# Integral inequalities of Simpsons type for ($\alpha,m$)-convex functions

Volume 9, Issue 12, pp 6364--6370 Publication Date: December 30, 2016
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### Authors

Ye Shuang - College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China. Yan Wang - College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China. Feng Qi - Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China. - Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China.

### Abstract

In this paper, we establish some integral inequalities of Simpson's type for ($\alpha,m$)-convex functions.

### Keywords

• Simpson's type integral inequality
• ($\alpha،m$)-convex function
• application mean inequality.

•  26A51
•  26D15
•  26E50
•  41A55

### References

• [1] R.-F. Bai, F. Qi, B.-Y. Xi, Hermite-Hadamard type inequalities for the m- and ($\alpha,m$)-logarithmically convex functions, Filomat, 27 (2013), 1--7

• [2] S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math., 33 (2002), 55--65

• [3] S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), 91--95

• [4] S. S. Dragomir, G. Toader, Some inequalities for m-convex functions, Studia Univ. Babeş-Bolyai Math., 38 (1993), 21--28

• [5] M. Klaričić Bakula, M. E. Özdemir, J. Pečarić, Hadamard type inequalities for m-convex and ($\alpha,m$)-convex functions, J. Inequal. Pure Appl. Math., 9 (2008), 12 pages

• [6] V. G. Miheşan, A generalization of the convexity, Seminar on Functional Equations, Approx. Convex, Cluj-Napoca, Romania (1993)

• [7] C. E. M. Pearce, J. Pečarić, Inequalities for differentiable mappings with application to special means and quadrature formula, Appl. Math. Lett., 13 (2000), 51--55

• [8] F. Qi, B.-Y. Xi, Some Hermite-Hadamard type inequalities for geometrically quasi-convex functions, Proc. Indian Acad. Sci. Math. Sci., 124 (2014), 333--342

• [9] F. Qi, T.-Y. Zhang, B.-Y. Xi, Hermite-Hadamard-type integral inequalities for functions whose first derivatives are convex, Ukrainian Math. J., 67 (2015), 625--640

• [10] G. Toader, Some generalizations of the convexity, in: Proceedings of the Colloquium on Approximation and Optimization (Cluj-Napoca, 1985), Univ. Cluj-Napoca, Cluj, 1985 (1985), 329--338

• [11] B.-Y. Xi, F. Qi, Hermite-Hadamard type inequalities for geometrically r-convex functions, Studia Sci. Math. Hungar., 51 (2014), 530--546

• [12] B.-Y. Xi, F. Qi, Inequalities of Hermite-Hadamard type for extended s-convex functions and applications to means, J. Nonlinear Convex Anal., 16 (2015), 873--890

• [13] B.-Y. Xi, T.-Y. Zhang, F. Qi, Some inequalities of Hermite-Hadamard type for m-harmonic-arithmetically convex functions, ScienceAsia, 41 (2015), 357--361