TOTALLY MONOTONE-FUNCTIONS WITH APPLICATIONS TO THE BERGMAN SPACE

Using a theorem of S. Bernstein [1] we prove a special case of the following maximum principle for the Bergman space conjectured by B. Korenblum [3]: There exists a number delta is-an-element-of (0, 1) such that 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 constant delta is greater than or equal to 1/square-root 3.