Inequalities for the beta function of n variables

Abstract

We present various inequalities for Euler's beta function of n variables. One of our theorems states that the inequalitieshold for all j c, -> 1 (i = 1,... , n; n > 3) with the best possible constants a n = 0 and b n = 1 -l/(n -1)!. This extends a recently published result of Dragomir et al., who investigated (*) for the special case n = 2.

Algebra generated by Toeplitz operators with $${\mathbb {T}}$$-invariant symbols

“…Note that, by [8,Corollary 3.5], the algebra T is generated, in fact, just by Toeplitz operators T a , with bounded measurable radial symbols a(r), and the single Toeplitz operator T / , where / :¼ / 1 ¼ n 1 n 2 . By [8,Section 3.3], the spectrum of T / coincides with Dð0; 1 2 Þ, the closed disk of radius 1 2 centered at the origin. The generating Toeplitz operators act on invariant subspace H p as follows: and where the closed connected set M 1 consists of all multiplicative functionals valued 0 on all sequences c 2 SOðZ þ Þ with lim p!1 cðpÞ ¼ 0.…”

“…The irreducible representations T , induced by the representations of the quotient algebra T =T \ K (which, by [8,Theorem 6.7], is isomorphic to CðM 1 Â Dð0; 1 2 ÞÞ), are all one-dimensional, parameterized by points ðl; fÞ 2 M 1 Â Dð0; 1 2 Þ, and defined on generators of the algebra T as follows: In this case, we have:…”

On some generalizations of the beta function in several variables

“…The extension of the beta function from two variables to n variables was introduced in the literature [1,2,6]. For n ≥ 3 integer, let E n−1 be the standard (n − 1)-simplex of R n−1 defined by…”

A Mean Value Inequality for Euler's Beta Function

“…In the recent past, several research papers have appeared providing various inequalities for the beta function and its relatives. We refer to [1][2][3][4][5], [7,9,10,11] and the references cited therein. For example, for all x, y > 0 we have that B(x, y) separates the geometric and arithmetic means of B(x, x) and B(y, y), that is,…”