Sharp inequalities and complete monotonicity for the Wallis ratio

Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity

Abstract In the expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, the author looks back and analyses some inequalities, the complete monotonicity of several functions involving ratios of two gamma or q-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio...

Universal Monotonicity of Eigenvalue Moments and Sharp Lieb-thirring Inequalities

for some constant Lσ,d ≥ L σ,d and are widely discussed in the literature (see e.g. [3, 9, 11]). A longstanding question is when (1.4) holds with Lσ,d = L cl σ,d. The most general result is due to Laptev and Weidl [10] who proved that Lσ,d = L cl σ,d for all σ ≥ 32 and d ≥ 1. Their proof is based on a dimensional reduction of Schrödinger operators with operator valued potentials which allows th...

Estimates for the constants of Landau and Lebesgue via some inequalities for the Wallis ratio

We establish several new inequalities of the constants of Landau and Lebesgue

Monotonicity and inequalities for the gamma function

Inequalities and asymptotic expansions associated with the Wallis sequence

We present the asymptotic expansions of functions involving the ratio of gamma functions and provide formulas for determining the coefficients of the asymptotic expansions. As consequences, we obtain the asymptotic expansions of the Wallis sequence. Also, we establish inequalities for the Wallis sequence.