Extremal problems in Bergman spaces

Date
2016-05
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Abstract

In this dissertation, we shall study non-linear extremal problems in Bergman space A2(D). We show the existence of the solutions and that the extremal functions are bounded. Further, we shall discuss special cases for polynomials, investigate the properties of the solution and provide a bound for the solution. These problems are equivalent formulations of B. Korenblum's conjecture, also known as Korenblum's Maximum Principle: for f, g∈A2(D), there is a constant c, 0<c<1 such that if |f(z)|≤|g(z)| for all z such that c<|z|<1, then ∥f∥2≤∥g∥2. The existence of such c was proved by W. Hayman but the exact value of the best possible value of c, denoted by κ, remains unknown for space A2(D).

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Keywords
Korenblum's maximum principle, Extremal problems
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