On a new singular direction of quasimeromorphic mappings

Acta Mathematica Scientia 2009,29B(5):1453–1460 http://actams.wipm.ac.cn

ON A NEW SINGULAR DIRECTION OF QUASIMEROMORPHIC MAPPINGS∗

)1,2

Wu Zhaojun (




Sun Daochun (

)2

1.Department of Mathematics, Xianning University, Xianning 437100, China 2.School of Mathematics, South China Normal University, Guangzhou 510631, China E-mail: [email protected]; [email protected]

Abstract By applying Ahlfors’ theory of covering surface, we establish a fundamental inequality for quasimeromorphic mapping in an angular domain. As an application, we prove the existence of a new singular direction for quasimeromorphic mapping f , namely, a precise S direction, for which the spherical characteristic function S(r, f ) is used as a comparison function. Key words quasimeromorphic function; S direction; spherical characteristic function; covering surface 2000 MR Subject Classification

1

30D30; 30D35

Introduction and Statement of Results

We consider complex valued functions f (z) that will be called quasimeromorphic function [1] if f is twice differentiable in x and y, and f (z) can be expressed as f = w(ψ), where w is analytic and ψ is a quasiconformal homeomorphism; it is just a generalization of quasiconformal functions (See Letho [2], Chapter 6, p239). Recently, Sun and Yang [3] introduced a class of quasimeromorphic functions and called them quasimeromorphic mappings. The definition is obtained from that of quasiconformal mappings by permitting multiple values. More precisely, we say that a continuous mapping f on a domain D in the extended complex plane C∞ := C ∪ {∞} is an K-quasimeromorphic mappings [4], if for any point z0 ∈ D, there exists a neighborhood U (⊂ D) of z0 and a positive integer n (dependent on z0 ), such that ⎧ ⎨ (f (z)) n1 , f (z0 ) = ∞, F (z) = ⎩ (f (z) − f (z0 )) n1 + f (z0 ), f (z0 ) = ∞ is a schlicht K-quasiconformal mapping. We call f a schlicht K-quasiconformal mapping [5] if for any rectangle R = {x + iy; a < x < b, c < y < d} in U , f (x + iy) is an absolutely continuous function of y for almost every x ∈ (a, b) and an absolutely continuous function of x for almost ∗ Received

December 25, 2006; revised August 9, 2008. Sponsored by the NSFC (10471048)

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every y ∈ (c, d); moreover, there exists K ≥ 1 such that f (z) = u(x, y) + iv(x, y) satisfies |fz | + |fz | ≤ K(|fz | − |fz |) a.e. in U . For an K-quasimeromorphic mapping f (z), let F (log r + iθ) = f (elog r+iθ ) = f (reiθ ), then F (log r + iθ) is also an K-quasimeromorphic mapping. Hence, for almost every θ ∈ [0, 2π], f (reiθ ) is an absolutely continuous function of r. It is obvious that the class of quasimeromorphic mappings is a subclass of the class of quasimeromorphic functions. For the results related to the value distribution of quasimeromorphic mappings and quasimeromorphic functions, we refer the reader to the articles [1], [3], [6]. In this article, we denote the Riemann sphere of diameter 1 by V . Here, we introduce some basic definitions and notation that are to be used later, (see [3]). The covering surface on sphere V , generated by f (z) = u(x, y) + iv(x, y), is denoted by Fr , and let S(r, f ) be the average covering times of Fr to V :   1 r 2π ux vy − vx uy |Fr | = rdθdr, S(r, f ) = |V | π 0 0 (1 + |f |2 )2 where |Fr | and |V | are the areas of Fr and V , respectively. We also call S(r, f ) the spherical characteristic function for f (z). Suppose that f (z) is an K- quasimeromorphic mapping on C, denote by n(r, θ, δ, a) the number of zero points of f (z) − a contained in {z : |z| < r, | arg z − θ| ≤ δ}, multiple zeros being counted only once. If lim S(r, f ) = ∞, then f (z) is called r→∞ a transcendental K-quasimeromorphic mapping. For a transcendental K-quasimeromorphic mapping f (z), we define its order as ρ = lim sup r→∞

log S(r, f ) . log r

In this article, we will study the singular direction of K-quasimeromorphic mappings, and extend some results for meromorphic functions to quasimeromorphic mappings. In 1928, Valiron introduced the notion of Borel direction, that is, a ray arg z = θ is called a Borel direction of order ρ for f if for every 0 < ε < π/2, lim sup r→∞

log n(r, θ, ε, a) ≥ρ log r

for all a in C∞ with at most two exceptions (see [7]). Note that the definition is only meaningful in the case of 0 < ρ < ∞. In this case, it is well known that f must have at least one Borel direction in [7]. When the order ρ = 0 or ∞, it is difficult to define the Borel direction as above. In this case, Zheng in [8] used T (r, f ) instead of log r and introduced a new singular direction, namely, the T -direction for f . Here we recall his definition as follows. Definition 1.1 A ray arg z = θ is called a T -direction for a meromorphic function f (z), if for every 0 < ε < π/2, N (r, θ, ε, a) lim sup >0 (1.1) T (r, f ) r→∞ for all a in C∞ with at most two exceptions. A radial arg z = θ is called a precise T -direction of f (z), if N (r, θ, ε, a) in (1.1) is in place of N (r, θ, ε, a). It was conjectured by Zheng in [8] that a meromorphic function f (z) has at least one T direction if T (r, f ) lim sup = +∞. (1.2) 2 r→∞ (log r)

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And with some additional assumptions, Zheng proved the existence of T -direction. Recently, Guo, Zheng and Ng [9] confirmed the Zheng’s conjecture. They proved the following theorem. Theorem A Let f (z) be a transcendental meromorphic function on C and satisfies (1.2), then f (z) has at least one T -direction. For the K-quasiconformal mapping, there is a difficult question. Let f (z) be a transcendental K-quasimeromorphic mapping on C and satisfy lim sup r→∞

S(r, f ) = +∞. log r

(1.3)

Whether there is a ray arg z = θ such that for every 0 < ε < π/2 and all a in C∞ with at most two exceptions, n(r, θ, ε, a) lim sup >0 (1.4) S(r, f ) r→∞ holds or not? The ray arg z = θ satisfying the condition in the above question called a precise S direction for f . A radial arg z = θ called an S direction of f (z) if, in (1.4), n(r, θ, ε, a) is in place of n(r, θ, ε, a). In order to prove the existence of precise S direction (or T -direction), one needs naturally an analogue of the second fundament theorem of Nevanlinna in an angular domain. In this article, we shall prove the following analogue of the fundament theorem by Sun and Yang [3] for quasimeromorphic mappings. Theorem 1.1 Let f (z) be a K-quasimeromorphic mapping on C, for an angular domain Ω(ϕ0 − δ, ϕ0 + δ), given q (q > 2) different points a1 , a2 , · · · , aq ∈ C∞ , for almost every τ : 0 < τ < δ < π, we have (q − 2)S(r, ϕ0 − τ, ϕ0 + τ, f ) q  16Kπh2 log r n(r, ϕ0 , δ, ai ) + ≤ (q − 2)(δ − τ ) i=1 +(q − 2)S(1, ϕ0 − τ, ϕ0 + τ, f ) + hL(1, f ) + hL(r, f ), where L(r, f ) is the spherical length of boundary of f (|z| < r). As an application of Theorem 1.1, we shall confirm the above question by proving the following theorem. Theorem 1.2 Let f (z) be a transcendental K-quasimeromorphic mapping on C and satisfies (1.3), then f (z) has at least one precise S direction. From Theorem 1.2, we can obtain the following corollaries. Corollary 1.1 [10] Let f (z) be a transcendental K-quasimeromorphic mapping on C and satisfy (1.3), then f (z) has at least one S direction. Corollary 1.2 Let f (z) be a transcendental K-quasimeromorphic mapping on C and satisfy (1.2), then f (z) has at least one precise T -direction which is the same as stated in Definition 1.1. Furthermore, f (z) has at least one T -direction. Corollary 1.2 partly improve a result by Li and Gu [11]. When K = 1 in Corollary 1.2, we may also obtain the following corollary. Corollary 1.3 Let f (z) be a transcendental meromorphic function on C and satisfy (1.2), then f (z) has at least one precise T -direction. Furthermore, f (z) has at least one T -direction.

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The Proof of Theorem 1.1 In order to prove Theorem 1.1, we need the following Lemma. Lemma 2.1 [12] Let F be a finite covering surface of a basic surface F0 , then σ + ≥ σ0 S − hL,

|F | ,L where σ + = max{0, σ}, σ and σ0 are the characteristics of F and F0 , respectively; S = |F 0| is the relative boundary of F , and h > 0 is a constant which depends on F0 only. Applying Lemma 2.1 and the similar method as that in [3] and [13] (or [14]), we give a proof of Theorem 1.1 as follows. Proof of Theorem 1.1 Without loss of generality, we may assume ϕ0 = 0. Consider f : D → G, where

D = Ω(−τ, τ ) ∩ {1 < |z| < r} − {f −1 (a1 ), · · · , f −1 (aq )}, G = C∞ − {a1 , · · · , aq }. Denote σ + = max{0, σ}, σ and σ0 = q − 2 are the characteristics of D and G, respectively. Furthermore, q  σ≤ [n(r, 0, τ, ai ) − n(r, 0, τ, ai )] + 1 − 2. i=1

σ+ ≤

q 

n(r, 0, τ, ai ) ≤

i=1

q 

S = S(r, −τ, τ, f ) − S(1, −τ, τ, f ),

n(r, 0, δ, ai ),

i=1

and L = L(r, −τ ) + L(r, τ ) + L(1, −τ, τ, f ) + L(r, −τ, τ, f ) ≤ L(r, −τ ) + L(r, τ ) + L(1, f ) + L(r, f ), where L(r, τ ) is the spherical length of the boundary of f (reiτ ). By using Lemma 2.1, this gives (q − 2)[S(r, −τ, τ, f ) − S(1, −τ, τ, f )] −

q 

n(r, 0, δ, ai ) − hL(1, f ) − hL(r, f )

i=1

≤ h[L(r, −τ ) + L(r, τ )].

(2.1)

Denote the left expression of (2.1) by A(r, τ ). Thus d[S(r, −τ, τ, f ) − S(1, −τ, τ, f )] d(A(r, τ )) = (q − 2) . dτ dτ We may as well suppose that f (reiτ ) is an absolutely continuous function of r, because f (reiθ ) is an absolutely continuous function of r for almost every θ ∈ [0, 2π]. For sufficiently large n and arbitrary ε > 0, we have ||f (reiτ )| − |fj || < ε,

r ∈ [rj−1 , rj ],

where |fj | = min{|f (reiτ )|, rj−1 ≤ r ≤ rj },

rj =

jr , j = 1, 2, · · · , 2n. n

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Hence, for any t ∈ [rj−1 , rj ], we have |f (teiτ )|2 ≤ |fj |2 + 2ε(ε + |fj |), so that √ 1 + |f (teiτ )|2 ≤ 2. 1 + |fj |2 Since L(r, τ ) = lim

n→∞

2n  j=1

|f (rj eiτ ) − f (rj−1 eiτ )|   1 + |f (rj eiτ )|2 1 + |f (rj−1 eiτ )|2

 1  u(rj cos τ, rj sin τ ) − u(rj−1 cos τ, rj−1 sin τ ) 2 n→∞ 1 + |fj | j=1   +i[v(rj cos τ, rj sin τ ) − v(rj−1 cos τ, rj−1 sin τ )] 2n | rj u dr − i rj v dr|  rj−1 r rj−1 r = lim 2 n→∞ 1 + |fj | j=1 ≤ lim

≤

2n 

2n   √ 2 lim n→∞

rj

rj−1

j=1

(u2r + vr2 )1/2 dr , 1 + |fj |2

we can derive that 2n   √ L(r, τ ) ≤ 2 lim n→∞

j=1

rj

rj−1

2n  rj  (u2r + vr2 )1/2 dr (u2r + vr2 )1/2 dt ≤ 2 lim . n→∞ 1 + |fj |2 1 + |f (teiτ )|2 j=1 rj−1

Because u2r + vr2 = [(ux cos τ + uy sin τ )2 + (vx cos τ + vy sin τ )2 ] ≤ 2(u2x + u2y + vx2 + vy2 ), from |fz | + |fz | ≤ K(|fz | − |fz |), we can derive that u2x + u2y + vx2 + vy2 ≤ 2K(uxvy − vx uy ). Hence

 √ L(r, τ ) ≤ 2 2K

1

r

(ux vy − vx uy )1/2 dt . 1 + |f (teiτ )|2

Applying the Schwarz inequality, we have  r

 r (u v − v u )1/2 (ux vy − vx uy )1/2 2  x y x y 2 2 [L(r, −τ ) + L(r, τ )] ≤ 16K ( dt) + ( dt) −iτ 2 1 + |f (te )| 1 + |f (teiτ )|2 1 1  r  r  r dt (ux vy − vx uy )1/2 2 (ux vy − vx uy )1/2 2  ≤ 16K ( ) tdt + ( ) tdt t 1 + |f (te−iτ )|2 1 + |f (teiτ )|2 1 1 1 d[S(r, −τ, τ, f ) − S(1, −τ, τ, f )] log r = 16Kπ dτ 16Kπ d(A(r, τ )) = log r q−2 dτ Note that A(r, τ ) is an increasing function of τ . There exists a τ0 such that, when 0 < τ ≤ τ0 , A(r, τ ) ≤ 0; when τ > τ0 , A(r, τ ) > 0, By (2.1) and the above expression, we have [A(r, τ )]2 ≤ h2 [L(r, −τ ) + L(r, τ )]2 ≤

16Kπh2 d(A(r, τ )) log r, q−2 dτ

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i.e., dτ ≤ h2 [L(r, −τ ) + L(r, τ )]2 ≤

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16Kπh2 d(A(r, τ )) log r. q − 2 [A(r, τ )]2

Integrating both sides of the inequality leads to 16Kπh2 log r. (q − 2)A(r, τ )

δ−τ ≤ Thus A(r, τ ) ≤

16Kπh2 log r. (q − 2)(δ − τ )

On the other hand, for τ > τ0 , the above inequality is obvious. To sum up, we get A(r, τ ) ≤ 2πh2 (q−2)(δ−τ ) log r for all τ . Replacing A(r, τ ) in the above inequality by its explicit expression, we have (q − 2)S(r, ϕ0 − τ, ϕ0 + τ, f ) q  16Kπh2 log r n(r, ϕ0 , δ, ai ) + ≤ (q − 2)(δ − τ ) i=1 +(q − 2)S(1, ϕ0 − τ, ϕ0 + τ, f ) + hL(1, f ) + hL(r, f ).

3

The proof of Theorem 1.2

To prove Theorem 1.2, we also need the following connection between S(r, f ) and L(r, f ). Lemma 3.1 [1], [15] Let f be an K-quasimeromorphic function in C. Then for arbitrary η, 0 < η < 1/2, 1−2η L(r, f ) ≤ [S(r, f )] 2 is valid for r∈EK , where EK is a set of finite logarithmic measure. Because quasimeromorphic mappings form a subset of quasimeromorphic functions, Lemma 3.1 is valid for quasimeromorphic mappings. Now, we are in the position to prove Theorem 1.2. ) Proof of Theorem 1.2 Since lim sup S(r,f log r = +∞, there exists an increasing sequence r→∞

{rn }, rn → ∞(rn → ∞) such that

lim

r→∞

S(rn , f ) = +∞. log rn

(3.1)

Using the finite covering theorem on [0, 2π], there exists surely some θ0 ∈ [0, 2π] such that for any 0 < τ < π/2, S(rn , θ0 − τ, θ0 + τ, f ) > 0. (3.2) lim sup S(rn , f ) rn →∞ Now, we prove that J : arg z = θ0 is a precise S direction. If the above statement is false, then there exist three distinct points a1 , a2 , a3 ∈ C∞ and a positive δ such that 3

lim sup i=1 r→∞

n(r, θ0 , δ, ai ) S(r, f )

= 0.

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Therefore, we have

3

lim

i=1

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n(rn , θ0 , δ, ai )

rn →∞

S(rn , f ) For any 0 < τ < δ, using Theorem 1.1, we have

= 0.

(3.3)

(q − 2)S(rn , θ0 − τ, θ0 + τ, f ) ≤

3 

n(rn , θ0 , δ, ai ) +

i=1

16Kπh2 log rn (q − 2)(δ − τ )

+(q − 2)S(1, θ0 − τ, θ0 + τ, f ) + hL(1, f ) + hL(rn , f ). Here, we can suppose that {rn } does not belong to EK stated in Lemma 3.1. Dividing by S(rn , f ) on both sides of the above inequality and taking supper limit, using (3.1)–(3.3) and Lemma 3.1, we can get a contradiction. Hence J is a precise S direction of f (z).

4

Concluding Remark

By using Theorem 1.1, we can derive the main result which was obtained by Deng [16], Yang and Liu [17]. Here we omit the details. Recently, Wu and Sun [18] proved that every quasimeromorphic function of finite positive order has at least one Borel direction of maximal kind, that is, a direction such that for any ε > 0 and any a ∈ C∞ , possibly except at most two values of a, we have n(r, θ, ε, a) lim sup > 0, U (r) r→∞ where U (r) is a type function of f (z). By using Theorem 1.1, we can derive also the existence of the Borel direction of maximal kind for a quasimeromorphic function with infinite order (or zero order) growth. Most recently, Wang and Gao [19] confirms the existence of T direction dealing with multiple values for an algebroid function w(z) under the condition (2) and an additional condition of the lower order to be finite. Then, they ask whether it is also the case without this additional condition. Wu and Sun [20] confirm this problem by proving the following Theorem 4.1. Theorem 4.1 Let w(z) be a ν-valued algebroid function defined on the whole complex plane and satisfy (1.2). Then, there at least exists a ray L : arg z = θ such that, for any given b ∈ C∞ , possibly with the exception of at most 2ν values of b, for an arbitrary small ε > 0, we have l) N (r, θ, ε, b) lim sup > 0, T (r, w) r→∞ l)

for any positive integer l ≥ 3. Here, N (r, θ0 , δ, a) is the intergrated counting function of nl) (r, θ0 , δ, a) and nl) (r, θ0 , δ, a) be the number distinct zeros with multiplicity ≤ l of w(z) − a

0 , δ)  {| in (θ z | ≤ r}. As the end of this article, we pose the following question. Question Let f (z) be a transcendental K-quasimeromorphic mapping on C and satisfies (1.3), whether there is a S direction dealing with multiple values? Let f (z) be a transcendental K-quasimeromorphic mapping on C and satisfies (1.2), whether there is a T direction dealing with multiple values?

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