Gamma函数的单调性与凹凸性

Monotonicity and convexity properties for the gamma function

设α∈R,β≥0,x∈(0,∞),f_(α,β,±1)(x)≡(e^(x)Г(x+β)/x^(x+β-α))^(±1),获得了函数f_(α,β,+1)的几何凹凸性,从而推广了f_(0,0,+1)的相应已知结果;将函数f_(α,β,±1)中的指数函数替换为双曲函数,通过研究其单调性与对数凹凸性,获得了关于gamma函数的一些不等式,从而改进了gamma函数相应的已知不等式。

Forα∈R,β≥0 and x∈(0,∞),let f_(α,β,±1)(x)≡(e^(x)Г(x+β)/x^(x+β-α))^(±1),In this paper,the authors study the geometric convexity of the function f_(α,β,+1),which is a generalization of the corresponding known result of f_(0,0,+1).They also study the monotonicity and logarithmic convexity properties of the gamma function combined with the hyperbolic functions instead of the exponential function in f_(α,β,±1).Moreover,the inequalities derived from these properties improve some related known results of the gamma function.