Some monotonicity and limit results for the regularised incomplete gamma function

Some monotonicity and limit results for the regularised incomplete gamma function

Letting P (u, x) denote the regularised incomplete gamma function, it is shown that for each α ≥ 0, P (x, x+ α) decreases as x increases on the positive real semiaxis, and P (x, x + α) converges to 1/2 as x tends to infinity. The statistical significance of these results is explored.

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.

Monotonicity and inequalities for the gamma function