An extension of an inequality for ratios of gamma functions

作者：Guo Bai Ni^*; Qi Feng

来源：Journal of Approximation Theory, 2011, 163(9): 1208-1216.

DOI：10.1016/j.jat.2011.04.003

• Extension
• Inequality
• Ratio
• Gamma function

In this paper, we prove that for x + y > 0 and y + 1 > 0 the inequality
[Gamma(x + y+1)/Gamma(y + 1)](1/x)/[Gamma(x + y +2)/Gamma(y + 1)](1/(x+1)) < (x + y/x + y + 1)(1/2)
is valid if x > 1 and reversed if x < 1 and that the power 1/2 is the best possible, where Gamma(x) is the Euler gamma function. This extends the result of [Y. Yu, An inequality for ratios of gamma functions, J. Math. Anal. Appl. 352 (2) (2009) 967-970] and resolves an open problem posed in [B.-N. Guo, F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math. 7 (2) (2003) 239-247].

• 出版日期2011-9
• 单位河南理工大学