Sharp 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...

Sharp inequalities for tangent function with applications

In the article, we present new bounds for the function [Formula: see text] on the interval [Formula: see text] and find sharp estimations for the Sine integral and the Catalan constant based on a new monotonicity criterion for the quotient of power series, which refine the Redheffer and Becker-Stark type inequalities for tangent function.

Weighted Norm Inequalities for the Local Sharp Maximal Function

Sharp Inequalities for the Psi Function and Harmonic Numbers

In this paper, two sharp inequalities for bounding the psi function ψ and the harmonic numbers Hn are established respectively, some results in [I. Muqattash and M. Yahdi, Infinite family of approximations of the Digamma function, Math. Comput. Modelling 43 (2006), 1329–1336.] are improved, and some remarks are given.

Sharp Weighted Inequalities for the Vector–valued Maximal Function

We prove in this paper some sharp weighted inequalities for the vector–valued maximal function Mq of Fefferman and Stein defined by Mqf(x) = ( ∞ ∑ i=1 (Mfi(x)) q )1/q , where M is the Hardy–Littlewood maximal function. As a consequence we derive the main result establishing that in the range 1 < q < p < ∞ there exists a constant C such that ∫ Rn Mqf(x) p w(x)dx ≤ C ∫ Rn |f(x)|qM [ p q ]+1 w(x)d...