Real zeros of derivatives of meromorphic functions and solutions of second order differential equations

Hellerstein, Simon; Shen, Li-Chien; Williamson, Jack

doi:10.1090/S0002-9947-1984-0752502-1

Real zeros of derivatives of meromorphic functions and solutions of second order differential equations
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Trans. Amer. Math. Soc. 285 (1984), 759-776 Request permission

Abstract:

We classify all functions $F$ meromorphic in the plane with only real zeros and real poles which satisfy the additional conditions that $F’$ has no zeros and $F''$ only real zeros. We apply this classification, in combination with some earlier results, to the study of the reality of zeros of solutions of the equation $w'' + H(z)w = 0,H$ entire.

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
W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
Simon Hellerstein and Jack Williamson, Derivatives of entire functions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227–249. MR 435393, DOI 10.1090/S0002-9947-1977-0435393-4
Simon Hellerstein and Jack Williamson, Derivatives of entire functions and a question of Pólya. II, Trans. Amer. Math. Soc. 234 (1977), no. 2, 497–503. MR 481004, DOI 10.1090/S0002-9947-1977-0481004-1
Simon Hellerstein and Jack Williamson, The zeros of the second derivative of the reciprocal of an entire function, Trans. Amer. Math. Soc. 263 (1981), no. 2, 501–513. MR 594422, DOI 10.1090/S0002-9947-1981-0594422-9
Simon Hellerstein, Li Chien Shen, and Jack Williamson, Reality of the zeros of an entire function and its derivatives, Trans. Amer. Math. Soc. 275 (1983), no. 1, 319–331. MR 678353, DOI 10.1090/S0002-9947-1983-0678353-3
Einar Hille, Ordinary differential equations in the complex domain, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. MR 0499382
B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1964. MR 0156975

The dependence of the growth of an entire function on the distribution of the zeros of its derivatives

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F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
G. Polya, On the zeros of the derivatives of a function and its analytic character, Bull. Amer. Math. Soc. 49 (1943), 178–191. MR 7781, DOI 10.1090/S0002-9904-1943-07853-6
John Rossi, The reciprocal of an entire function of infinite order and the distribution of the zeros of its second derivative, Trans. Amer. Math. Soc. 270 (1982), no. 2, 667–683. MR 645337, DOI 10.1090/S0002-9947-1982-0645337-X

On the dependence of the growth of an entire function on the distribution of the zeros of its derivatives and a conjecture of Pólya

The theory of functions

Hans Wittich, Neuere Untersuchungen über eindeutige analytische Funktionen, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 8, Springer-Verlag, Berlin-New York, 1968 (German). Zweite, korrigierte Auflage. MR 0244490