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On transcendental meromorphic functions with radially distributed values. (English) Zbl 1081.30032

Let f(z) be a transcendental meromorphic function. For an unbounded subset X of the complex plane , the author denotes by n(r,X,f=a) and n ¯(r,X,f=a) the number of the roots repeated according to multiplicity and distinct roots of f(z)-a=0, a ¯, in X{z:|z|<r}. The integrated counting functions N(r,X,f=a) and N ¯(r,X,f=a) are defined in the usual manner. The author considers q pair {α j ,β j } of real numbers such that -πα 1 <β 1 α 2 <β 2 α q <β q π and defines ω=max j {π/(β j -α j )}. The author obtains several theorems in this paper. One of the main result is the following. Let f(z) be a transcendental meromorphic function of finite lower order μ. Suppose that f(z) satisfies δ:=δ(a,f (p) )>0 for some a ¯ and an integer p. If for q pair {α j ,β j } of real numbers given above and for an integer k>0, it holds

n(r,Y,f=0)+n ¯(r,Y,f (k) =1)=o(T(dr,f)),d1,

for Y= j=1 q {z:αargzβ j } and j=1 q (α j+1 -β j )<(4/β)arcsinδ/2, α q+1 =2π+α 1 , where β=max{ω,μ}, then λ(f)ω. The methods for the proofs are from the Nevanlinna theory in the angular domains, and Baernstein’s theorem on the spread relation. The author comments on a singular direction in terms of the Nevanlinna characteristic function. A radial argz=0 is called T direction of f(z) provided that given any b ¯, possibly with the exception of at most two values for arbitrary small ε>0 it holds

lim sup r N(r,Z,f=b) T(r,f)>0,

where Z={z:θ-ε<argz<θ+ε}.


MSC:
30D35Distribution of values (one complex variable); Nevanlinna theory