Some New Integral Inequalities for $n$-Times Differentiable Trigonometrically Convex Functions

Year 2020, Volume: 3 Issue: 3, 109 - 114, 29.09.2020

Kerim Bekar

https://doi.org/10.32323/ujma.652513

Abstract

In this manuscript, by using an integral identity together with both the Hölder, Hölder-İşcan and the Power-mean integral inequalities we obtain several new inequalities for $n$-time differentiable trigonometrically convex functions.                                                                                                                                                                                                                                                                                                                                                                                                                 

Keywords

Convex function, trigonometrically convex function, Hölder Integral inequality, Power-Mean Integral inequality, Hölder-İşcan integral inequality

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 J. Math, 41 (4), 2010, 353–359.
• [2] 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.
• [3] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
• [4] D.Y. Hwang, Some Inequalities for n-time Differentiable Mappings and Applications, Kyung. Math. Jour., 43 (2003), 335–343.
• [5] İ. İşcan, S. Turhan, S. Maden, Some Hermite-Hadamard-Fejer type inequalities for Harmonically convex functions via Fractional Integral, New Trends Math. Sci., 4 (2) (2016), 1-10.
• [6] İ. İşcan, M. Kunt, Hermite-Hadamard-Fejer type inequalities for quasi-geometrically convex functions via fractional integrals, J. Math., Volume 2016, Article ID 6523041, 7 pages.
• [7] İ. İşcan, New refinements for integral and sum forms of H¨older inequality, J. Inequal. Appl., (2019) 2019:304, 11 pages.
• [8] 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.
• [9] H. Kadakal, Hermite-Hadamard type inequalities for trigonometrically convex functions, Sci. Stud. Res. Ser. Math. Info., 28(2), (2018), 19-28.
• [10] S. Maden, H. Kadakal, M. Kadakal and İ. İşcan, Some new integral inequalities for n-times differentiable convex and concave functions. J. Non. Sci. Appl., 10 12,(2017), 6141-6148.
• [11] S. Özcan, Some Integral Inequalities for Harmonically (a; s)-Convex Functions, J. Funct. Spaces, 2019, Article ID 2394021, 8 pages (2019).
• [12] S. Özcan, and İ. İşcan, Some new Hermite-Hadamard type inequalities for s-convex functions and their applications, J. Inequal. Appl., Article number: 2019:201 (2019).
• [13] J.E. Pecaric, F. Porschan and Y.L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press Inc., 1992.
• [14] 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.
• [15] 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, http://dx.doi.org/10.1155/2012/980438.