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On the zeros of the derivatives of some entire functions of finite order

Published online by Cambridge University Press:  20 January 2009

Robert M. Gethner
Robert M. Gethner
Affiliation:
Northern Illinois University, Dekalb, Illinois 60115, USA
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Given a function f meromorphic in the plane, and complex numbers w and a, call w an a-point of f(k) if f(k)(w) = a. Denote by Λ(a,f) the set of z∈ℂ such that every neighborhood of z contains a-points of infinitely many of the functions f(k). Adapting the terminology of Pólya [16], who introduced the sets Λ(a,f) in [15],we call Λ(a,f) the final set of f with respect to a.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 28 , Issue 3 , October 1985 , pp. 381 - 407
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

REFERENCES

1.Abi-Khuzam, F. F., The asymptotic behavior of a Lindelõf function and its Taylor coefficients, J. of Math. Anal and Appl. 93 (1983), 495–526.CrossRefGoogle Scholar
2.Cartwright, M. L., Integral Functions (Cambridge University Press, 1962).Google Scholar
3.Debruijn, N. G., Asymptotic Methods in Analysis (North Holland Pub. Co., Amsterdam, 1958).Google Scholar
4.Edrei, A., Zeros of successive derivatives of entire functions of the form h(z) exp (— e z), Trans. Amer. Math. Soc. 259 (1980), 207–226.Google Scholar
5.Edrei, A., Zeros of successive derivatives of certain entire functions of infinite order, Functions, series, operators, Vol. I, II (Budapest, 1980), 449–484;Google Scholar
Colloq. Math. Soc. János Bolyai 35 (North-Holland, Amsterdam-New York, 1983).Google Scholar
6.Gethner, R. M., A Pólya “shire” theorem for entire functions, (Ph.D. thesis, University of Wisconsin, 1982).Google Scholar
7.Hardy, G. H., On the function P p(x), Quart. J. Math. Oxford Ser. (2), 37 (1906), 146–172.Google Scholar
8.Hayman, W. K., A generalisation of Stirling's formula, J. Reine Agnew. Math. 196 (1956), 67–95.CrossRefGoogle Scholar
9.Hayman, W. K., Meromorphic Functions (Clarendon Press, Oxford, 1964).Google Scholar
10.Heins, M. H., Selected Topics in the Theory of Functions of a Complex Variable (Holt, Rinehart, and Winston, New York, 1962).Google Scholar
11.Levin, B. JA., Distribution of Zeros of Entire Functions, Rev. Ed. (Amer. Math. Soc, Providence, 1980).Google Scholar
12.Lindelöf, E., Mémoire sur la théorie des fonctions entiéres de genre fini, Acta Soc. Sci. Fenn. 31 (1901), 1–77.Google Scholar
13.Mcleod, R. M., On the zeros of the derivatives of some entire functions (Ph.D. thesis, The Rice Institute, Houston, 1955).Google Scholar
14.Mcleod, R. M., On the zeros of the derivatives of some entire functions, Trans. Amer. Math. Soc. 91 (1959), 354–367.CrossRefGoogle Scholar
15.Pólya, G., Über die Nullstellen sukzessiver Derivierten, Math. Z. 12 (1922), 36–60.CrossRefGoogle Scholar
16.Pólya, G., On the zeros of the derivatives of a function and its analytic character, Bull. Amer. Math. Soc. 49 (1943), 178–191.CrossRefGoogle Scholar
17.Saks, S. and Zygmund, A., Analytic Functions (Elsevier Pub. Co., New York, 1971).Google Scholar
18.Titchmarsh, E. C., The Theory of Functions (Oxford University Press, 1939).Google Scholar