A Note on the Growth of Meromorphic Functions with a Radially Distributed Value

and Applied Analysis 3 with ω max{π/ βj − αj : 1 ≤ j ≤ q}, suppose that lim sup r→∞ logn ( r, Y, f a ( f ′ n b ) log r ≤ ρ 1.7 with a positive number ρ, finite complex numbers a/ 0 and b, and Y ⋃q j 1{z : αj ≤ arg z ≤ βj}, for any positive integer n ≥ 2, and that q ∑ j 1 ( αj 1 − βj ) < 4 σ arcsin √ δ (∞, f ′) 2 , αq 1 α1 2π 1.8 with σ max{ω, ρ, μ}. Then the order λ of f z has the estimation λ ≤ max{ω, ρ}. Now there arises a natural question. Question 1. What can be said if f ′ in Theorems A and B is replaced by the kth derivative f k ? In this paper, we will prove the following results which generalize Theorems A and B. Theorem 1.1. Let f z be a transcendental meromorphic function with δ ∞, f k > 0 for a positive integer k in C and let Lj : arg z θj j 1, 2, . . . , q be a finite number of rays issued from the origin such that −π ≤ θ1 < θ2 < · · · < θq < π, θq 1 θ1 2π 1.9 with ω max{π/ θj 1 − θj : 1 ≤ j ≤ q}. Set Y C \ ⋃q j 1 Lj . If f z satisfies lim sup r→∞ logn ( r, Y, f a ( f k )n b ) log r ≤ ρ 1.10 with a positive number ρ, finite complex numbers a/ 0 and b, for any positive integer n ≥ k 1, then the order λ of f z has the estimation λ ≤ max{ω, ρ}. Remark 1.2. Let k 1. Then by Theorem 1.1 we get Theorem A. Corollary 1.3. Let f z be a transcendental entire function, let the notations θj (j 1, 2, . . . , q 1), ω, and Y be defined as in Theorem 1.1, and suppose that the function f z fulfills the same condition 1.10 as in Theorem 1.1. Then the order λ of f z has the estimation λ ≤ max{ω, ρ}. Theorem 1.4. Let f z be a transcendental meromorphic function of finite lower order μ with δ ∞, f k > 0 for a positive integer k in C. For q pairs of real numbers {αj , βj} such that −π ≤ α1 < β1 ≤ α2 < β2 ≤ · · · ≤ αq < βq ≤ π 1.11 4 Abstract and Applied Analysis with ω max{π/ βj − αj : 1 ≤ j ≤ q}, suppose that lim sup r→∞ logn ( r, Y, f a ( f k )n b ) log r ≤ ρ 1.12 with a positive number ρ, finite complex numbers a/ 0 and b, and Y ⋃q j 1{z : αj ≤ arg z ≤ βj}, for any positive integer n ≥ k 1, and that q ∑ j 1 ( αj 1 − βj ) < 4 σ arcsin √ δ (∞, f k ) 2 , αq 1 α1 2π 1.13 with σ max{ω, ρ, μ}. Then the order λ of f z has the estimation λ ≤ max{ω, ρ}. Remark 1.5. Let k 1. Then by Theorem 1.4 we get Theorem B. Corollary 1.6. Let f z be a transcendental entire function of finite lower order μ, let the notations αj , βj (j 1, 2, . . . , q), ω, σ, and Y be defined as in Theorem 1.4, and suppose that the function f z fulfills the same conditions 1.12 and 1.13 as in Theorem 1.4. Then the order λ of f z has the estimation λ ≤ max{ω, ρ}. In order to prove our results, we require the Nevanlinna theory of meromorphic functions in an angular domain. For the sake of convenience, we recall some notations and definitions. Let f be ameromorphic function on the angular domainΩ α, β {z : α ≤ arg z ≤ β}, where 0 < β − α ≤ 2π . Nevanlinna et al. 12, 13 introduced the following notations: Aα,β ( r, f ) ω π ∫ r 1 ( 1 tω − t ω r2ω ) { log ∣ ∣ ∣f ( te )∣ ∣ ∣ log ∣ ∣ ∣f ( te )∣ ∣ ∣ }dt t , Bα,β ( r, f ) 2ω πrω ∫β α log ∣ ∣ ∣f ( re )∣ ∣ ∣ sinω θ − α dθ, Cα,β ( r, f ) 2 ∑ 1<|bm|<r ( 1 |bm| − |bm| ω r2ω ) sinω θm − α , 1.14 where ω π/ β − α and bm |bm|em are the poles of f in Ω α, β appearing according to their multiplicities. The function Cα,β r, f is called the angular counting function counting multiplicities of the poles of f in Ω α, β , and Cα,β r, f is called the angular reduced counting function ignoring multiplicities of the poles of f in Ω α, β . Further, Nevanlinna’s angular characteristic function Sα,β r, f is defined as follows: Sα,β ( r, f ) Aα,β ( r, f ) Bα,β ( r, f ) Cα,β ( r, f ) . 1.15 Throughout the paper, we denote by R r, ∗ a quantity satisfying R r, ∗ O{log rT r, ∗ }, ∀r / ∈ E, 1.16 Abstract and Applied Analysis 5 where E denotes a set of positive real numbers with finite linear measure. It is not necessarily the same for every occurrence in the context. 2. Some Lemmas In this section we present some lemmas which will be needed in the sequel. Lemma 2.1 see 5, 12–14 . Let f be meromorphic in C. Then in Ω α, β for an arbitrary finite complex number a, we have Sα,β (and Applied Analysis 5 where E denotes a set of positive real numbers with finite linear measure. It is not necessarily the same for every occurrence in the context. 2. Some Lemmas In this section we present some lemmas which will be needed in the sequel. Lemma 2.1 see 5, 12–14 . Let f be meromorphic in C. Then in Ω α, β for an arbitrary finite complex number a, we have Sα,β (