Inequalities and asymptotic expansions for the gamma function

Inequalities for the Gamma Function

We prove the following two theorems: (i) Let Mr(a, b) be the rth power mean of a and b. The inequality Mr(Γ(x), Γ(1/x)) ≥ 1 holds for all x ∈ (0,∞) if and only if r ≥ 1/C − π2/(6C2), where C denotes Euler’s constant. This refines results established by W. Gautschi (1974) and the author (1997). (ii) The inequalities xα(x−1)−C < Γ(x) < xβ(x−1)−C (∗) are valid for all x ∈ (0, 1) if and only if α ≤...

Asymptotic Expansions of Gamma and Related Functions, Binomial Coefficients, Inequalities and Means

We give an overview of the use of asymptotic expansions of gamma and related functions — ratio of gamma functions, powers, digamma and polygamma functions. The aim is to introduce a general theory which can unify various particular formulas for factorial functions and binomial coefficients. The connection with inequalities for gamma function is established. Also, a systematic approach to asympt...

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.

Monotonicity and inequalities for the gamma function

Asymptotic Expansions of the Gamma Function Associated with the Windschitl and Smith Formulas

In this paper, we develop the Windschitl and Smith formulas for the gamma function to complete asymptotic expansions and provide explicit formulas for determining the coefficients of these asymptotic expansions. Furthermore, we establish new asymptotic expansions for the ratio of gamma functions Γ(x + 1)/Γ(x + 1 2 ).