Data di Pubblicazione:
2015
Citazione:
Inequalities for the beta function / L. Grenié, G. Molteni. - In: MATHEMATICAL INEQUALITIES & APPLICATIONS. - ISSN 1331-4343. - 18:4(2015), pp. 1427-1442.
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.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Completely monotone function; Euler Beta function; Explicit bounds; Mathematics (all); Applied Mathematics
Elenco autori:
L. Grenié, G. Molteni
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