ON BOREL DIRECTION CONCERNING SMALL FUNCTIONS

Authors

  • T. Y. PETER CHERN Department of Applied Mathematics, Kaohsiung Polytechnic Institute, Ta-Hsu Hsiang, Kaoh­siung County, Taiwan 84008 R. O. C.

DOI:

https://doi.org/10.5556/j.tkjm.29.1998.4289

Keywords:

Borel direction, small function, finiter positive order

Abstract

In this paper, we shall prove Theorem 1

Let $f$ be nonconstant meromorphic  in $\mathbb{C}$ with finite positive order $\lambda$, $\lambda(r)$ be a proximate order of $f$ and $U(r, f)=r^{\lambda(r)}$, then for each number $\alpha$,$0<\alpha<\pi/2$, there exists a number $\phi_0$ with $0\le \phi_0 < 2\pi$ such that the inequality

\[ \limsup_{r\to\infty}\sum_{i=1}^3 n(r, \phi_0, \alpha, f=a_i(z))/U(r, f)>0,\]

holds for any three distinct meromorphic function $a_i(z)(i=1, 2, 3)$ with $T(r,a_i)=o(U(r, f))$ as $r\to\infty$.

References

M. Diernacki, "Sur les directions de Borel des functions meromorphes," Acta Math., 56 (1930), 197-204.

C. T. Chuang, Sigular Direction of Meromorphic Functions (in Chinese), Science Press, Beijing, 1982.

H. Milloux, "Le theoreme de Picard, suites de functions holomorphes; functions meromor­phes et functions entieres.'' J . de Math., 3 (1924), 345-401.

X. Pang, "On the singular direction of meromorphic function" (in Chinese), Advances in Mathematics (China), 16 (3) (1987), 309-315

M. Tsuji, "On Borel's directions of meromorphic functions of finite order, III, Kodai Math Sem. Rep., {1950), 104-108.

J M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publ., N. Y., 1975.

G. Valiron, "Recherches sur le theoreme de M. Borel dans la theorie des functions mero­morphes.'' Acta Math. 52 (1928), 67-92.

G. Valiron, Directions de Borel des functions meromorphes, Memor. Sci. Math., Fasc. 89, Press, 1938.

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Published

2021-05-25

How to Cite

CHERN, T. Y. P. (2021). ON BOREL DIRECTION CONCERNING SMALL FUNCTIONS. Tamkang Journal of Mathematics, 29(1), 13-16. https://doi.org/10.5556/j.tkjm.29.1998.4289

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