Totally monotone functions with applications to the Bergman space
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- by B. Korenblum, R. O’Neil, K. Richards and K. Zhu PDF
- Trans. Amer. Math. Soc. 337 (1993), 795-806 Request permission
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 $|f(z)| \leq |g(z)|$ on $\delta \leq |z| < 1$ then ${\left \| f \right \|_2} \leq {\left \| g \right \|_2}$, where ${\left \| {} \right \|_2}$ 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 $1/\sqrt 3$.References
- Serge Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 (1929), no. 1, 1–66 (French). MR 1555269, DOI 10.1007/BF02547400 L. Carleson, Private communication with the first-named author.
- Boris Korenblum, A maximum principle for the Bergman space, Publ. Mat. 35 (1991), no. 2, 479–486. MR 1201570, DOI 10.5565/PUBLMAT_{3}5291_{1}2
- B. I. Korenblyum, On two theorems from the theory of absolutely monotonic functions, Uspehi Matem. Nauk (N.S.) 6 (1951), no. 4(44), 172–175. MR 0043852
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 337 (1993), 795-806
- MSC: Primary 30D15; Secondary 26A48, 30C80, 30H05
- DOI: https://doi.org/10.1090/S0002-9947-1993-1118827-0
- MathSciNet review: 1118827