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), 517528
MSC (2000):
Primary 30D15, 31A05
Published electronically:
June 21, 2001
MathSciNet review:
1862132
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: 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.
W.
K. Hayman, Subharmonic functions. Vol. 2, London Mathematical
Society Monographs, vol. 20, Academic Press Inc. [Harcourt Brace
Jovanovich Publishers], London, 1989. MR 1049148
(91f:31001)
 2.
P.
D. Barry, The minimum modulus of small integral and subharmonic
functions, Proc. London Math. Soc. (3) 12 (1962),
445–495. MR 0139741
(25 #3172)
 3.
P.
C. Fenton, The infimum of small subharmonic
functions, Proc. Amer. Math. Soc.
78 (1980), no. 1,
43–47. MR
548081 (80j:31001), http://dx.doi.org/10.1090/S00029939198005480816
 4.
A.
A. Gol′dberg, The minimum modulus of a meromorphic function
of slow growth, Mat. Zametki 25 (1979), no. 6,
835–844, 956 (Russian). MR 540239
(81c:30058)
 5.
P.
C. Fenton, The minimum of small entire
functions, Proc. Amer. Math. Soc.
81 (1981), no. 4,
557–561. MR
601729 (82c:30040), http://dx.doi.org/10.1090/S00029939198106017290
 6.
P.
C. Fenton, The minimum modulus of certain small
entire functions, Trans. Amer. Math. Soc.
271 (1982), no. 1,
183–195. MR
648085 (83h:30022), http://dx.doi.org/10.1090/S00029947198206480855
 7.
P.
D. Barry, On integral functions which grow little more rapidly than
do polynomials, Proc. Roy. Irish Acad. Sect. A 82
(1982), no. 1, 55–95. MR 669467
(84i:30035)
 8.
P.
D. Barry, Some theorems related to the
𝑐𝑜𝑠𝜋𝜌 theorem, Proc. London
Math. Soc. (3) 21 (1970), 334–360. MR 0283223
(44 #456)
 9.
Chyzhykov I. E. Asymptotic properties of meromorphic in the halfplane or in the unit disk functions, Thesis, Lviv, 1998, 156 pp. (in Ukrainian)
 10.
W.
K. Hayman and P.
B. Kennedy, Subharmonic functions. Vol. I, Academic Press
[Harcourt Brace Jovanovich Publishers], London, 1976. London Mathematical
Society Monographs, No. 9. MR 0460672
(57 #665)
 11.
Akhiezer N. I. Elements of elliptic functions theory, GITTL, MoscowLeningrad, 1948, 292 pp. (in Russian)
 1.
 Hayman W.K. Subharmonic functions, Vol 2. London Math. Soc. Monographs, 20, Academic Press, 1989, pp. ixxvi and 285875. 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, 445495. MR 25:3172
 3.
 Fenton P.C. The infimum of small subharmonic functions, Proc. Amer. Math. Soc. 78 (1980), no. 1, 4347. MR 80j:31001
 4.
 Goldberg A. A. On the minimum modulus of a meromorphic function of slow growth, Mat. Zametki. 25 (1979), no. 6, 835844. (in Russian) Engl. trans. in Math. Notes (1979), 432437. MR 81c:30058
 5.
 Fenton P.C. The minimum of small entire functions, Proc. Amer. Math. Soc. 81 (1981), no. 4, 557561. MR 82c:30040
 6.
 Fenton P.C. The minimum modulus of certain small entire functions, Proc. Amer. Math. Soc. 271 (1982), no. 1, 183195. 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, 5595. MR 84i:30035
 8.
 Barry P.D. Some theorems related to the theorem, Proc. London Math. Soc.(3) 21 (1970), no. 2, 334360. MR 44:456
 9.
 Chyzhykov I. E. Asymptotic properties of meromorphic in the halfplane 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, MoscowLeningrad, 1948, 292 pp. (in Russian)
Similar Articles
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:
http://dx.doi.org/10.1090/S0002993901061883
PII:
S 00029939(01)061883
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 # 9900089
Communicated by:
Juha M. Heinonen
Article copyright:
© Copyright 2001 American Mathematical Society
