A counterexample to Khabibullin's conjecture for integral inequalities

Abstract

Khabibullin's conjecture for integral inequalities has two numeric parameters $n$ and $\alpha$ in its statement, $n$ being a positive integer and $\alpha$ being a positive real number. This conjecture is already proved in the case where $n>0$ and $0<\alpha\leq 1/2$. However, for $\alpha>1/2$ it is not always valid. In this paper a counterexample is constructed for $n=2$ and $\alpha=2$. Then Khabibullin's conjecture is reformulated in a way suitable for all $\alpha>0$.