On Khabibullin's conjecture about pair of integral inequalities

Abstract

Khabibullin's conjecture is a statement about a pair of integral inequalities, where one inequality implies the other. They depend on two parameters $n\geqslant 2$, $n\in \mathbb{N}$, and $\alpha\in\mathbb{R}_+$. These inequalities were originally introduced by Khabibullin [6] in his survey regarding Paley problem in $\mathbb{C}_n$ and related topics about meromorphic functions. It is possible to express the inequalities in three equivalent forms. The first statement is in terms of logarithmically convex functions, the second statement is in terms of increasing functions, and the third statement is in terms of non-negative functions. In this paper we work solely with the second variant of the hypothesis. It is well established that the conjecture is true whenever $0\leqslant \alpha\leqslant 1/2$ for all $n$. Several proofs exist in the literature among which one is given by the author [2] and it relates the integral inequalities with the general theory of Laplace transform. But it was not known if the statement was true when $\alpha>1/2$ until Sharipov [8] showed that the conjecture fails when $\alpha=2$, $n=2$. However the question of whether this conjecture holds for at least some $n\geqslant 2$ and $\alpha>1/2$ remained an open problem. In this paper we aim to solve this question. Motivated by Sharipov's approach, we develop a method of constructing a counterexample for the more general case $n\geqslant 2$ and $\alpha>1/2$. By an explicit counterexample we show that Khabibullin's conjecture does not hold in general.