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
(1)
then
(2)
This statement of the Khabibullin's conjecture completes his survey.[2]
Relation to Euler's Beta function
Note that the product in the right hand side of the inequality (2) is related to the Euler's Beta function:
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
References
^Khabibullin B.N. (1999). "Paley problem for plurisubharmonic functions of finite lower order". Sbornik: Mathematics. 190 (2): 309–321.
^Khabibullin BN (2002). "The representation of a meromorphic function as the quotient of entire functions and Paley problem in : a survey of some results". Mat. Fizika, analiz, geometria. 9 (2): 146–167. arXiv:math.CV/0502433.