Khabibullin's conjecture on integral inequalities

Khabibullin's conjecture is a conjecture in mathematics related to Paley's problem[1] for plurisubharmonic functions and to various extremal problems in the theory of entire functions of several variables. The conjecture was named after its proposer, B. N. Khabibullin.

There are three versions of the conjecture, one in terms of logarithmically convex functions, one in terms of increasing functions, and one in terms of non-negative functions. The conjecture has implications in the study of complex functions and is related to Euler's Beta function. While the conjecture is known to hold for certain conditions, counterexamples have also been found.

The first statement in terms of logarithmically convex functions

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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

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The product in the right hand side of the inequality (2) is related to the Euler's Beta function  :

 

Discussion

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For each fixed   the function

 

turns the inequalities (1) and (2) 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  .[3]

The second statement in terms of increasing functions

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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

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

 

then

 

See also

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References

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  1. ^ Khabibullin B.N. (1999). "Paley problem for plurisubharmonic functions of finite lower order". Sbornik: Mathematics. 190 (2): 309–321. Bibcode:1999SbMat.190..309K. doi:10.1070/SM1999v190n02ABEH000387. S2CID 250806401.
  2. ^ 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.
  3. ^ Sharipov, R. A. (2010). "A Counterexample to Khabibullin's Conjecture for Integral Inequalities". Ufa Mathematical Journal. 2 (4): 99–107. arXiv:1008.2738. Bibcode:2010arXiv1008.2738S.