Tatjana Bajić. "Opial inequalities for a conformable ∆-fractional calculus on time scales." Mathematica Moravica 28, no. 2 (2024): 17–32. https://doi.org/10.5937/matmor2402017m.

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In this paper, an Opial-type inequality is introduced on time scale for a conformal ∆-fractional differentiable function of order a, a ∈ (0, 1]. In the case where the certain weight functions are included, one generalization of the Opial inequality is proved using conformal ∆-fractional calculus on time scales. Moreover, for n times conformal ∆-fractional differentiable function on time scale, n ∈ N, an Opial inequality is obtained. In particular, through examples, the main results from the paper are compared with classical ones on generalized time scales. At the end of the paper, we indicate

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5

Koliha, J. J., and J. Pecaric. "Weighted Opial inequalities." Tamkang Journal of Mathematics 33, no. 1 (2002): 83–92. http://dx.doi.org/10.5556/j.tkjm.33.2002.308.

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This paper presents a class of very general weighted Opial type inequalities. The notivation comes from the monograph of Agarwal and Pang (Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Acad., Dordrecht 1995) and the work of Anastassiou and Pecaric (J. Math. Anal. Appl. 239 (1999), 402-418). Assuming only a very general inequality, we extend the latter paper in several directions. A new result generalizing the original Opial's inequality is obtained, and applications to fractional derivatives are given.

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6

SARIKAYA, Mehmet Zeki, and Candan Can Bilişik. "Opial-Jensen and functional inequalities for convex functions." Journal of Fractional Calculus and Nonlinear Systems 3, no. 2 (2022): 27–36. http://dx.doi.org/10.48185/jfcns.v3i2.553.

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The main of this article are presenting generalized Opial type inequalities which will be defined as theOpial-Jensen inequality for convex function. Further, new Opial type inequalities will be given for functionalsdefined with the help of the Opial inequalities.

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7

PACHPATTE, B. G. "OPIAL TYPE INEQUALITY IN SEVERAL VARIABLES." Tamkang Journal of Mathematics 22, no. 1 (1991): 7–11. http://dx.doi.org/10.5556/j.tkjm.22.1991.4562.

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   In the present note we establish a new integral inequality of the Opial type mvolving a function of $n$ variables and its partial derivative. A corresponding result on the discrete analogue of the main result is also given.   

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8

Deng, Yinbin. "The opial inequality in R N." Acta Mathematica Scientia 21, no. 4 (2001): 572–76. http://dx.doi.org/10.1016/s0252-9602(17)30447-2.

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9

Kai-Chen, Hsu, and Tseng Kuei-Lin. "Some New Discrete Inequalities of Opial and Lasota's Type." Journal of Progressive Research in Mathematics 4, no. 2 (2015): 294–302. https://doi.org/10.5281/zenodo.3980472.

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10

Pachpatte, B. G. "On Opial type inequalities in two independent variables." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 100, no. 3-4 (1985): 263–70. http://dx.doi.org/10.1017/s0308210500013809.

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SynopsisThe aim of the present paper is to establish some new integral inequalities of Opial type in two independent variables. Our results are the two independent variable generalizations of some of the inequalities recently established by the present author and in special cases yield the two independent variable analogue of the Opial inequality and its generalization given by G. S. Yang.

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11

Agarwal, Ravi, Martin Bohner, Donal O’Regan, Mahmoud Osman, and Samir Saker. "A General Dynamic Inequality of Opial Type." Applied Mathematics & Information Sciences 10, no. 3 (2016): 875–79. http://dx.doi.org/10.18576/amis/100306.

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12

Li, Qiao-Luan, and Wing-Sum Cheung. "An Opial-Type Inequality on Time Scales." Abstract and Applied Analysis 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/534083.

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We establish some new Opial-type inequalities involving higher order delta derivatives on time scales. These extend some known results in the continuous case in the literature and provide new estimates in the setting of time scales.

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13

Alzer, Horst. "Note on a discrete Opial-type inequality." Archiv der Mathematik 65, no. 3 (1995): 267–70. http://dx.doi.org/10.1007/bf01195098.

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14

Nosheen, Ammara, Anum Saba, Khuram Ali Khan, and Michael Kikomba Kahungu. "q,h-Opial-Type Inequalities via Hahn Operators." Discrete Dynamics in Nature and Society 2022 (October 26, 2022): 1–12. http://dx.doi.org/10.1155/2022/2650126.

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In this paper, the well-known Hölder’s inequality is proved via Hahn differential and integral operators, which is a helping tool to establish some Opial-type inequalities via Hahn’s calculus. The weight functions involved in these Opial-type inequalities are positive and monotone. In search of applications, some new as well as some existing inequalities in the literature are obtained by applying suitable limits.

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15

Anthonio, Yisa Oluwatoyin, Abimbola Abolarinwa, and Kamilu Rauf. "Some Results on Pachpatte-Type of Opial Inequality." Pan-American Journal of Mathematics 1 (July 7, 2022): 3. http://dx.doi.org/10.28919/cpr-pajm/1-3.

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16

Brown, Richard C., and Michael Plum. "An Opial-type inequality with an integral boundary condition." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2060 (2005): 2635–51. http://dx.doi.org/10.1098/rspa.2005.1449.

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We determine the best constant K and extremals of the Opial-type inequality , where y is required to satisfy the boundary condition . The techniques employed differ from those utilized recently by Denzler to solve this problem, and also from those used originally to prove the classical inequality; but they also yield a new proof of that inequality.

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17

Saker, S. H., D. M. Abdou, and I. Kubiaczyk. "Opial and Pólya Type Inequalities Via Convexity." Fasciculi Mathematici 60, no. 1 (2018): 145–59. http://dx.doi.org/10.1515/fascmath-2018-0009.

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Abstract In this paper, we prove some new dynamic inequalities related to Opial and Pólya type inequalities on a time scale 𝕋. We will derive the integral and discrete inequalities of Pólya’s type as special cases and also derive several classical integral inequalities of Opial’s type that has been obtained in the literature as special cases. The main results will be proved by using the chain rule, Hölder’s inequality and Jensen’s inequality, Taylor formula on time scales.

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18

Yang, Gou-Sheng, and Tien-Shou Huang. "On some inequalities related to Opial-Type inequality in two variables." Tamkang Journal of Mathematics 33, no. 4 (2002): 379–86. http://dx.doi.org/10.5556/j.tkjm.33.2002.287.

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19

Agarwal, R. P., and P. Y. H. Pang. "Remarks on the Generalizations of Opial′s Inequality." Journal of Mathematical Analysis and Applications 190, no. 2 (1995): 559–77. http://dx.doi.org/10.1006/jmaa.1995.1091.

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20

Pang, P. Y. H., and R. P. Agarwal. "On an Opial Type Inequality Due to Fink." Journal of Mathematical Analysis and Applications 196, no. 2 (1995): 748–53. http://dx.doi.org/10.1006/jmaa.1995.1438.

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21

PACHPATTE, B. G. "A NOTE ON GENERALIZED OPIAL TYPE INEQUALITIES." Tamkang Journal of Mathematics 24, no. 2 (1993): 229–35. http://dx.doi.org/10.5556/j.tkjm.24.1993.4494.

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   In the present note we establish some new inequalities involv- ing integrals of functions and their derivatives which in the special cases yield the well known Opial inequality and some of its generalizations.   

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22

Li, Horng Jaan, and Cheh Chih Yeh. "Inequalities for a function involving its integral and derivative." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 1 (1995): 133–51. http://dx.doi.org/10.1017/s0308210500030791.

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We give a concise approach to generalising the inequalities of Wirtinger, Hardy, Weyl and Opial by using the well-known inequality: if X and Y are non-negative, thenfor p > 1 (0 < p < 1), respectively.

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23

Rauf, K., and Y. O. Anthonio. "Results on an integral inequality of the opial- type." Global Journal of Pure and Applied Sciences 23, no. 1 (2017): 151. http://dx.doi.org/10.4314/gjpas.v23i1.15.

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24

Andrić, Maja, Ana Barbir, Sajid Iqbal, and Josip Pečarić. "An Opial-type integral inequality and exponentially convex functions." Fractional Differential Calculus, no. 1 (2015): 25–42. http://dx.doi.org/10.7153/fdc-05-03.

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25

George, A. Anastassiou. "Psi -Hilfer and Hilfer fractional self adjoint operator analytic inequalities." Asia Mathematika 5, no. 1 (2021): 83–102. https://doi.org/10.5281/zenodo.4723438.

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We give here  \(\psi\)-Hilfer and Hilfer fractional self-adjoint operator inner product comparison, Poincare, Sobolev, and Opial type inequalities. At first, we give right and left  \(\psi\)-Hilfer fractional representation formulae in the self-adjoint operator sense. Operator inequalities are based on the self-adjoint operator order over a Hilbert space.

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26

Saker, Samir H. "Some Opial Dynamic Inequalities Involving Higher Order Derivatives on Time Scales." Discrete Dynamics in Nature and Society 2012 (2012): 1–22. http://dx.doi.org/10.1155/2012/157301.

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We will prove some new Opial dynamic inequalities involving higher order derivatives on time scales. The results will be proved by making use of Hölder's inequality, a simple consequence of Keller's chain rule and Taylor monomials on time scales. Some continuous and discrete inequalities will be derived from our results as special cases.

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27

Andrić, Maja, Josip Pečarić, and Ivan Perić. "An Opial-Type inequality for fractional derivatives of two functions." Fractional Differential Calculus, no. 1 (2013): 55–68. http://dx.doi.org/10.7153/fdc-03-04.

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28

Vivas-Cortez, Miguel, Francisco Martínez, Juan E. Nápoles Valdes, and Jorge E. Hernández. "On Opial-type inequality for a generalized fractional integral operator." Demonstratio Mathematica 55, no. 1 (2022): 695–709. http://dx.doi.org/10.1515/dema-2022-0149.

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Abstract This article is aimed at establishing some results concerning integral inequalities of the Opial type in the fractional calculus scenario. Specifically, a generalized definition of a fractional integral operator is introduced from a new Raina-type special function, and with certain results proposed in previous publications and the choice of the parameters involved, the established results in the work are obtained. In addition, some criteria are established to obtain the aforementioned inequalities based on other integral operators. Finally, a more generalized definition is suggested,

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29

He, X. G. "A Short Proof of a Generalization on Opial′s Inequality." Journal of Mathematical Analysis and Applications 182, no. 1 (1994): 299–300. http://dx.doi.org/10.1006/jmaa.1994.1086.

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30

Xu, Han, Li Sha, and Li Qiaoluan. "Opial Type Inequalities for Conformable Fractional Derivative and Integral of Two Functions." Journal of Progressive Research in Mathematics 12, no. 3 (2017): 1924–31. https://doi.org/10.5281/zenodo.3975409.

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31

Saker, Samir H., and Mohammed A. Arahet. "Distributions of Zeros of Solutions for Third-Order Differential Equations with Variable Coefficients." Mathematical Problems in Engineering 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/158460.

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For the third-order linear differential equations of the formr(t)x′′(t)′+p(t)x′(t)+q(t)x(t)=0, we will establish lower bounds for the distance between zeros of a solution and/or its derivatives. The main results will be proved by making use of Hardy’s inequality and some generalizations of Opial and Wirtinger type inequalities.

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32

Andrić, Maja, Josip Pečarić, and Ivan Perić. "A multiple Opial type inequality for the Riemann-Liouville fractional derivatives." Journal of Mathematical Inequalities, no. 1 (2013): 139–50. http://dx.doi.org/10.7153/jmi-07-13.

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33

Andrić, Maja, Ana Barbir, Ghulam Farid, and Josip Pečarić. "Opial-type inequality due to Agarwal–Pang and fractional differential inequalities." Integral Transforms and Special Functions 25, no. 4 (2013): 324–35. http://dx.doi.org/10.1080/10652469.2013.851079.

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34

Gov, Esra, and Orkun Tasbozan. "Some quantum estimates of opial inequality and some of its generalizations." New Trends in Mathematical Science 1, no. 6 (2018): 76–84. http://dx.doi.org/10.20852/ntmsci.2018.247.

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35

Alzer, H. "An Opial-type inequality involving higher-order derivatives of two functions." Applied Mathematics Letters 10, no. 4 (1997): 123–28. http://dx.doi.org/10.1016/s0893-9659(97)00071-2.

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36

Goroncy, Agnieszka, and Udo Kamps. "Relations for m-generalized order statistics via an Opial-type inequality." Journal of Statistical Planning and Inference 142, no. 6 (2012): 1457–63. http://dx.doi.org/10.1016/j.jspi.2011.12.026.

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37

Shi, Da, Ghulam Farid, Abd Elmotaleb A. M. A. Elamin, Wajida Akram, Abdullah A. Alahmari, and B. A. Younis. "Generalizations of some $ q $-integral inequalities of Hölder, Ostrowski and Grüss type." AIMS Mathematics 8, no. 10 (2023): 23459–71. http://dx.doi.org/10.3934/math.20231192.

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<abstract><p>This paper investigates some well-known inequalities for $ q $-$ h $-integrals. These include Hölder, Ostrowski, Grüss and Opial type inequalities. Refinement of the Hadamard inequality for $ q $-$ h $-integrals is also established by applying the definition of strongly convex functions. From main theorems, $ q $-Hölder, $ q $-Ostrowski and $ q $-Grüss inequalities can be obtained in particular cases.</p></abstract>

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38

Saker, Samir H. "Lyapunov's Type Inequalities for Fourth-Order Differential Equations." Abstract and Applied Analysis 2012 (2012): 1–25. http://dx.doi.org/10.1155/2012/795825.

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For a fourth-order differential equation, we will establish some new Lyapunov-type inequalities, which give lower bounds of the distance between zeros of a nontrivial solution and also lower bounds of the distance between zeros of a solution and/or its derivatives. The main results will be proved by making use of Hardy’s inequality and some generalizations of Opial-Wirtinger-type inequalities involving higher-order derivatives. Some examples are considered to illustrate the main results.

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39

Chouhan, Amit. "On certain new CAUCHY–TYPE fracitioanl integral inequalities and OPIAL–TYPE fractional derivative inequalities." Tamkang Journal of Mathematics 46, no. 1 (2015): 67–73. http://dx.doi.org/10.5556/j.tkjm.46.2015.1586.

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The aim of this paper is to establish several new fractional integral and derivative inequalities for non-negative and integrable functions. These inequalities related to the extension of general Cauchy type inequalities and involving Saigo, Riemann-Louville type fractional integral operators together with multiple Erdelyi-Kober operator. Furthermore the Opial-type fractional derivative inequality involving H-function is also established. The generosity of H-function could leads to several new inequalities that are of great interest of future research.

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40

El-Deeb, Ahmed A., and Dumitru Baleanu. "New Weighted Opial-Type Inequalities on Time Scales for Convex Functions." Symmetry 12, no. 5 (2020): 842. http://dx.doi.org/10.3390/sym12050842.

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Our work is based on the multiple inequalities illustrated in 1967 by E. K. Godunova and V. I. Levin, in 1990 by Hwang and Yang and in 1993 by B. G. Pachpatte. With the help of the dynamic Jensen and Hölder inequality, we generalize a number of those inequalities to a general time scale. In addition to these generalizations, some integral and discrete inequalities will be obtained as special cases of our results.

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41

Pečarić, Josip, and Ilko Brnetić. "Note on the Generalization of the Godunova–Levin–Opial Inequality in Several Independent Variables." Journal of Mathematical Analysis and Applications 215, no. 1 (1997): 274–82. http://dx.doi.org/10.1006/jmaa.1997.5529.

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42

Lin, Faa-Jeng, Chao-Fu Chang, Yu-Cheng Huang, and Tzu-Ming Su. "A Deep Reinforcement Learning Method for Economic Power Dispatch of Microgrid in OPAL-RT Environment." Technologies 11, no. 4 (2023): 96. http://dx.doi.org/10.3390/technologies11040096.

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This paper focuses on the economic power dispatch (EPD) operation of a microgrid in an OPAL-RT environment. First, a long short-term memory (LSTM) network is proposed to forecast the load information of a microgrid to determine the output of a power generator and the charging/discharging control strategy of a battery energy storage system (BESS). Then, a deep reinforcement learning method, the deep deterministic policy gradient (DDPG), is utilized to develop the power dispatch of a microgrid to minimize the total energy expense while considering power constraints, load uncertainties and electr

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43

Dube, Deepali Y., and Hiren G. Patel. "Suppressing the Noise in Measured Signals for the Control of Helicopters." Fluctuation and Noise Letters 18, no. 01 (2019): 1950002. http://dx.doi.org/10.1142/s0219477519500020.

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This paper concerns with the non-linear system having multiple-inputs multiple-outputs (MIMO). The plant mainly comprises: bench-top helicopter, tail and main rotor of a helicopter system. The dynamics are presented with control methodologies where a conventional strategy proves the instability of the system while the deadbeat and sliding mode control with linear matrix inequality regulates the future estimates. There have been disturbances like presence of unwanted ripples in the output of the non-linear systems (in case of stability also after 100[Formula: see text]s) and in the tracking of

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44

Mirković, Tatjana Z., Slobodan B. Tričković, and Miomir S. Stanković. "Opial inequality in q-calculus." Journal of Inequalities and Applications 2018, no. 1 (2018). http://dx.doi.org/10.1186/s13660-018-1928-z.

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45

Anthonio, Yisa Oluwatoyin, Kamilu Rauf, Abdullai Ayinla Abdurasid, and Oluwaseun Raphael Aderele. "Multivariate Opial-type Inequalities on Time Scales." Earthline Journal of Mathematical Sciences, January 27, 2023, 13–26. http://dx.doi.org/10.34198/ejms.12123.1326.

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Opial inequality was developed to provide bounds for integral of functions and their derivatives. It has become an indispensable tool in the theory of mathematical analysis due to its usefulness. A refined Jensen inequality for multivariate functions is employed to establish new Opial-type inequalities for convex functions of several variables on time scale.

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46

Nasiruzzaman, Md, Aiman Mukheimer, and M. Mursaleen. "Some Opial-type integral inequalities via $(p,q)$-calculus." Journal of Inequalities and Applications 2019, no. 1 (2019). http://dx.doi.org/10.1186/s13660-019-2247-8.

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AbstractIn this paper, we introduce a new Opial-type inequality by using $(p,q)$(p,q)-calculus and establish some integral inequalities. We find a $(p,q)$(p,q)-generalization of a Steffensens-type integral inequality and some other inequalities.

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47

Pachpatte, B. G. "ON SOME NEW GENERALIZATIONS OF OPIAL INEQUALITY." Demonstratio Mathematica 19, no. 2 (1986). http://dx.doi.org/10.1515/dema-1986-0203.

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48

Bosch, Paul, Ana Portilla, Jose M. Rodriguez, and Jose M. Sigarreta. "On a generalization of the Opial inequality." Demonstratio Mathematica 57, no. 1 (2024). http://dx.doi.org/10.1515/dema-2023-0149.

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Abstract Inequalities are essential in pure and applied mathematics. In particular, Opial’s inequality and its generalizations have been playing an important role in the study of the existence and uniqueness of initial and boundary value problems. In this work, some new Opial-type inequalities are given and applied to generalized Riemann-Liouville-type integral operators.

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49

BARBIR, ANA, KRISTINA KRULIĆ HIMMELREICH, and JOSIP PREČARIĆ. "GENERAL OPIAL TYPE INEQUALITY FOR QUOTIENT OF FUNCTIONS." Sarajevo Journal of Mathematics, 2016. http://dx.doi.org/10.5644/sjm.12.2.06.

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50

Pečarić, Josip, and llko Brnetić. "NOTE ON GENERALIZATION OF GODUNOVA-LEVIN-OPIAL INEQUALITY." Demonstratio Mathematica 30, no. 3 (1997). http://dx.doi.org/10.1515/dema-1997-0310.

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