Monotonicity and sharp inequalities related to gamma function

Monotonicity and inequalities for the gamma function

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...

Supplements to Known Monotonicity Results and Inequalities for the Gamma and Incomplete Gamma Functions

In particular they proved that for x > 0 and α = 0 the function [Γ(1 + 1/x)]x decreases with x, while when α=1 the function x[Γ(1+1/x)]x increases.Moreover they also showed that the values α= 0 and α= 1, in the properties mentioned above, cannot be improved if x ∈ (0,+∞). In this paper we continue the investigation on the monotonicity properties for the gamma function proving, in Section 2, the...

Monotonicity and Convexity for the Gamma Function

Let a and b be given real numbers with 0 ≤ a < b < a + 1. Then the function θa,b(x) = [Γ(x + b)/Γ(x + a)]1/(b−a) − x is strictly convex and decreasing on (−a,∞) with θa,b(∞) = a+b−1 2 and θa,b(−a) = a, where Γ denotes the Euler’s gamma function.

Integral representations and complete monotonicity related to the remainder of Burnside's formula for the gamma function

In the paper, the authors establish integral representations of some functions related to the remainder of Burnside’s formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions. These results extend and generalize some known conclusions. 1. Motivation and main results A function f is said to be completely monotonic on an interval I if f ha...