B. Korenblum et al., TOTALLY MONOTONE-FUNCTIONS WITH APPLICATIONS TO THE BERGMAN SPACE, Transactions of the American Mathematical Society, 337(2), 1993, pp. 795-806
Using a theorem of S. Bernstein [1] we prove a special case of the fol
lowing maximum principle for the Bergman space conjectured by B. Koren
blum [3]: There exists a number delta is-an-element-of (0, 1) such tha
t if f and g are analytic functions on the open unit disk D with \f(z)
\ less-than-or-equal-to \g(z)\ on delta less-than-or-equal-to \z\ < 1
then \\f\\2 less-than-or-equal-to \\g\\2, where \\ \\2 is the L2 norm
with respect to area measure on D. We prove the above conjecture when
either f or g is a monomial; in this case we show that the optimal con
stant delta is greater than or equal to 1/square-root 3.