Korenblum-Type Extremal Problems in Bergman Spaces

We shall study non-linear extremal problems in Bergman space ${\mathcal{A}}^2(\mathbb{D})$. We show the existence of the solution and that the extremal functions are bounded. Further, we shall discuss special cases for polynomials, investigate the properties of the solution and provide a bound for the solution. This problem is an equivalent formulation of B. Korenblum’s conjecture, also known as Korenblum’s Maximum Principle: for $f$, $g\in $\mathcal{A}$^2($\mathbb{D}$)$,thereisaconstant$c$,$0<c<1$suchthatif$|f(z)|\leq |g(z)|$forall$z$suchthat$c<|z|<1$,then$|f|_2\leq |g|_2$\mathrm{.}Theexistenceofsuch$c$ was proved by W. Hayman but the exact value of the best possible value of $c$, denoted by $\kappa$, remains unknown.