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...

. The Kneser graph K(n; k) has as vertices all the k-subsets of a fixed n-set and has as edges the pairs fA; Bg of vertices such that A and B are disjoint. It is known that these graphs are Hamiltonian if \Gamma n\Gamma1 k\Gamma1 \Delta \Gamma n\Gammak k \Delta for n 2k + 1. We determine asymptotically for fixed k the minimum value n = e(k) for which this inequality holds. In addition we give an asymptotic formula for the solution of k\Gamma(n)\Gamma(n \Gamma 2k + 1) = \Gamma 2 (n \Gamma k + 1) for n 2k + 1, as k !1, when n and k are not restricted to take integer values. We also show that for all prime powers q and n 2k, k 1, the q-analogues K q (n; k) are Hamiltonian by consideration of the analogous inequality for q-binomial coefficients. 1. Introduction. The Kneser graph K(n; k) is the graph whose vertices are the k-subsets of the set [n] = f1; 2; : : : ; ng and whose edges are the pairs fA; Bg of k-subsets such that A and B are disjoint [15]. If n = 2k + 1, K(n;...

This paper is mainly a survey of published results. We recall the definition of positive definite and (conditionally) negative definite functions on abelian semigroups with involution, and we consider three main examples: Rk, [0, ∞ [ k, N0–the first with the inverse involution and the two others with the identical involution. Schoenberg’s theorem explains the possibility of constructing rotation invariant positive definite and conditionally negative definite functions on euclidean spaces via completely monotonic functions and Bernstein functions. It is therefore important to be able to decide complete monotonicity of a given function. We combine complete monotonicity with complex analysis via the relation to Stieltjes functions and Pick functions and we give a survey of the many interesting relations between these classes of functions and completely monotonic functions, logarithmically completely monotonic functions and Bernstein functions. In Section 6 it is proved that log x − Ψ(x) and Ψ ′ (x) are logarithmically completely monotonic (where Ψ(x) = Γ ′ (x)/Γ(x)), and these results are new as far as we know. We end with a list of completely monotonic functions related to the Gamma function.

Abstract. In the article, a notion “logarithmically absolutely monotonic function” is introduced, an inclusion that a logarithmically absolutely monotonic function is also absolutely monotonic is revealed, the logarithmically complete monotonicity and the logarithmically absolute monotonicity of the function α x+β 1+ are proved, where α and β are given real parameters, a new proof x for the inclusion that a logarithmically completely monotonic function is also completely monotonic is given, and an open problem is posed.

Abstract. We prove some properties of completely monotonic functions and apply them to obtain results on gamma and q-gamma functions. 1.

k (x) = O max 1, (α 2 + β 2) 1/4}) max x∈[−1,1] (1 − x)α+ 1 2 (1 + x) β+1 2 [Erdélyi et al.,Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614]. Here we will confirm this conjecture in the ultraspherical case α = β ≥ 1+√2, even in a stronger 4 form by giving very explicit upper bounds. We also show that δ2 − x2 2 α (1 − x) ( P (α,α)