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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Totally monotone functions with applications to the Bergman space


Authors: B. Korenblum, R. O’Neil, K. Richards and K. Zhu
Journal: Trans. Amer. Math. Soc. 337 (1993), 795-806
MSC: Primary 30D15; Secondary 26A48, 30C80, 30H05
MathSciNet review: 1118827
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Abstract: 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 \in (0,1)$ such that if $ f$ and $ g$ are analytic functions on the open unit disk $ {\mathbf{D}}$ with $ \vert f(z)\vert \leq \vert g(z)\vert$ on $ \delta \leq \vert z\vert < 1$ then $ {\left\Vert f \right\Vert _2} \leq {\left\Vert g \right\Vert _2}$, where $ {\left\Vert {} \right\Vert _2}$ is the $ {L^2}$ norm with respect to area measure on $ {\mathbf{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/\sqrt 3 $.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1118827-0
PII: S 0002-9947(1993)1118827-0
Article copyright: © Copyright 1993 American Mathematical Society