In mathematics, Khabibullin's conjecture, named after B. N. Khabibullin, is related to Paley

Raymond Paley

Raymond Edward Alan Christopher Paley was an English mathematician. Paley was born in Bournemouth, England. He was educated at Eton. From there he entered Trinity College, Cambridge where he showed himself the most brilliant student among a remarkable collection of fellow undergraduates...

's problem for plurisubharmonic functions and to various extremal problems in the theory of entire function

Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

s of several variables.

The first statement in terms of logarithmically convex functions

Khabibullin's conjecture (version 1, 1992). 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 . Nowadays it is even unknown if the conjecture is true for and for at least one
.

The second statement in terms of increasing functions

Khabibullin's conjecture (version 2). 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 (version 3). Let be a non-negative continuous function on the half-line and . If

then