We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, star-shaped, and superadditive functions which are related to Γ and ψ. Euler’s gamma function Γ(x) =

Abstract. A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw’s double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of two gamma functions and originating from establishment of the best upper and lower bounds in Kershaw’s double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in R n−1 and R n respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized. 1.

The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

We prove that certain functions involving the gamma and q-gamma function are monotone. We also prove that (x m /(x)) (m+1) is completely monotonic. We conjecture that (x m / (m\Gamma1 (x)) (m) is completely monotonic for m 2, we prove it, with help from Maple, for 2 m 16. We give a very useful Maple proceedure to verify this for higher values of m. A stronger result is also formulated where we conjecture that the power series coefficients of a certain function are all positive. Running Title: Gamma Function Inequalities Mathematics Subject Classification. Primary 33B15. Secondary 26D07, 26D10. Key words and phrases. gamma function, digamma function, inequalities, complete monotonicity. 1. Introduction. Inequalities of functions involving gamma functions have been of interest since the 1950's when inequalities by Gautchi, Erber and Kershaw were established. For references and generalizations we refer the interested reader to [5], [13], [14], [15], [16], and to Alzer's p...

Abstract. In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel’s, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, Lazarević-Lupa¸s’s, Kershaw’s and Elezović-Giordano-Pečarić’s inequalities, claim, monotonic and convex properties. On the other hand, we introduce some related advances on the topic of bounding the ratio of two gamma functions in recent years. Contents

Abstract. In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some well-known results for the volume Ωn of the unit ball B n ⊂ R n,thesurface area ωn−1 of the unit sphere S n−1, and some related constants. 1.