On meromorphic functions with regions free of poles and zeros

Introduction.In this note we improve one of the results of Edrei and Fuchs [2].We shall adopt the terminology, notations and conventions of [2].We shall write, for instance, [2, Lemma 5] to denote Lemma 5 of [2].The aim of this investigation is to prove the following THEOREM.Let the s B-regular curves L,: z=te ta J«> (f^fo>O, /=1, 2, -, s; divide \z\^U into s sectors, each of which has opening^c>0.Suppose that all but a finite number of zeros and poles of the meromorphic function f(z) lie on the curves L 3 .If some τ (rφθ, τφoo) is a deficient value (in the sense of R. Nevanlinna) of the function f(z), then the order λ of f{z) does not exceed λ ίy where 2B+1 \ 2(β+l)/' c 2ff+l \ ι eB(B+l) Q 2 sin -4 2 1 J.