Further Inequalities Associated with the Classical Gamma Function

Further Inequalities for the Gamma Function 599

For X > 0 and k > 0 we present a method which permits us to obtain inequalities of the type (it + a)x_l < T(k + \)/T(k + 1) < (k + ß)x'x, with the usual notation for the gamma function, where a and ß are independent of k. Some examples are also given which improve well-known inequalities. Finally, we are also able to show in some cases that the values a and ß in the inequalities that we obtain ...

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 ≤...

Inequalities for Gamma Function Ratios

Write R(x, y) = Γ(x + y) Γ(x). Inequalities for this ratio have interesting applications, and have been considered by a number of writers over a long period. In a Monthly article [7], Wendel showed that x(x + y) y−1 ≤ R(x, y) ≤ x y for 0 ≤ y ≤ 1. (1) Wendel's method was an ingenious application of Hölder's inequality to the integral definition of the gamma function. Note that both inequalities ...

Monotonicity and inequalities for the gamma function

Some inequalities for the gamma function

In this paper are established some inequalities involving the Euler gamma function. We use the ideas and methods that were used by J. Sándor in his paper [2].