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On the Growth of Entire Functions Bounded on Large Sets

Published online by Cambridge University Press:  20 November 2018

Lowell J. Hansen
Lowell J. Hansen*
Affiliation:
Wayne State University, Detroit, Michigan
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There have been many indications of a relationship between the rate of growth of an entire function and the “size” of the set, E(c), where the modulus of the function is larger than the constant, c. Theorems of this type include the classical theorem of Wiman on functions of bounded minimum modulus, the Phragmén-Lindelöf Theorem, the Denjoy-Carleman-Ahlfors Theorem, and its many subsequent improvements. These theorems can all be understood as quantitative versions of the statement that if ƒ is an entire function such that, for some c > 0, the set E(c) is ‘'small”, then the maximum modulus function M(R, f) is forced to grow rapidly with R.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 6 , 01 December 1977 , pp. 1287 - 1291
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Ahlfors, L. V., Untersuchungen zur Théorie der konformen Abbildung una der ganzen Funktionen, Acta Soc. Sci. Fenn. (2) A, 9 (1930), 1–40.Google Scholar
2. Al-Katifi, W., On the asymptotic values and paths of certain integral and meromorphie junctions, Proc. London Math. Soc. (3) 16 (1966), 599–634.Google Scholar
3. Hansen, L. J., Hardy classes and ranges of functions, Michigan Math. J. 17 (1970), 235–248.Google Scholar
4. Hansen, L. J. On the growth of entire functions which are large on small sets, Notices of Amer. Math. Soc, June 1976, (Abstract).Google Scholar
5. Hansen, L. J. and Hayman, W. K., On the growth of functions omitting large sets, Journal d'Analyse Math. 30 (1976), 208–214.Google Scholar
6. Hayman, W. K., Research problems in function theory, Proceedings of the Symposium on Complex Analysis, Canterbury 1973 (edited by J. Clunie and W. K. Hayman), London Math. Soc. Lecture Notes Series #12 (Cambridge Univ. Press, 1974), 143–180.Google Scholar
7. Kennedy, P. B., A class of integral functions bounded on certain curves, Proc. London Math. Soc. (3) 6 (1956), 518–547.Google Scholar
8. Titchmarsh, E. C., Theory of functions (2nd Edition, Oxford, 1939).Google Scholar
9. Tsuji, M., A theorem on the majoration of harmonic measure and its applications, Tôhoku Math. J. (2) 8(1951), 13–23.Google Scholar