Senlin Guo

Complete monotonicity of some functions involving polygamma functions

In the present paper, we establish necessary and sufficient conditions for the functions x^@a|@j^(^i^)(x+@b)| and @a|@j^(^i^)(x+@b)|-x|@j^(^i^+^1^)(x+@b)| respectively to be monotonic and completely monotonic on (0,~), where i@?N, @a>0 and @b>=0 are scalars, and @j^(^i^)(x) are polygamma functions.

Supplements to a class of logarithmically completely monotonic functions associated with the gamma function

Abstract In this article, the authors present a necessary condition, a sufficient condition and a necessary and sufficient condition for a class of functions associated with the gamma function to be logarithmically completely monotonic. As a consequence of these results, supplements to the recent investigation by Chen and Qi [C.-P. Chen, F. Qi, Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl. 321 (2006) 405–411] are provided and a new Keckic–Vasic type inequality is concluded.

Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic

In this article, necessary and sufficient conditions under which two classes of functions are, respectively, logarithmically completely monotonic and completely monotonic are presented. From these conditions, several two-sided inequalities for the ratio of two gamma functions are deduced.

A class of logarithmically completely monotonic functions

The main object of this work is to give some conditions for a class of functions to be logarithmically completely monotonic. Our result is shown to be an extension of a result which was proven in the recent literature on this subject.

A certain function class related to the class of logarithmically completely monotonic functions

In this article, we introduce and investigate the notion of strongly logarithmically completely monotonic functions. Among other results, we present some properties and relationships involving this function class and several closely-related function classes.

Some exact constants for the approximation of the quantity in the Wallis' formula

In this article, a sharp two-sided bounding inequality and some best constants for the approximation of the quantity associated with the Wallis’ formula are presented.MSC:41A44, 26D20, 33B15.

Some properties of a class of functions related to completely monotonic functions

In this article, we present several properties of the composition of functions which are related to the completely monotonic and absolutely monotonic functions. Relevant connections of the results derived in this article with those in earlier investigations are also indicated.

A class of logarithmically completely monotonic functions related to the gamma function with applications

In this article, a necessary and sufficient condition and a necessary condition are established for a class of functions involving the gamma function to be logarithmically completely monotonic on . As applications of the necessary and sufficient condition, several two-sided bounding inequalities for the psi and polygamma functions and the ratio of two gamma functions are derived.

A class of k-log-convex functions and their applications to some special functions

Abstract Let a and b be two real numbers and f be a positive and differentiable function on an interval I. The authors establish the i-log-convex or i-log-concave properties for i∈ℕ of the function [f(bx)] a /[f(ax)] b for ax∈I and bx∈I when the function u k−1[ln f(u)](k) for k∈ℕ is monotonic and apply these properties to deduce some known and new conclusions related to some special functions, such as the gamma function, Riemanns zeta function, complete elliptic integrals, exponential mean, and extended mean values.

A Function Involving Gamma Function and Having Logarithmically Absolute Convexity

In this paper, the logarithmically complete monotonicity, logarithmically absolute monotonicity and logarithmically absolute convexity of the function [Γ(1+tx)] s /[Γ(1+sx)] t for x, s, t∈ℝ such that 1+sx>0 and 1+tx>0 with s≠t are verified, some known results are generalized.