The constant of embedding between two Bergman spaces of the upper-half plane

About a conjecture of Lieb-Solovej

We prove that for any holomorphic function on the upper-halfplane for and the constant of embedding between two bergman spaces of the upper-half plane is sharp whenever is an integer and we prove that it holds when we also prove that when restricted to powers of the bergman kernel, the conjecture holds.We next study the case where is close to hereafter and we transfer the conjecture to the unitdisc where we show that the conjecture holds when restricted to analytic monomials.Finally, we overview the bounds we obtain in our attempts to prove the conjecture.