On meromorphic functions with regions free of poles and zeros (Q5899737)

The author proves the following theorem: Let the s B-regular curves \(L_ j:\) \(z=te^{i\alpha_ j(t)}\), \(t\geq t_ 0>0\), \(j=1,2,...,s\); \(\alpha_ 1(t)<\alpha_ 2(t)<...<\alpha_ s(t)<\alpha_ 1(t)+2\pi =\alpha_{s+1}(t)\) divide \(| z| \geq t_ 0\) into s sectors, each of which has opening \(\geq c>0\). Suppose that all but a finite number of zeros and poles of the meromorphic function f(z) lie on the curves \(L_ j\). If some \(z_ 0\in {\mathbb{C}}\), \(z_ 0\neq 0\) is a deficient value in the sense of Nevanlinna of the function f(z), then the order \(\lambda\) of f(z) does not exceed \(\lambda_ 1\), where quantity \(\lambda_ 1=\lambda_ 1(B,c)\) is determined in the paper.