Better Results for Trigonometrically Convex Functions via Hölder-İşcan and Improved Power-Mean Inequalities

Year 2020, Volume: 3 Issue: 1, 38 - 43, 25.03.2020

Mahir Kadakal

https://doi.org/10.32323/ujma.571525

Cited By: 4

Abstract

In this paper, using Hölder-İşcan and improved power-mean integral inequalities and together with an integral identity, we obtain Hadamard type inequalities for functions whose second derivatives in absolute value at certain power are trigonometrically convex functions. In addition, we prove that our results give better approach than previous results.

Keywords

Convex function, trigonometrically convex function, Hölder-İşcan integral inequality, improved power-mean integral inequality

References

• [1] S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11(5) (1998), 91-95.
• [2] S.S. Dragomir, Refinements of the Hermite-Hadamard integral inequality for log-convex functions, Aust. Math. Soc. Gaz. 28(3) (2001), 129-134.
• [3] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Its Applications, RGMIA Monograph 2002.
• [4] S.S. Dragomir, J. Pecaric and LE.Persson, Some inequalities of Hadamard Type, Soochow Journal of Mathematics, 21(3) (2001), pp. 335-341.
• [5] J. Hadamard, Etude sur les proprietes des fonctions entieres en particulier d’une fonction consideree par Riemann, J. Math. Pures Appl. 58 (1893), 171-215.
• [6] İ. İşcan, New refinements for integral and sum forms of H¨older inequality, Journal of Inequalities and Applications, (2019) 2019:304
• [7] H. Kadakal, Hermite-Hadamard type inequalities for trigonometrically convex functions, Scientific Studies and Research. Series Mathematics and Informatics, 28(2) (2018), 19-28.
• [8] M. Kadakal, İ. İşcan, H. Kadakal and K. Bekar, On improvements of some integral inequalities, Researchgate, DOI: 10.13140/RG.2.2.15052.46724, Preprint, January 2019.
• [9] 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) (2017), 63-68.
• [10] S. Maden, H. Kadakal, M. Kadakal and İ. İşcan, Some new integral inequalities for n-times differentiable convex and concave functions, Journal of Nonlinear Sciences and Applications, 10(12) (2017), 6141-6148.
• [11] 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, F´ısicas y Naturales. Serie A. Matematicas, 113(2) (2019), 1059-1080
• [12] F. Usta, H. Budak and M. Z. Sarıkaya, Some New Chebyshev Type Inequalities Utilizing Generalized Fractional Integral Operators, AIMS Mathematics, 5(2) (2020) 1147-1161.
• [13] F. Usta, H. Budak, M. Z. Sarıkaya and E. Set, On generalization of trapezoid type inequalities for s-convex functions with generalized fractional integral operators, Filomat, 32(6) (2018), 2153–2171.
• [14] S. Varosanec, On h-convexity, J. Math. Anal. Appl. 326 (2007) 303-311.
• [15] G. Zabandan, A new refinement of the Hermite-Hadamard inequality for convex functions, J. Inequal. Pure Appl. Math. 10(2) (2009), Article ID 45.