Korenblum maximum principle in mixed norm spaces

It is well known that the Korenblum maximum principle holds in Bergman spaces A(p) if and only if p >= 1. In this note, we improve this result by proving that the Korenblum maximum principle holds in mixed norm spaces H(p,q,alpha )when 1 <= p <= q < infinity and does not hold when 0 < q < 1. As an immediate consequence, we obtain that the Korenblum maximum principle holds in weighted Bergman spaces A(gamma)(p) if and only if p >= 1.