Document Type : Research Paper

Authors

1 Adıyaman University, Faculty of Science and Arts, Department of Mathematics, Adıyaman, Turkey.

2 Ağrı İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, 04100, Ağrı, Turkey.

3 Ordu University, Faculty of Science and Letters, Department of Mathematics,Ordu, Turkey.

Abstract

In this study, new Hermite-Hadamard type inequalities are generated for geometric-arithmetic functions with the help of an integral equation proved for differentiable functions. In proofs, some classical integral inequalities, such as H\"{o}lder's inequality, basic definitions and known mathematical analysis procedures are used. The third part of the study includes various applications confirming the accuracy of the generated results. A brief conclusion of the study has been given in the last part of the paper.

Keywords

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[1] G.D. Anderson, M.K. Vamanamurthy and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl., {335 (2007), pp. 1294-1308.
[2] A.O. Akdemir, M.E. Ozdemir, M. Avci Ardic and A. Yalcin, Some new generalizations for $GA-$convex functions, Filomat, 31 (4) (2017), pp. 1009-1016.
[3] M. Avci Ardic, A.O. Akdemir and K. Yildiz, On some new inequalities via $GG-$convexity and $GA-$convexity, Filomat, 32 (16) (2018).
[4] M. Avci Ardic, A.O. Akdemir and E. Set, New Ostrowski like inequalities for $GG-$convex and $GA-$convex functions, Mathematical Inequalities and Applications, 19(4) (2016), pp. 1159-1168.
[5] H.A. Coban, I. Iscan and M. Kunt, New Ostrowski type inequalities for $GA-$convex functions, New Trends in Mathematical Sciences, 4 (4) (2016), pp. 1-11.
[6] S.S. Dragomir, Inequalities of Hermite-Hadamard type for $GA-$ convex functions, Annales Mathematicae Silesianae (2018), DOI: 10.2478/amsil-2018-0001.
[7] I. Iscan, Jensen-Mercer inequality for $GA-$convex functions and some related inequalities, Journal of Inequalities and Applications, (2020) 2020:212.
[8] Y. Khurshid, M.A. Khan and Y.-M. Chu, Conformable fractional integral inequalities for $GG-$ and $GA-$convex functions, AIMS Mathematics, 5(5) (2020), pp. 5012-5030.
[9] M. Kunt and I. Iscan, Fractional Hermite-Hadamard-Fejer type inequalities for $GA-$convex functions, Turkish Journal of Inequalities, 2 (1) (2018), pp. 1-20.
[10] M.A. Latif, New Hermite-Hadamard type integral inequalities for $GA-$convex functions with applications, Analysis, 34 (4) (2014), pp. 379-389.
[11] C.P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl., 3 (2) (2000), pp. 155-167.
[12] C.P. Niculescu, Convexity according to means, Math. Inequal. Appl., 6 (4) (2003), pp. 571-579.
[13] R.A. Satnoianu, Improved $GA-$convexity inequalities, Journal of Inequalities in Pure and Applied Mathematics, 3 (5) (2002), Article 82.
[14] S.-H. Wang and X.-T. Shi, Hermite-Hadamard type inequalities for $n-$time differentiable and $GA-$convex functions with applications to means, Journal of Analysis & Number Theory, 4 (1) (2006), pp. 15-22.
[15] X.-M. Zhang, Y.-M. Chu and X.-H. Zhang, The Hermite-Hadamard type inequality of $GA-$convex functions and its application, Journal of Inequalities and Applications, Volume 2010, Article ID 507560, 11 pages.
[16] T.-Y. Zhang, A.-P. Ji and F. Qi, Some inequalities of Hermite-Hadamard type for $GA-$convex functions with applications to means, Le Matematiche, Vol. LXVIII (2013) - Fasc. I, pp. 229-239.