Horst Alzer

On some inequalities for the gamma and psi functions

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?.

On some inequalities for the incomplete gamma function

Let p ¬= 1 be a positive real number. We determine all real numbers α = α(p) and β = β(p) such that the inequalities formula math. formula math. are valid for all x > 0. And, we determine all real numbers a and b such that - log(1 - e -ax ) ≤ √ x ∞ e-t/t ≤ - log(1 - e -bx ) hold for all > 0.

Some gamma function inequalities

A class of completely monotonic functions are presented involving the gamma function as well as the derivative of the psi function. As a consequence, new upper and lower bounds for the ratio F(x + 1)/F(x + s) are obtained and compared with related bounds given in part by J. D. Keckic and P. M. Vasic. Our results are further applied to obtain functions which are Laplace transforms of infinitely divisible probability measures.

Inequalities for the gamma and q-gamma functions

We present several sharp inequalities for the classical gamma and q-gamma functions. Some inequalities involve the psi function and its q-analogue. Our results improve, complement, and generalize some known (nonsharp) estimates.

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 ≤ 1−C and β ≥ (π2/6−C)/2, while (∗) holds for all x ∈ (1,∞) if and only if α ≤ (π2/6− C)/2 and β ≥ 1. These bounds for Γ(x) improve those given by G. D. Anderson an S.-L. Qiu (1997).

Monotonicity properties of the gamma function

Abstract Let G c ( x ) = log Γ ( x ) − x log x + x − 1 2 log ( 2 π ) + 1 2 ψ ( x + c ) ( x > 0 ; c ≥ 0 ) . We prove that G a is completely monotonic on ( 0 , ∞ ) if and only if a ≥ 1 / 3 . Also, − G b is completely monotonic on ( 0 , ∞ ) if and only if b = 0 . An application of this result reveals that the best possible nonnegative constants α , β in 2 π x x exp ( − x − 1 2 ψ ( x + α ) ) Γ ( x ) 2 π x x exp ( − x − 1 2 ψ ( x + β ) ) ( x > 0 ) are given by α = 1 / 3 and β = 0 .

A subadditive property of the gamma function

Abstract Let Δ=minx⩾0Γ(2x)/Γ(x) and α ∗ = log 2/ log Δ=−0.946850… . We prove that the function x↦(Γ(x))α is subadditive on (0,∞) if and only if α ∗ ⩽α⩽0 .

Sharp Bounds for the Ratio ofq - Gamma Functions

Let Γq (0 < q ≠ 1) be the q–gamma function and let s ∈ (0, 1) be a real number. We determine the largest number α = α(q, s) and the smallest number β = β(q, s) such that the inequalities hold for all positive real numbers x. Our result refines and extends recently published inequalities by Ismail and Muldoon (1994).

Inequalities for the constants of Landau and Lebesgue

The constants of Landau and Lebesgue are defined for all integers n ≥ 0 by Gn = Σk=0n1/16k )2 and Ln = 12/π ∫-ππ| sin((n+½)t)/sin(½t)| dt, respectively. We establish sharp inequalities for Gn and Ln/2 in terms of the logarithmic derivative of the gamma function. Further, we prove that the sequence (ΔGn) is completely monotonic, we provide best possible upper and lower bounds for the ratios (Gn-1 + Gn+1)/Gn and (L(n-1)/2+L(n+1)/2)/Ln/2, and we present sharp bounds for Ln/2/Gn and Ln/2 - Gn.

Inequalities for the polygamma functions

Let F_n(x; c) = (\Psi^{(n)}(x))^2 - c \Psi^{(n - 1)}(x) \Psi^{(n + 1)}(x) \quad (x > 0), where