Inequalities for the Beta function

Inequalities for the Beta function

Let g(x) := (e/x)xΓ(x+ 1) and F(x,y) := g(x)g(y)/g(x + y) . Let D x,y be the k th differential in Taylor’s expansion of logF(x,y) . We prove that when (x,y) ∈ R+ one has (−1)k−1D x,y (X ,Y ) > 0 for every X ,Y ∈ R+ , and that when k is even the conclusion holds for every X ,Y ∈ R with (X ,Y ) = (0,0) . This implies that Taylor’s polynomials for logF provide upper and lower bounds for logF accor...

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 ≤...

On Some Inequalities for the Gamma Function

We present some elementary proofs of well-known inequalities for the gamma function and for the ratio of two gamma functions. The paper is purely expository and it is based on the talk that the first author gave during the memorial conference in Patras, 2012.

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 ...

Monotonicity and inequalities for the gamma function