Some new integral inequalities for n-times differentiable log-convex functions

Year 2017, Volume: 5 Issue: 2, 10 - 15, 30.03.2017

İmdat Iscan Huriye Kadakal Mahir Kadakal

https://izlik.org/JA43CZ55XU

Abstract

 In this work, by using an integral identity together with both the Ho¨lder and the Power-Mean integral inequality we establish several new inequalities for n-time differentiable log-convex functions.

Keywords

Convex function , Log-Convex function

References

• [1] M. Alomari and M. Darus, “On The Hadamard’s Inequality for Log-Convex Functions on the Coordinates”, Hindawi Publishing Corporation Journal of Inequalities and Applications, Volume 2009, Article ID 283147, 13 pages, doi:10.1155/2009/283147.
• [2] M. A. Ardıc¸ and M. Emin ¨Ozdemir, “Inequalities for log-convex functions vıa three times differentiability”, arXiv:1405.7480v1 [math.CA] 29 May 2014.
• [3] S. P. Bai, S.-H. Wang and F. Qi, “Some Hermite-Hadamard type inequalities for n-time differentiable ( α ,m)-convex functions”, Jour. of Ineq. and Appl., 2012, 2012:267.
• [4] 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.
• [5] P. Cerone, S.S. Dragomir, J. Roumeliotis and J. Sunde, “A new generalization of the trapezoid formula for n-time differentiable mappings and applications”, Demonstratio Math., 33 (4) (2000), 719–736.
• [6] S. S. Dragomir and C.E.M. Pearce, “Selected Topics on Hermite-Hadamard Inequalities and Applications”, RGMIA Monographs, Victoria University, 2000, online: http://www.staxo.vu.edu.au/RGMIA/monographs/hermite hadamard.html.
• [7] S.S. Dragomir, “New jensen’s type inequalities for differentiable log-convex functions of selfadjoint operators in Hilbert spaces”, Sarajevo Journal of Mathematics, Vol.7 (19) (2011), 67-80.
• [8] D. Y. Hwang, “Some Inequalities for n-time Differentiable Mappings and Applications”, Kyung. Math. Jour., 43 (2003), 335–343.
• [9] I. Is¸can, “Ostrowski type inequalities for p-convex functions”, New Trends in Mathematical Sciences, 4 (3) (2016), 140-150.
• [10] I. Is¸can and S. Turhan, “Generalized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral”, Moroccan J. Pure and Appl. Anal.(MJPAA), Volume 2(1) (2016), 34-46. [11] I. Is¸can, “Hermite-Hadamard type inequalities for harmonically convex functions”, Hacettepe Journal of Mathematics and Statistics, Volume 43 (6) (2014), 935–942.
• [12] 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.
• [13] S. Maden, H. Kadakal, M. Kadakal and ˙ I. ˙Is¸can, “Some new integral inequalities for n-times differentiable convex and concave functions”, https://www.researchgate.net/publication/312529563, (Submitted).
• [14] M. Mansour, M. A. Obaid, “A Generalization of Some Inequalities for the log-Convex Functions”, International Mathematical Forum, 5, 2010, no. 65, 3243 – 3249. [15] C. P. Niculescu, “The Hermite–Hadamard inequality for log-convex functions”, Nonlinear Analysis 75 (2012) 662–669.
• [16] J. Park, “Some Hermite-Hadamard-like Type Inequalities for Logarithmically Convex Functions”, Int. Journal of Math. Analysis, Vol. 7, 2013, no. 45, 2217-2233.
• [17] 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.
• [18] G. S. Yang, K. L. Tseng and H. T. Wang, “A note on ıntegral inequalıtıes of Hadamard type for log-convex and log-concave functıons”, Taıwanese Journal of Mathematıcs, Vol. 16, No. 2, pp. 479-496, April 2012.
• [19] C. Yıldız, “New inequalities of the Hermite-Hadamard type for n-time differentiable functions which are quasiconvex”, Journal of Mathematical Inequalities, 10, 3(2016), 703-711.
• [20] X. Zhang, W. Jiang, “Some properties of log-convex function and applications for the exponential function”, Computers and Mathematics with Applications 63 (2012) 1111–1116.