Characteristic Functions and Borel Exceptional Values of E-Valued Meromorphic Functions

and Applied Analysis 3 Let n r, f or n r, ∞̂ denote the number of poles of f z in |z| ≤ r and let n r, a, f denote the number of a-points of f z in |z| ≤ r, counting with multiplicities. Define the volume function associated with E-valued meromorphic function f z by V ( r, ∞̂, f V r, f 1 2π ∫ Cr log ∣∣ ∣ ∣ r ξ ∣∣ ∣ ∣Δ log ∥ ∥f ξ ∥ ∥dx ∧ dy, ξ x iy, V ( r, a, f ) 1 2π ∫ Cr log ∣ ∣ ∣ ∣ r ξ ∣ ∣ ∣ ∣Δ log ∥ ∥f ξ − a∥dx ∧ dy, ξ x iy, 1.5 and the counting function of finite or infinite a-points by N ( r, f ) n ( 0, f ) log r ∫ r 0 n ( t, f ) − n0, f t dt, 1.6 N r, ∞̂ n 0, ∞̂ log r ∫ r 0 n t, ∞̂ − n 0, ∞̂ t dt, 1.7 N ( r, a, f ) n ( 0, a, f ) log r ∫ r 0 n ( t, a, f ) − n0, a, f t dt, 1.8 respectively. Next, we define m ( r, f ) m ( r, ∞̂, f 1 2π ∫2π 0 log ∥∥f ( re ∥∥dθ, m r, a m ( r, a, f ) 1 2π ∫2π 0 log 1 ∥f ( reiθ ) − a∥ T ( r, f ) m ( r, f ) N ( r, f ) . 1.9 Let n r, f or n r, ∞̂ denote the number of poles of f z in |z| ≤ r, and let n r, a, f denote the number of a-points of f z in |z| ≤ r, ignoring multiplicities. Similarly, we can define the counting functions N r, f , N r, ∞̂ , and N r, a, f of n r, f , n r, ∞̂ , and n r, a, f . If f z is an E-valued meromorphic function in the whole complex plane, then the order and the lower order of f z are defined by λ ( f ) lim sup r→ ∞ log T ( r, f ) log r , μ ( f ) lim inf r→ ∞ log T ( r, f ) log r . 1.10 4 Abstract and Applied Analysis If f z is an E-valued meromorphic function in CR, 0 < R < ∞, then the order and the lower order of f z are defined by λ ( f ) lim sup r→R− log T ( r, f ) log 1/ R − r , μ ( f ) lim inf r→R− log T ( r, f ) log 1/ R − r . 1.11 Lemma 1.1. Let B x be a positive and continuous function in 0, ∞ which satisfies lim supx→ ∞ logB x / logx ∞. Then there exists a continuously differentiable function ρ x , which satisfies the following conditions. i ρ x is continuous and nondecreasing for x ≥ x0 x0 > 0 and tends to ∞ as x → ∞. ii The functionU x x x x ≥ x0 satisfies the following: lim x→ ∞ logU X logU x 1, X x x logU x . 1.12 iii lim supx→ ∞ logB x / logU x 1. Lemma 1.1 is due to K. L. Hiong also Qinglai Xiong and ρ x is called the proximate order of Hiong. A simple proof of the existence of ρ r was given by Chuang 7 . Suppose that f z is an E-valued meromorphic function of infinite order in the unit disk C1. Let x 1/ 1 − r and X 1/ 1 − R . From ii and iii in Lemma 1.1, we have lim r→ 1− logU 1/ 1 − R logU 1/ 1 − r 1, R r logU 1/ 1 − r 1 logU 1/ 1 − r 1 , lim sup x→ 1− log T ( r, f ) logU 1/ 1 − r 1. 1.13 Here, the functions ρ 1/ 1 − r and U 1/ 1 − r are called the proximate order and type function of f z , respectively. Definition 1.2. An E-valued meromorphic function f z in CR, 0 < R ≤ ∞ is of compact projection, if for any given ε > 0, ‖Pn f z − f z ‖ < ε has sufficiently larg n in any fixed compact subset D ⊂ CR. Throughout this paper, we say that f z is an E-valued meromorphic function meaning that f z is of compact projection. C.-G. Hu and Q. Hu 3 established the following Nevanlinna’s first and second main theorems of E-valued meromorphic functions. Abstract and Applied Analysis 5 Theorem 1.3. Let f z be a nonconstant E-valued meromorphic function in CR, 0 < R ≤ ∞. Then for 0 < r < R, a ∈ E, f z /≡a,and Applied Analysis 5 Theorem 1.3. Let f z be a nonconstant E-valued meromorphic function in CR, 0 < R ≤ ∞. Then for 0 < r < R, a ∈ E, f z /≡a, T ( r, f ) V r, a N r, a m r, a log ∥ ∥cq a ∥ ∥ ε r, a . 1.14 Here, ε r, a is a function satisfying that |ε r, a | ≤ log ‖a‖ log 2, ε r, 0 ≡ 0, 1.15 and cq a ∈ E is the coefficient of the first term in the Laurent series at the point a. Theorem 1.4. Let f z be a nonconstant E-valued meromorphic function in CR, 0 < R ≤ ∞ and a k ∈ E ∪ {∞̂} k 1, 2, . . . , q be q ≥ 3 distinct points. Then for 0 < r < R,