The Hermite-Hadamard type inequality and its estimations via generalized convex functions of Raina type
Keywords:Convex function, m-convex function, s-type convex function, Hölder's inequality, improved power-mean integral inequality
The theory of convexity plays an important role in various branches of science and engineering. The main objective of this work is to introduce the idea of a generalized convex function by unifying s-type m-convex function and Raina type function. In addition, some beautiful algebraic properties and examples are discussed. Applying this new definition, we explore a new sort of Hermite-Hadamard inequality. Furthermore, to enhance the paper we investigate several new estimations of Hermite-Hadamard type inequality. The concepts and tools of this paper may invigorate and revitalize for additional research in this mesmerizing and absorbing field of mathematics.
Hardy, G.H., Little, J.E. and Polya, G. Inequalities. Cambridge, UK. Cambridge University Press. Cambridge mathematical library, (1952).
Kadakal, H. Hermite-Hadamard type inequalities for trigonometrically convex functions. Scientific Studies and Research. Series Mathematic and İnformatics, 28(2), 19-28, (2018).
Niculescu, C.P. and Persson, L.E. Convex functions and their applications.Springer, New York, (2006).
Tariq, M. New Hermite-Hadamard type inequalities via p-harmonic exponential type convexity and applications. Universal Journal of Mathematics and Applications, 4(2), 59-69, (2021).
Özcan, S. and İşcan, İ. Some new Hermite-Hadamard type integral inequalities for the s-convex functions and theirs applications. Journal of Inequalities and Applications, 2019(201), 1-14, (2019).
Tariq, M., Nasir, J., Sahoo, S.K. and Mallah, A.A. A note on some Ostrowski type inequalities via generalized exponentially convex function. Journal of Mathematical Analysis and Modeling, 2(2), 1-15, (2021).
Khan, M.A., Chu, Y.M., Khan, T.U. and Khan, J. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications. Open Math, 15(1), 1414-1430, (2017).
Butt, S.I., Tariq, M., Aslam, A., Ahmad, H. and Nofel, T.H. Hermite-Hadamard type inequalities via generalized harmonic exponential convexity. Journal of Function Spaces, 1-12, (2021).
Xi, B.Y. and Qi, F. Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means. Journal of Function Spaces, 1-14, (2012).
Butt, S.I., Kashuri, A., Tariq, M., Nasir, J., Aslam, A. and Geo, W. Hermite-Hadamard-type inequalities via n-polynomial exponential-type convexity and their applications. Advances in Difference Equations, 508, (2020).
Butt, S.I., Kashuri, A., Tariq, M., Nasir, J., Aslam, A. and Geo, W. n-polynomial exponential-type p-convex function with some related inequalities and their application. Heliyon, 6(11), (2020).
Hadamard, J. Étude sur les propriétés des fonctions entiéres en particulier d’une fonction considéréé par Riemann. Journal of Mathematics. Pures. Appl, 58, 171-215, (1893).
Raina, R.K. On generalized Wright’s hypergeometric functions and fractional calculus operators. East Asian Mathematical Journal, 21(2), 191-203, (2005).
Cortez, M.J.V., Liko, R., Kashuri, A. and Hernández, J.E.H. New quantum estimates of trapezium-type inequalities for generalized φ- convex functions. Mathematics, 7(11), 1047-1066, (2019).
Cortez, M.J.V., Kashuri, A. and Hernández, J.E.H. Trapezium-type inequalities for Raina’s fractional integrals operator using generalized convex functions.Symmetry, 12(6), 1034, (2020).
Rashid, S., İşcan, İ., Baleanu, D. and Chu, Y.M. Generation of new fractional inequalities via n-polynomials s-type convexity with applications. Advances in Difference Equations, Article ID 264.
Barani, A., Ghazanfari, A.G. and Dragomir, S.S. Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex. Journal of Inequalities and Applications, 2012(1), 1-9, (2012).
Toplu, T., Kadakal, M. and İşcan, İ. On n-polynomial convexity and some related inequalities. AIMS Math, 5(2), 1304-1318, (2020).
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