Abstract. In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two gamma functions to be logarithmically completely monotonic, and some double inequalities are deduced for bounding divided differences of polygamma functions. 1.

Abstract. In this paper, we prove that for x + y> 0 and y + 1> 0 the inequality [Γ(x + y + 1)/Γ(y + 1)] 1/x s x + y [Γ(x + y + 2)/Γ(y + 1)] 1/(x+1) x + y + 1 is valid if x> 1 and reversed if x < 1, where Γ(x) is the Euler gamma function. This completely extends the result in [Y. Yu, An inequality for ratios of gamma

Abstract. The main aim of this paper is to prove that the double inequality