Khabibullin's conjecture on integral inequalities


In mathematics, Khabibullin's conjecture, named after B. N. Khabibullin, is related to Paley's problem for plurisubharmonic functions and to various extremal problems in the theory of entire functions of several variables.

The first statement in terms of logarithmically convex functions

Khabibullin's conjecture. Let be a non-negative increasing function on the half-line such that. Assume that is a convex function of. Let,, and. If
then
This statement of the Khabibullin's conjecture completes his survey.

Relation to Euler's Beta function

Note that the product in the right hand side of the inequality is related to the Euler's Beta function :

Discussion

For each fixed the function
turns the inequalities and to equalities.
The Khabibullin's conjecture is valid for without the assumption of convexity of. Meanwhile, one can show that this conjecture is not valid without some convexity conditions for. In 2010, R. A. Sharipov showed that the conjecture fails in the case and for.

The second statement in terms of increasing functions

Khabibullin's conjecture. Let be a non-negative increasing function on the half-line and. If
then

The third statement in terms of non-negative functions

Khabibullin's conjecture. Let be a non-negative continuous function on the half-line and. If
then