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We obtain sufficient conditions under which the Julia lines of entire functions of slow growth do not have finite exceptional values. Let f be a function meromorphic and transcendental in C (in what follows, meromorphic). We use the standard notation of the Nevanlinna value-distribution theory (see, e.g., [1, pp. 23 -37]): T ( r, f ) is the Nevanlinna characteristic function, n ( r, a, f ) is the counting function of points a of the function f, N ( r, a, f ) is the Nevanlinna counting function, and f *is called a Julia line of the function f if f takes any value from the extended complex plane infinitely many times, except, possibly, two values inside each anglewhere ε > 0 is an arbitrary number. If the function f takes a value a ∈ C only finitely many times inside the angle { z : θ -ε < arg z < θ + ε } for certain ε > 0, then we say that f has an exceptional value on the Julia line l θ = { z : arg z = θ }.It is well known that if f is an entire transcendental (in what follows, entire) function, then it has at least one Julia line. If f is a meromorphic function and lim sup , ln r T r f r →∞ ( ) 2 = + ∞ (see [2, 3]) or if f is a meromorphic function and (see [4, 5]) lim sup z →∞then f has at least one Julia line. The subsequent results in this direction were obtained by Ostrovskii, Valiron, Cartwright, Anderson, Clunie, Toppila, Gold'berg, Sheremeta, and others (see [6], Chap. 5).Following Hayman [7], we define an ε-set as any countable set of disks K ( a n , r n ) = { z : | z -a n | < r n }, n ∈ N, that do not contain the origin of coordinates and for which the sum of angles at which they are seen from
We obtain sufficient conditions under which the Julia lines of entire functions of slow growth do not have finite exceptional values. Let f be a function meromorphic and transcendental in C (in what follows, meromorphic). We use the standard notation of the Nevanlinna value-distribution theory (see, e.g., [1, pp. 23 -37]): T ( r, f ) is the Nevanlinna characteristic function, n ( r, a, f ) is the counting function of points a of the function f, N ( r, a, f ) is the Nevanlinna counting function, and f *is called a Julia line of the function f if f takes any value from the extended complex plane infinitely many times, except, possibly, two values inside each anglewhere ε > 0 is an arbitrary number. If the function f takes a value a ∈ C only finitely many times inside the angle { z : θ -ε < arg z < θ + ε } for certain ε > 0, then we say that f has an exceptional value on the Julia line l θ = { z : arg z = θ }.It is well known that if f is an entire transcendental (in what follows, entire) function, then it has at least one Julia line. If f is a meromorphic function and lim sup , ln r T r f r →∞ ( ) 2 = + ∞ (see [2, 3]) or if f is a meromorphic function and (see [4, 5]) lim sup z →∞then f has at least one Julia line. The subsequent results in this direction were obtained by Ostrovskii, Valiron, Cartwright, Anderson, Clunie, Toppila, Gold'berg, Sheremeta, and others (see [6], Chap. 5).Following Hayman [7], we define an ε-set as any countable set of disks K ( a n , r n ) = { z : | z -a n | < r n }, n ∈ N, that do not contain the origin of coordinates and for which the sum of angles at which they are seen from
UDC 517.53For the entire Dirichlet series f (z) = X 1 n=0 ane zλn , we establish necessary and sufficient conditions on the coefficients an and exponents λn under which the function f has the Paley effect, i.e., the condition lim supis satisfied, where M f (r) and T f (r) are the maximum modulus and the Nevanlinna characteristic of the function f, respectively.
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