About a conjecture regarding plurisubharmonic functions

Abstract

In this work we present Khabibullin's conjecture in its different equivalent forms. Applying the concept of the integral operator, we transform the original conjecture into a new form which proves to be helpful in studying it by means of the Laplace transform. Using Laplace transform of integral inequalities, we are able to show the uniqueness of a solution that satisfies both inequalities with identity. Furthermore we provide a new proof of Khabibullin's theorem by methods of the Laplace transform and contour integration from complex analysis. However, this method of transform fails to prove the conjecture and a brief reasoning is provided.