Some new inequalities for (α,m1,m2 )-GA convex functions

Year 2020, , 652 - 664, 01.08.2020

Mahir Kadakal

https://doi.org/10.16984/saufenbilder.699212

Abstract

In this manuscript, firstly we introduce and study the concept of (α,m_1,m_2 )-Geometric-Arithmetically (GA) convex functions and some algebraic properties of such type functions. Then, we obtain Hermite-Hadamard type integral inequalities for the newly introduced class of functions by using an identity together with Hölder integral inequality, power-mean integral inequality and Hölder-İşcan integral inequality giving a better approach than Hölder integral inequality. Inequalities have been obtained with the help of Gamma function. In addition, results were obtained according to the special cases of α, m_1 and m_2.

Keywords

(alpha m1 m2)-GA convex function, Hölder and power-mean inequalities, Hölder-İşcan inequality, Hermite-Hadamard integral inequality, power-mean inequality

References

• M.K. Bakula, M.E. Özdemir, and J. Pečarić, “Hadamard type inequalities for m-convex and (α,m)-convex functions,” J. Inequal. Pure Appl. Math. 9(4), Art. 96, 12 pages, 2008.
• S.S. Dragomir, “Inequalities of Hermite-Hadamard type for GA-convex functions,” Annales Mathematicae Silesianae. 32(1). Sciendo, 2018.
• S.S. Dragomir and RP. Agarwal, “Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula,” Appl. Math. Lett. 11, 91-95, 1998.
• S.S. Dragomir and C.E.M. Pearce, “Selected Topics on Hermite-Hadamard Inequalities and Its Applications,” RGMIA Monograph, 2002.
• S.S. Dragomir, J. Pečarić and LE.Persson, “Some inequalities of Hadamard Type,” Soochow Journal of Mathematics, 21(3), pp. 335-341, 2001.
• J. Hadamard, “Étude sur les propriétés des fonctions entières en particulier d’une fonction considérée par Riemann,” J. Math. Pures Appl. 58, 171-215, 1893.
• İ. İşcan, “New refinements for integral and sum forms of Hölder inequality,” 2019:304, 11 pages, 2019.
• İ. İşcan and M. Kunt, “Hermite-Hadamard-Fejer type inequalities for quasi-geometrically convex functions via fractional integrals, Journal of Mathematics,” Volume 2016, Article ID 6523041, 7 pages, 2016.
• A.P. Ji, T.Y. Zhang, F. Qi, “Integral inequalities of Hermite-Hadamard type for (α,m)-GA-convex functions,” arXiv preprint arXiv:1306.0852, 4 June 2013.
• H. Kadakal, “Hermite-Hadamard type inequalities for trigonometrically convex functions,” Scientific Studies and Research. Series Mathematics and Informatics, 28(2), 19-28, 2018.
• H. Kadakal, “New Inequalities for Strongly r-Convex Functions,” Journal of Function Spaces, Volume 2019, Article ID 1219237, 10 pages, 2019.
• H. Kadakal, “(m_1,m_2 )-convexity and some new Hermite-Hadamard type inequalities,” International Journal of Mathematical Modelling and Computations, (Accepted for publication), 2020.
• H. Kadakal, “(α,m_1,m_2 )-convexity and some inequalities of Hermite-Hadamard type,” Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 2128-2142, 2019.
• M. Kadakal, “(m_1,m_2 )-geometric arithmetically convex functions and related inequalities,” Mathematical Sciences and Applications E-Notes, (Submitted to journal), 2020.
• M. Kadakal, H. Kadakal and İ. İşcan, “Some new integral inequalities for n-times differentiable s-convex functions in the first sense,” Turkish Journal of Analysis and Number Theory, 5(2), 63-68, 2017.
• V.G. Miheşan, “A generalization of the convexity,” Seminar on Functional Equations, Approx. Convex, Cluj-Napoca, (Romania), 1993.
• C.P. Niculescu, “Convexity according to the geometric mean,” Math. Inequal. Appl. 3(2), 155-167, 2000.
• C.P. Niculescu, “Convexity according to means,” Math. Inequal. Appl. 6 (4), 571-579, 2003.
• S. Özcan, “Some Integral Inequalities for Harmonically (α,s)-Convex Functions, Journal of Function Spaces,” 2019, Article ID 2394021, 8 pages 2019.
• S. Özcan, and İ. İşcan, “Some new Hermite-Hadamard type inequalities for s-convex functions and their applications,” Journal of Inequalities and Applications, Article number: 2019:201, 2019.
• Y. Shuang, Yin, H.P. and Qi, F., “Hermite-Hadamard type integral inequalities for geometric-arithmetically s-convex functions,” Analysis, 33, 197-208, 2013.
• G. Toader, “Some generalizations of the convexity,” Proc. Colloq. Approx. Optim., Univ. Cluj Napoca, Cluj-Napoca, 329-338, 1985.
• F. Usta, H. Budak and M.Z. Sarıkaya, “Montgomery identities and Ostrowski type inequalities for fractional integral operators,” Revista de la Real Academia de Ciencias Exactas, F13 ̆053'fsicas y Naturales. Serie A. Matemáticas, 113(2), 1059-1080, 2019.
• F. Usta, H. Budak and M.Z. Sarıkaya, “Some New Chebyshev Type Inequalities Utilizing Generalized Fractional Integral Operators,” AIMS Mathematics, 5(2), 1147-1161, 2020.
• S. Varošanec, “On h-convexity,”J. Math. Anal. Appl. 326, 303-311, 2007.