Mean growth of inner functions
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- by Domingo A. Herrero
- Proc. Amer. Math. Soc. 37 (1973), 175-180
- DOI: https://doi.org/10.1090/S0002-9939-1973-0308403-4
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Abstract:
Let A be a closed subset of the closed unit disc. It is shown that there exists a “universal growth function” $\varphi (r,A)$ such that $1 - \smallint _0^{2\pi }|q(r{e^{ix}}){|^2}dx/2\pi = O(\varphi (r,A))$ for all inner functions $q(z)$ whose zeroes lie in $A \cap \{ |z| < 1\}$ and whose singularities in the unit circle lie on $A \cap \{ |z| = 1\}$, if and only if the Lebesgue measure of $A \cap \{ |z| = 1\}$ is zero.References
- Domingo Antonio Herrero, Inner function operators, Notas Ci. Ser. M Mat. 9 (1971), no. 1, 7–41. MR 634685
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962. MR 133008
- J. E. Littlewood, Lectures on the Theory of Functions, Oxford University Press, 1944. MR 12121
- D. J. Newman and Harold S. Shapiro, The Taylor coefficients of inner functions, Michigan Math. J. 9 (1962), 249–255. MR 148874
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 37 (1973), 175-180
- MSC: Primary 30A76
- DOI: https://doi.org/10.1090/S0002-9939-1973-0308403-4
- MathSciNet review: 0308403