On a Conjecture of Zheng Jianhua
DOI:
https://doi.org/10.5644/SJM.03.2.03Keywords:
$T$ direction, Nevanlinna direction, small meromorphic function, Ahlfors characteristic functionAbstract
In this paper, a new proof of the existence of the $T$ direction is given. Furthermore, we prove that for a meromorphic function $f$ with finite lower order, satisfying $\limsup\limits_{r\rightarrow \infty} \frac{T(r,f)}{\log^{2}r}=\infty,$ there exists a $T$ direction concerning small functions.
1991 Mathematics Subject Classification. Primary 30D30, Secondary 30D35
Downloads
References
G. Julia, Sur quelques proprietes nouvelles des fonctions entieres on meromorphes, Ann. Ecole Norm. Sup., 36 (1919), 93–125.
A. Ostrowsk¨ı, Ueber Folgen analytischer Functionen und einige Verscharfungen des Picardschen Statzes, Math. Zeit., 24 (1924), 215–258
G. Valiron, Recherches sur le theoreme de M. Borel dans la theorie des fonctions meromorphes, Acta Math., 52 (1928), 67–92.
G. Valiron, Directions de Borel des fonctions meromorphes, Mem. Sci. Math. Fasc., 89, 1938.
J. H. Zheng, On transcendental meromorphic functions with radially distributed values, Sci. China, Ser. A., 33 (2003), 539–550.
H. Guo, J. H. Zheng, and T. W. Ng, On a new singular direction of meromorphic functions, Bull. Aust. Math. Soc., 69 (2004), 277–287.
L. Yang, Value Distribution Theory, Science Press, Beijing, 1982, in Chinese; Springer-Verlag, 1993, in English.
X. L. Zhang, A fundamental inequality for meromorphic functions in an angular domain and its application, Acta Math. Sin, New Series, 10 (1994), 308–314.
X. L. Zhang, The deficiency and Nevanlinna direction of the meromorphic functions, J. Beijing Inst. Technol., 8 (1999), 14–18.
Y. N. Lu and G. H. Zhang, On Nevanlinna direction of meromorphic function, Sci. China, Ser. A., 13 (1983), 215–224.
M. Tsuji, Potential theory in modern function theory, Maruzen Co. LTD, Tokyo, 1959.
J. H. Zheng, On value distribution of meromorphic functions with respect to arguments, Preprint, August 12, 2005.
D. C. Sun, Meromorphic function without singular direction, Chin. J. Math., 2 (1982), 195–200.
Z. J. Wu and D. C. Sun, On the singular direction of zero order meromorphic functions, Chin. J. Math., 26 (2006), 399–403.
