Khabibullin's conjecture on integral inequalities

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In mathematics, Khabibullin's conjecture, named after B. N. Khabibullin, is related to Paley's problem[1] 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[edit]

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[edit]

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

Discussion[edit]

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[edit]

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[edit]

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

then

References[edit]

  1. ^ Khabibullin B.N. (1999). "Paley problem for plurisubharmonic functions of finite lower order". Sbornik: Mathematics. 190 (2): 309–321. 
  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/0502433free to read. 
  3. ^ Sharipov, R. A. (2010). "A Counterexample to Khabibullin's Conjecture for Integral Inequalities". Ufa Mathematical Journal,. 2 (4): 99–107. arXiv:1008.2738free to read.