MONOTONICITY RESULTS AND ASSOCIATED INEQUALITIES FOR K-GAMMA FUNCTION

Monotonicity and inequalities for the gamma function

Monotonicity Results for the Gamma Function

The function [Γ(x+1)] 1/x x+1 is strictly decreasing on [1,∞), the function [Γ(x+1)]1/x √ x is strictly increasing on [2,∞), and the function [Γ(x+1)] 1/x √ x+1 is strictly increasing on [1,∞), respectively. From these, some inequalities, for example, the Minc-Sathre inequality, are deduced, and two open problems posed by the second author are solved partially.

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.

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 ≤...