Entire functions with almost radially distributed values

Kimura, Shigeru

doi:10.1090/S0002-9939-1978-0486525-X

Entire functions with almost radially distributed values
HTML articles powered by AMS MathViewer

by Shigeru Kimura PDF

Proc. Amer. Math. Soc. 71 (1978), 73-78 Request permission

Abstract:

Let $f(z)$ be an entire function of finite lower order. Assume that there exist a positive number h and an unbounded sequence $\{ {w_n}\} _{n = 1}^\infty$ such that all roots of the equations $f(z) = {w_n}(n = 1,2, \ldots )$ lie in $\{ z;|\operatorname {Im} z| < h\}$. Then $f(z)$ is a polynomial of degree not greater than two. The hypothesis of the finiteness of lower order of $f(z)$ cannot be removed.

References

Albert Edrei, Meromorphic functions with three radially distributed values, Trans. Amer. Math. Soc. 78 (1955), 276–293. MR 67982, DOI 10.1090/S0002-9947-1955-0067982-9
Albert Edrei, The deficiencies of meromorphic functions of finite lower order, Duke Math. J. 31 (1964), 1–21. MR 158079
Albert Edrei and Wolfgang H. J. Fuchs, Bounds for the number of deficient values of certain classes of meromorphic functions, Proc. London Math. Soc. (3) 12 (1962), 315–344. MR 138765, DOI 10.1112/plms/s3-12.1.315
W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
Maurice Heins, On a notion of convexity connected with a method of Carleman, J. Analyse Math. 7 (1959), 53–77. MR 114000, DOI 10.1007/BF02787680
D. F. Shea, On the Valiron deficiencies of meromorphic functions of finite order, Trans. Amer. Math. Soc. 124 (1966), 201–227. MR 200458, DOI 10.1090/S0002-9947-1966-0200458-1