Critical points of real entire functions

and a conjecture of Pólya

Author:
Young-One Kim

Journal:
Proc. Amer. Math. Soc. **124** (1996), 819-830

MSC (1991):
Primary 30D15, 30D35

DOI:
https://doi.org/10.1090/S0002-9939-96-03083-3

MathSciNet review:
1301508

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a nonconstant real entire function of genus and assume that all the zeros of are distributed in some infinite strip , . It is shown that (1) if has nonreal zeros in the region , and has nonreal zeros in the same region, and if the points and are located outside the Jensen disks of , then has exactly critical zeros in the closed interval , (2) if is at most of order , , and minimal type, then for each positive constant there is a positive integer such that for all has only real zeros in the region , and (3) if is of order less than , then has just as many critical points as couples of nonreal zeros.

**[CCS1]**T. Craven, G. Csordas, and W. Smith,*The zeros of derivatives of entire functions and the Pólya--Wiman conjecture*, Ann. of Math. (2)**125**(1987), 405--431. MR**88a:30007****[CCS2]**------,*Zeros of derivatives of entire functions*, Proc. Amer. Math. Soc.**101**(1987), 323--326. MR**88k:30024****[G]**W. Gontcharoff,*Recherches sur les dérivées successives des fonctions analytiques*, Ann. École Norm.**47**(1930), 1--78.**[HSW]**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), 319--331. MR**84a:30050****[HW1]**S. Hellerstein and J. Williamson,*Derivatives of entire functions and a question of Pólya*, Trans. Amer. Math. Soc.**227**(1977), 227--249. MR**55:8353****[HW2]**------,*Derivatives of entire functions and a question of Pólya. II*, Trans. Amer. Math. Soc.**234**(1977), 497--503. MR**58:1151****[K1]**Y. O. Kim,*A proof of the Pólya--Wiman conjecture*, Proc. Amer. Math. Soc.**109**(1990), 1045--1052. MR**90k:30049****[K2]**------,*On a theorem of Craven, Csordas and Smith*, Complex Variables**22**(1993), 207--209. MR**94m:30052****[L]**B. Ja. Levin,*Distribution of Zeros of Entire Functions*, Transl. Math. Monographs, vol. 5, Amer. Math. Soc., Providence, RI, 1964. MR**28:217****[LO]**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*, Amer. Math. Soc. Transl. Ser. 2, vol. 32, Amer. Math. Soc., Providence, RI, 1963, pp. 323--357, MR**24:A833****[P1]**G. Pólya,*Some problems connected with Fourier's work on transcendental equations*, Quart. J. Math. Oxford Ser. 1 (1930), 21--34.**[P2]**------,*On the zeros of derivatives of a function and its analytic character*, Bull. Amer. Math. Soc.**49**(1943), 178--191. MR**4:192d****[S]**T. Sheil-Small,*On the zeros of the derivatives of real entire functions and Wiman's conjecture*, Ann. of Math. (2)**129**(1989), 179--193. MR**90a:30084**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
30D15,
30D35

Retrieve articles in all journals with MSC (1991): 30D15, 30D35

Additional Information

**Young-One Kim**

Affiliation:
Department of Mathematics, College of Natural Sciences, Sejong University, Seoul 133–747, Korea

DOI:
https://doi.org/10.1090/S0002-9939-96-03083-3

Keywords:
P\'{o}lya--Wiman conjecture,
Laguerre--P\'{o}lya class,
Fourier critical point

Received by editor(s):
March 28, 1994

Received by editor(s) in revised form:
September 7, 1994

Additional Notes:
This research is supported by the research grant of the Ministry of Education, Republic of Korea, and SNU–GARC.

Dedicated:
To the memory of Professor Jongsik Kim

Communicated by:
Albert Baernstein II

Article copyright:
© Copyright 1996
American Mathematical Society