Novel Ostrowski type inequalities via exponentially (m1, m2)−convex functions and their applications

Saad Ihsan Butt, Bahtiyar Bayraktar, Jamshed Nasir

Abstract


In this work, we introduce and explore the new concept of convexity class, namely exponential (m1, m2)−convex functions along with some examples. In extension, we perform an artistic analysis of the properties of this class. We give the formulation of the new quadrature type identity. Based on this identity, we obtain some integral inequalities for the introduced convex functions via fractional operators tool. In addition, specific means of different positive real numbers and some new limits for the q−digamma function are also presented.

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M. Alomari, M. Darus, S. S. Dragomir, P. Cerone, Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense, Applied Mathematics Letters 23 (2010), no. 9, 1071-1076. DOI:10.1016/j.aml.2010.04.038

M. Alomari, M. Darus, Some Ostrowski type inequalities for quasi-convex functions with applications to special means, RGMIA Res. Rep. Coll. 13 (2010), no. 2.

M. U. Awan, M. A. Noor, K. I. Noor, Hermite-Hadamard inequalities for exponentially convex functions, Appl. Math. Inf. Sci. 12 (2018), no. 2, 405-409. DOI: 10.12785/amis/120215

B. Bayraktar, S. Butt, S. Shaukat, J. N_apoles, New Hadamard Type Inequalities Via (s;m1;m2)-Convex Functions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31 (2021), no. 4, 597-612. DOI:10.35634/Vm210405

B. Bayraktar, J.E. N_apoles Valdes, Integral inequalities for mappings whose derivatives are (h;m; s)convex mod_ed of second type via Katugampola integrals, Annals of the University of Craiova, Mathematics and Computer Science Series 49, (2022), no 2, 371-383. DOI: 10.52846/ami.v49i2.1596

S.I. Butt, J. Nasir, M.A. Dokuyucu, A.O. Akdemir, E. Set, Some Ostrowski-Mercer type inequalities for di_erentiable convex functions via fractional integral operators with strong kernels, Applied and computational mathematics 21 (2022), no. 3, 329-348.

S. I. Butt, S. Yousaf, A. O. Akdemir, M. A. Dokuyucu, New Hadamard-type integral inequalities via a general form of fractional integral operators, Chaos, Solitons and Fractals 148 (2021), 111025. DOI: 10.1016/j.chaos.2021.111025

P. Cerone, S. S. Dragomir, Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions, Demonstratio Math. XXXVII (2004), no. 2, 299-308.

S.S. Dragomir, On the Ostrowski's integral inequality for mappings with bounded variation and applications, Math. Ineq. Appl. 1 (1998), no. 2, 59-66.

I. Ișcan, New refinements for integral and sum forms of Hölder inequality, Journal of Inequalities and Applications 2019 (2019), no. 1, 1-11.

H. Kadakal, (α,m1,m2)-convexity and some inequalities of Hermite-Hadamard type, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68 (2019), no. 2, 2128-2142. DOI: 10.31801/cfsuasmas.511184

M. Kadakal, I. Ișcan, H. Kadakal, K. Bekar, On improvements of some integral inequalities, Researchgate, Preprint, January, (2019). DOI: 10.13140/RG.2.2. 15052.46724.

A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.

T. Lara, E. Rosales, J. L. Sánchez, New properties of m-convex functions, International Journal of Mathematical Analysis 9 (2015), no. 15, 735-742. DOI: 10.12988/ijma.2015.412389

D. S. Mitrinovič, J. Pečarič, A. M. Fink, Inequalities involving functions and their integrals and derivatives, Springer Science and Business Media 53, (1991).

J. E. Nápoles, B. Bayraktar, On The Generalized Inequalities Of The Hermite – Hadamard Type, FILOMAT 35 (2021), no. 14, 4917-4924. https://doi.org/10.2298/FIL2114917N

J. E. Nápoles, B. Bayraktar, S. I. Butt, New integral inequalities of Hermite-Hadamard type in a generalized context, Punjab University Journal Of Mathematics, 53 (2021), no. 11, 765-777. Doi:10.52280/Pujm.2021.531101

J. E. Nápoles, F. Rabossi, A. D. Samaniego, CONVEX FUNCTIONS: ARIADNE'S THREAD OR CHARLOTTE'S SPIDERWEB?, Advanced Mathematical Models and Applications 5 (2020), no. 2, 176-191.

M. A. Noor, K. I. Noor, M. U. Awan, Fractional Ostrowski inequalities for s-Godunova-Levin functions, International Journal of Analysis and Applications 5 (2014), no. 2, 167-173.

M. E. Özdemir, S. I. Butt, B. Bayraktar, J. Nasir, Several integral inequalities for (α,s,m)-convex functions, AIMS Mathematics 5 (2020), no. 4, 3906-3921. DOI: 10.3934/math.202053

B. G. Pachpatte, On an inequality of Ostrowski type in three independent variables, J. Math. Anal. Appl. 249 (2000), 583-591. DOI:10.1006/jmaa.2000.6913

A. Rafiq, N. A. Mir, F. Ahmad, Weighted Chebysev-Ostrowski type inequalities, Applied Math. Mechanics (English Edition) 28 (2007), no. 7, 901-906.

S. Rashid, M. A. Noor, K. I. Noor, Fractional exponentially m-convex functions and inequalities, International Journal of Analysis and Applications 17 (2019), no. 3, 464-478.

E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, Computers and Mathematics with Applications 63 (2012), no. 7, 1147-1154. DOI:10.1016/j.camwa.2011.12.023

M. E. Özdemir, H. Kavurmaci, E. Set, Ostrowski's type inequalities for (α,m)-convex function, Kyungpook Mathematical Journal 50 (2010), no. 3, 371-378. DOI: 10.5666/KMJ.2010.50.3.371

G. Toader, Some generalizations of the convexity, Proceedings of The Colloquium On Approximation and Optimization, Univ. Cluj-Napoca, Cluj-Napoca, 1984, 329-338.

G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, 1995.

J. Xie, M. A. Ali, H. Budak, M. Fekan, T. Sitthiwirattham, Fractional Hermite-Hadamard inequality, Simpson's and Ostrowski's type inequalities for convex functions with respect to a pair of functions, Rocky Mountain J. Math. 53 (2023), no. 2, 611-628.




DOI: https://doi.org/10.52846/ami.v51i2.1888