Radially Distributed Values of Holomorphic Curves
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Radially Distributed Values of Holomorphic Curves Nan Wu1
Received: 25 September 2016 / Revised: 14 November 2016 / Accepted: 3 January 2017 © The Author(s) 2017. This article is an open access publication
Abstract Using the spread relation we investigate the growth of transcendental holomorphic curves when they have radially distributed small holomorphic curves. Keywords Holomorphic curves · Spread relation · Radial distribution Mathematics Subject Classification Primary 30D10; Secondary 30D20 · 30B10 · 34M05
1 Introduction Let f (z) be a transcendental meromorphic function on the complex plane C and let a(z) be a meromorphic function satisfying T (r, a) = o(T (r, f )), r → ∞. Then a(z) is called a small function with respect to f (z). We shall call a small function a(z) of a Nevanlinna deficient function of f (z) if and only if δ(a(z), f ) = lim inf
m r,
r →∞
1 f (z)−a(z)
T (r, f )
= 1 − lim sup r →∞
N (r, a(z), f ) > 0. T (r, f )
Communicated by James K. Langley.
B 1
Nan Wu [email protected] Department of Mathematics, School of Science, China University of Mining and Technology (Beijing), Beijing 100083, People’s Republic of China
123
N. Wu
Recall the Valiron deficiency (a(z), f ) = lim sup
m r,
1 f (z)−a(z)
T (r, f ) N (r, a(z), f ) = 1 − lim inf > 0. r →∞ T (r, f ) r →∞
Throughout the paper, we shall adopt the standard notation used in Nevanlinna theory (see [6]). Suppose that f is a meromorphic function, D = D(α1 , . . . , αn ) is a system of rays D = ∪nj=1 {z : arg z = α j }, α1 < α2 < · · · αn+1 = α1 + 2π, ω = ω(D) = max{π/(α j+1 − α j ), 1 ≤ j ≤ n}, and G(ε, D) = C\ ∪nj=1 {z : | arg z − α j | < ε}. We say that the a-points of f are attracted to D if for any ε > 0, n(r, a, D, ε, f ) = o(T (r, f )), r → ∞, where n(r, a, D, ε, f ) denotes the number of zeros of f (z) − a(z) in the region G(ε, D). In 1992, Yang and Li [14] considered the distribution of arguments of the a(z)points of f (z) for a small function a(z) with respect to f (z) following Goldberg’s work [4] and gave the following result. Theorem A ([14]). Suppose that D is some system of rays and f is a meromorphic function of finite order λ > ω. If (b(z), f ) = 0 and b(z)-points of f are attracted to D for a small function b(z), then δ(a(z), f ) = 0 for any small function a(z). In fact, the work of Yang and Li [14] stems from the study of the growth of meromorphic functions with radially distributed values. A value on the extended complex plane C = C ∪ {∞} is called a radially distributed value of a transcendental meromorphic function if most of the points at which the value is assumed to be distributed lie closely along a finite number of rays from the origin. Further, we recall the results obtained by Wu [12] and Zheng [15]. In 1993, Wu [12] used the Nevanlinna theory of meromorphic functions in angular domains to study this problem and obtained the following theorem. Theorem B ([12]). Let f (z) be a meromorphic function of finite lower order μ in C. Suppose that arg z = θ j ( j = 1, 2, . . . , q; 1 ≤ q < ∞; 0 ≤ θ1
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