Let g(x):= (e/x)xΓ(x+1) and F(x,y):= g(x)g(y)/g(x+y). Let Dx,y (k) be the k th differential in Taylor's expansion of logF(x,y) . We prove that when (x,y) ∈ R+ 2 one has (-1)k-1Dx,y (k) (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 according to the parity of their degree. The formula connecting the Beta function to the Gamma function shows that the bounds for F are actually bounds for Beta.

(2015). Inequalities for the beta function [journal article - articolo]. In MATHEMATICAL INEQUALITIES & APPLICATIONS. Retrieved from http://hdl.handle.net/10446/57993

Inequalities for the beta function

Grenié, Loic Andre Henri;
2015-01-01

Abstract

Let g(x):= (e/x)xΓ(x+1) and F(x,y):= g(x)g(y)/g(x+y). Let Dx,y (k) be the k th differential in Taylor's expansion of logF(x,y) . We prove that when (x,y) ∈ R+ 2 one has (-1)k-1Dx,y (k) (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 according to the parity of their degree. The formula connecting the Beta function to the Gamma function shows that the bounds for F are actually bounds for Beta.
articolo
2015
Grenie', Loïc André Henri; Molteni, Giuseppe
(2015). Inequalities for the beta function [journal article - articolo]. In MATHEMATICAL INEQUALITIES & APPLICATIONS. Retrieved from http://hdl.handle.net/10446/57993
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10446/57993
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