The best bounds in Gautschi's inequality

Abstract

Abstract. Different approach to both Gautschi's inequalities (1) and (2) is given. This results in obtaining the best upper bound in (1) and the best lower bound in (2). The main result is the proof of the convexity of the function [Γ(x+t)/Γ(x+s)] 1/(t−s) for |t−s| < 1 . Several new very simple inequalities for digamma function, like ψ (x) < exp(−ψ (x)) or ψ (x + 1) < log(x + e −γ ) are also proved.Mathematics subject classification (1991): 33B15, 26D07.

Smale’s 17th problem: Average polynomial time to compute affine and projective solutions

“…for example [EGP00]). We also use that be the number of arithmetic operations required to perform i loops of the algorithm.…”

Remarks on a paper by Chao-Ping Chen and Feng Qi

“…The required inequality (1.3) follows from the fact that the sequence Q n (α) is strictly decreasing. This, in turn, is an immediate consequence of a result of N. Elezović, C. Giordano and J. Pecarić, [8] who showed that the function This completes the proof of (1.3).…”

Three classes of logarithmically completely monotonic functions involving gamma and psi functions

“…In order to establish the best bounds in Kershaw's inequality (1), among other things, the papers [3,5,13,18] established the following monotonicity and convexity property of z s,t (x): The function z s,t (x) is either convex and decreasing for |t − s| < 1 or concave and increasing for |t − s| > 1. This result was further generalized in the papers [10,11].…”