Derivatives of meromorphic functions of finite order
Authors:
Werner P. Kohs and Jack Williamson
Journal:
Trans. Amer. Math. Soc. 306 (1988), 765772
MSC:
Primary 30D35
MathSciNet review:
933316
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Abstract: Let be a nonentire, meromorphic function of finite order with only real zeros and real poles such that has no zeros. We classify all such real and all such strictly nonreal whose poles are of bounded multiplicities. We also give examples of such which are strictly nonreal and whose poles are of unbounded multiplicities.
 [1]
Lars
V. Ahlfors, Complex analysis, 3rd ed., McGrawHill Book Co.,
New York, 1978. An introduction to the theory of analytic functions of one
complex variable; International Series in Pure and Applied Mathematics. MR 510197
(80c:30001)
 [2]
Albert
Edrei, Meromorphic functions with three
radially distributed values, Trans. Amer. Math.
Soc. 78 (1955),
276–293. MR 0067982
(16,808d), http://dx.doi.org/10.1090/S00029947195500679829
 [3]
Simon
Hellerstein and Jack
Williamson, Derivatives of entire functions and a
question of Pólya, Trans. Amer. Math.
Soc. 227 (1977),
227–249. MR 0435393
(55 #8353), http://dx.doi.org/10.1090/S00029947197704353934
 [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 0481004
(58 #1151), http://dx.doi.org/10.1090/S00029947197704810041
 [5]
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
(81k:30036), http://dx.doi.org/10.1090/S00029947198105944229
 [6]
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
(84a:30050), http://dx.doi.org/10.1090/S00029947198306783533
 [7]
Simon
Hellerstein, LiChien
Shen, and Jack
Williamson, Real zeros of derivatives of
meromorphic functions and solutions of second order differential
equations, Trans. Amer. Math. Soc.
285 (1984), no. 2,
759–776. MR
752502 (85j:30065), http://dx.doi.org/10.1090/S00029947198407525021
 [8]
A.
Hinkkanen and J.
Rossi, On a problem of Hellerstein, Shen and
Williamson, Proc. Amer. Math. Soc.
92 (1984), no. 1,
72–74. MR
749894 (86c:30061), http://dx.doi.org/10.1090/S00029939198407498941
 [9]
W. P. Kohs, Ph. D. Thesis, University of Hawaii, 1985.
 [10]
B.
Ja. Levin, Distribution of zeros of entire functions, American
Mathematical Society, Providence, R.I., 1964. MR 0156975
(28 #217)
 [11]
B.
Ja. Levin and I.
V. Ostrovskiĭ, The dependence of the growth of an entire
function on the distribution of zeros of its derivatives, Sibirsk.
Mat. Ž. 1 (1960), 427–455 (Russian). MR 0130979
(24 #A833)
 [1]
 L. Ahlfors, Complex analysis, McGrawHill, 1979. MR 510197 (80c:30001)
 [2]
 A. Edrei, Meromorphic functions with three radially distributed values, Trans. Amer. Math. Soc. 78 (1955), 276293. MR 0067982 (16:808d)
 [3]
 S. Hellerstein and J. Williamson, Derivatives of entire functions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227249. MR 0435393 (55:8353)
 [4]
 , Derivatives of entire functions and a question of Pólya. II, Trans. Amer. Math. Soc. 234 (1977), 497503. MR 0481004 (58:1151)
 [5]
 , The zeros of the second derivative of the reciprocal of an entire function, Trans. Amer. Math. Soc. 263 (1981), 501513. MR 594422 (81k:30036)
 [6]
 S. Hellerstein, L. C. Shen and J. Williamson, Reality of the zeros of an entire function and its derivatives, Trans. Amer. Math. Soc. 275 (1983), 319331. MR 678353 (84a:30050)
 [7]
 , Real zeros of derivatives of meromorphic functions and solutions of second order differential equations, Trans. Amer. Math. Soc. 285 (1984), 759776. MR 752502 (85j:30065)
 [8]
 A. Hinkkanen and J. Rossi, On a problem of Hellerstein, Shen and Williamson, Proc. Amer. Math. Soc. 92 (1984), 7274. MR 749894 (86c:30061)
 [9]
 W. P. Kohs, Ph. D. Thesis, University of Hawaii, 1985.
 [10]
 B. Ja. Levin, Distribution of zeros of entire functions, Transl. Math. Mono., vol. 5, Amer. Math. Soc., Providence, R. I., 1964. MR 0156975 (28:217)
 [11]
 B. Ja. Levin and I. V. Ostrovskii, On the dependence of the growth of an entire function on the distribution of the zeros of its derivatives, Sibirsk. Mat. Zh. 1 (1960), 427455; English transl., Amer. Math. Soc. Transl. (2) 32 (1963), 323357. MR 0130979 (24:A833)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719880933316X
PII:
S 00029947(1988)0933316X
Keywords:
Meromorphic functions,
derivative,
zeros
Article copyright:
© Copyright 1988
American Mathematical Society
