Survey Article: Bellman function method and sharp inequalities for martingales

Sharp Maximal Inequalities for Conditionally Symmetric Martingales and Brownian Motion

Let B = {Bt)t>0 be a standard Brownian motion. For c > 0, k > 0 , let T(c, k) = inî{t > 0: maxs<í Bs cBt > k} , T"(c,k)= inf{r>0: max^, \BS\ c\B,\ > k} . We show that for c > 0 and k > 0, both T(c, k) and T*{c, k) axe finite almost everywhere. Moreover, T(c, k) and T*(c, k) e L if and only if c < pKp 1) for p > 1 , and for all c > 0 when p < 1 . These results have analogues for simple random wa...

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.

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 Isoperimetric Inequalities via the Abp Method

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of R. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarka...

Weighted Norm Inequalities for the Local Sharp Maximal Function