In 1991, Korenblum presented his conjecture on
Bergman spaces. He speculated that if $f(z)$ and $g(z)$ are two
holomorphic functions on the unit disc in the complex plane, then
there exists a number $0<c<1$ such that the condition
"$|f(z)|\geq|g(z)|$" could implies "$||f||_{2}\geq||g||_{2}$", where
$||.||_{2}$ is the Bergman norm. This maximum principle was
confirmed in 1999 by Hayman. It is not only an analogous property
with $H^{p}$ spaces but also inspires numerous research in other
function spaces. In this article we introduce the development of
this problem and the related research results from which the idea
originally came from Korenblum's maximum principle. In the end we
give some discussions about the circumstances when the functions are
constrained in the form of $\prod(z-a_{i})$ for $ -1 \leq a_{i} \leq
1$.

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