Some New Integral Inequalities for n-times Differentiable s-Convex Functions in the First Sense

Mahir Kadakal^1, , Huriye Kadakal² and İmdat İşcan¹

¹Department of Mathematics, Faculty of Sciences and Arts, Giresun University-Giresun-TÜRKİYE

²Institute of Science, Ordu University-Ordu-TÜRKİYE

Cite this paper:
Mahir Kadakal, Huriye Kadakal and İmdat İşcan. Some New Integral Inequalities for n-times Differentiable s-Convex Functions in the First Sense. Turkish Journal of Analysis and Number Theory. 2017; 5(2):63-68. doi: 10.12691/tjant-5-2-4

Abstract

In this work, by using an integral identity together with both the Hölder and the Power-Mean integral inequality we establish several new inequalities for n-time differentiable -convex functions in the first sense.

Keywords:
convex function - convex function in the first sense hölder integral inequality and power-mean integral inequality

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	M. Alomari, M. Darus and S. S. Dragomir, “New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are quasi-convex”, Tamkang Journal of Mathematics, 41 (4), 2010, 353-359.
[2]	S.-P. Bai, S.-H. Wang And F. Qi, “Some Hermite-Hadamard type inequalities for n-time differentiable (a,m) -convex functions”, Jour. of Ineq. and Appl., 2012, 2012: 267.
[3]	P. Cerone, S. S. Dragomir and J. Roumeliotis, “Some Ostrowski type inequalities for n-time differentiable mappings and applications”, Demonstratio Math., 32 (4) (1999), 697-712.
[4]	P. Cerone, S. S. Dragomir and J. Roumeliotis and J. ˇsunde, “A new generalization of the trapezoid formula for n-time differentiable mappings and applications”, Demonstratio Math., 33 (4) (2000), 719-736.
[5]	S. S. Dragomir and C. E. M. Pearce, “Selected Topics on Hermite-Hadamard Inequalities and Applications”, RGMIA Monographs, Victoria University, 2000.
[6]	D.-Y. Hwang, “Some Inequalities for n-time Differentiable Mappings and Applications”, Kyung. Math. Jour., 43 (2003), 335-343.
[7]	D. A. Ion, “Some estimates on the Hermite-Hadamard inequalities through quasi-convex functions”, Annals of University of Craiova, Math. Comp. Sci. Ser., 34 (2007), 82-87.
[8]	J. L. W. V. Jensen, “On konvexe funktioner og uligheder mellem middlvaerdier”, Nyt. Tidsskr. Math. B., 16, 49-69, 1905.
[9]	W.-D. Jiang, D.-W. Niu, Y. Hua and F. Qi, “Generalizations of Hermite-Hadamard inequality to n-time differentiable function which are s -convex in the second sense”, Analysis (Munich), 32 (2012), 209-220.
[10]	U. S. Kirmacı, M. K. Bakula, M. E. Özdemir and J. Pe´Cari´C, “Hadamard-type inequalities for s -convex functions”, Appl. Math. and Comp., 193 (2007), 26-35.
[11]	M. E. Özdemir and U. S. Kırmacı, “Two new theorem on mappings uniformly continuous and convex with applications to quadrature rules and means”, Appl. Math. and Comp., 143 (2003), 269-274.
[12]	M. E. Özdemir, Ç. Yildiz, “New Inequalities for n-time differentiable functions”, Arxiv: 1402. 4959v1.
[13]	M. E. Özdemir, Ç. Yıldız, “New Inequalities for Hermite-Hadamard and Simpson Type with Applications”, Tamkang J. of Math., 44, 2, 209-216, 2013.
[14]	M. E. Özdemir, Ç. Yıldız, A. O. Akdemir And E. Set, “New Inequalities of Hadamard Type for Quasi-Convex Functions”, AIP Conference Proceedings, 1470, 99-101 (2012).
[15]	J. E. Peˇcari´c, F. Porschan and Y. L. Tong, “Convex Functions, Partial Orderings, and Statistical Applications”, Academic Press Inc., 1992.
[16]	A. Saglam, M. Z. Sarıkaya and H. Yıldırım, “Some new inequalities of Hermite-Hadamard’s type”, Kyung. Math. Jour., 50 (2010), 399-410.
[17]	E. Set, M. E. Özdemir and S. S. Dragomir, “On Hadamard-Type Inequalities Involving Several Kinds of Convexity”, Jour. of Ineq. and Appl., 2010, 286845.
[18]	S. H. Wang, B.-Y. Xi and F. Qi, “Some new inequalities of Hermite-Hadamard type for n-time differentiable functions which are m-convex”, Analysis (Munich), 32 (2012), 247-262.
[19]	B.-Y. Xi and F. Qi, “Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means”, J. Funct. Spaces Appl., (2012).
[20]	S. Maden, H. Kadakal, M. Kadakal and İ. İşcan, “Some new integral inequalities for n-times differentiable convex and concave functions”. https://www.researchgate.net/publication/312529563, (Submitted)
[21]	S. S. Dragomir, S. Fitzpatrick, “The Hadamard’s inequality for s-convex functions in the second sense”, Demonstratio Math., 32, No. 4 (1999), 687-696.