An addition to the -theorem for subharmonic and entire functions of zero lower order

Author:
I. E. Chyzhykov

Journal:
Proc. Amer. Math. Soc. **130** (2002), 517-528

MSC (2000):
Primary 30D15, 31A05

DOI:
https://doi.org/10.1090/S0002-9939-01-06188-3

Published electronically:
June 21, 2001

MathSciNet review:
1862132

Full-text PDF

Abstract | References | Similar Articles | Additional Information

We obtain a sharp asymptotic relation between the infimum and the maximum on a circle of a subharmonic function of zero lower order. An example is constructed, which shows the sharpness of the relation in the class of entire functions of zero order such that , where as .

**1.**Hayman W.K.*Subharmonic functions, Vol 2.*London Math. Soc. Monographs, 20, Academic Press, 1989, pp. i-xxvi and 285-875. MR**91f:31001****2.**Barry P.D.*The minimum modulus of small integral and subharmonic functions,*Proc. London Math. Soc. (3)**12**(1962), no. 47, 445-495. MR**25:3172****3.**Fenton P.C.*The infimum of small subharmonic functions,*Proc. Amer. Math. Soc.**78**(1980), no. 1, 43-47. MR**80j:31001****4.**Goldberg A. A.*On the minimum modulus of a meromorphic function of slow growth,*Mat. Zametki.**25**(1979), no. 6, 835-844. (in Russian) Engl. trans. in Math. Notes (1979), 432-437. MR**81c:30058****5.**Fenton P.C.*The minimum of small entire functions,*Proc. Amer. Math. Soc.**81**(1981), no. 4, 557-561. MR**82c:30040****6.**Fenton P.C.*The minimum modulus of certain small entire functions,*Proc. Amer. Math. Soc.**271**(1982), no. 1, 183-195. MR**83h:30022****7.**Barry P.D.*On integral functions which grow little more rapidly than do polynomials,*Proc. Roy. Irish Acad.**82A**(1982) no. 1, 55-95. MR**84i:30035****8.**Barry P.D.*Some theorems related to the -theorem,*Proc. London Math. Soc.(3)**21**(1970), no. 2, 334-360. MR**44:456****9.**Chyzhykov I. E.*Asymptotic properties of meromorphic in the half-plane or in the unit disk functions,*Thesis, Lviv, 1998, 156 pp. (in Ukrainian)**10.**Hayman W.K., Kennedy P.B.*Subharmonic functions, Vol 1,*London Math. Soc. Monographs, 9, Academic Press, 1976. MR**57:665****11.**Akhiezer N. I.*Elements of elliptic functions theory,*GITTL, Moscow-Leningrad, 1948, 292 pp. (in Russian)

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
30D15,
31A05

Retrieve articles in all journals with MSC (2000): 30D15, 31A05

Additional Information

**I. E. Chyzhykov**

Affiliation:
Department of Mechanics and Mathematics, Lviv National University, Universytetska 1, Lviv, 79000, Ukraine

Email:
matstud@franko.lviv.ua

DOI:
https://doi.org/10.1090/S0002-9939-01-06188-3

Keywords:
Subharmonic function,
$\cos\pi\rho$-theorem,
entire function,
minimum modulus

Received by editor(s):
July 5, 2000

Published electronically:
June 21, 2001

Additional Notes:
The author was supported in part by INTAS, Grant # 99-00089

Communicated by:
Juha M. Heinonen

Article copyright:
© Copyright 2001
American Mathematical Society