运用分析方法, 给出了AH凸函数的Hermite-Hadamard型不等式, 并把这个积分不等式应用到Gamma函数理论上, 得到了新的Kershaw型不等式. 最后, 还提出了两个公开问题.
This paper establishes Hermite-Hadamard type inequalities involving AH convex functions using analytical methods. As applications of the integral inequalities to Gamma function, a Kershaw-type of Gamma function is proved, which strengthens some previous results.
[1] 石焕南. 受控理论与解析不等式[M]. 哈尔滨: 哈尔滨工业大学出版社, 2012.
[2] 张小明, 褚玉明. 解析不等式新论[M]. 哈尔滨: 哈尔滨工业大学出版社, 2009.
[3] Niculescu C P, Persson L E. Convex functions and their applications [M]. New York: Springer-Verlag, 2006: 76-110.
[4] Anderson G D, Vamanamurthy M K, Vuorinen M. Generalized convexity and inequalities [J]. J Math Anal Appl, 2007, 335(3): 1294-1308.
[5] Hardy G H, Littlewood J E, Polya G. Inequalities [M]. 2nd ed. Cambridge: Cambridge University Press, 1952.
[6] Gautschi W. Some elementary inequalities relating to the Gamma and incomplete Gamma function [J]. J Math Phy, 1959, 38(1): 77-81.
[7] Qi F. A new lower bound in the second Kershaw’s double inequality [J]. Journal of Computational and Applied Mathematics, 2008, 214(2): 610-616.
[8] Qi F, Guo S, Chen S. A new upper bound in the second Kershaw’s double inequality and its generalizations [J]. J Comp Appl Math, 2008, 220(1/2): 111-118.
[9] Qi F. A class of logarithmically completely monotonic functions and application to the best bounds in the second Gautschi-Kershaw’s inequality [J]. Journal of Computational and Applied Mathematics, 2009, 224(2): 538-543.
[10] Chen C P, Batir N. Some inequalities and monotonicity properties associated with the Gamma and Psi functions [J]. Applied Mathematics and Computation, 2012, 218(17): 8217-8225.
[11] Koumandos S, Lamprecht M. Complete monotonicity and related properties of some special functions [J]. Mathematics of Computation, 2013, 82: 1097-1120.
[12] 匡继昌. 常用不等式[M]. 4版. 济南: 山东科学技术出版社, 2010.
