saujs

Sakarya University Journal of Science

2147-835X

Sakarya University

10.16984/saufenbilder.699212

Mathematical Sciences

Matematik

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

https://orcid.org/0000-0002-0240-918X

Kadakal

Mahir

Giresun University

08 01 2020

24 4 652 664 03 05 2020 05 10 2020

1997

Sakarya University Journal of Science

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.

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

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.