On the Hilbert matrix norm on positively indexed weighted Bergman spaces.

The Hilbert matrix is bounded on weighted Bergman spaces A α p if and only if 1 < α + 2 < p with the conjectured norm π / sin (α + 2) π p . In the case of positively indexed weighted Bergman spaces, that is, in the case when α > 0 , the conjecture was confirmed for α 0 ≤ p , where α 0 is a unique zero of the function Φ α (x) = 2 x 2 - 4 (α + 2) + 1 x + 2 α + 2 x + α + 2 on the interval α + 2 , 2 (α + 2) . In this note we prove, that if α > 0 , then the conjecture is valid for all 3 α 4 + 2 + 3 α 4 + 2 2 - α + 2 2 ≤ p. This improves the best previously known result for all α > 1 2 . [ABSTRACT FROM AUTHOR]