Schwarz type Lemmas and a Landau type theorem of functions satisfying the biharmonic equation

Accepted Manuscript Schwarz type Lemmas and a Landau type theorem of functions satisfying the biharmonic equation

Shaolin Chen, Jian-Feng Zhu

PII: DOI: Reference:

S0007-4497(19)30022-3 https://doi.org/10.1016/j.bulsci.2019.01.015 BULSCI 2781

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Bulletin des Sciences Mathématiques

Received date:

1 April 2018

Please cite this article in press as: S. Chen, J.-F. Zhu, Schwarz type Lemmas and a Landau type theorem of functions satisfying the biharmonic equation, Bull. Sci. math. (2019), https://doi.org/10.1016/j.bulsci.2019.01.015

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SCHWARZ TYPE LEMMAS AND A LANDAU TYPE THEOREM OF FUNCTIONS SATISFYING THE BIHARMONIC EQUATION SHAOLIN CHEN AND JIAN-FENG ZHU Abstract. For g ∈ C(D), ϕ ∈ C(T) and f ∗ ∈ C(T), let BHϕ,g,f ∗ (D) be the class of all complex-valued functions f ∈ C 4 (D) satisfying Δ(Δf ) = g in the unit disk D with Δf = ϕ in the unit circle T and f = f ∗ in T, where Δf is the Laplacian of f . In this article, we will show that several Schwarz type lemmas for f ∈ BHϕ,g,f ∗ (D), and, by applying these results, we will establish a Landau type theorem on a subclass of BHϕ,g,f ∗ (D).

1. Preliminaries and main results This paper continues the study of previous work of [11] and is mainly motivated by the articles of Kalaj [21], and the monograph of [28]. In order to state our main results, we need to recall some basic definitions and some results which motivate the present work. Let C ∼ = R2 be the complex plane. For a ∈ C and r > 0, we let D(a, r) = {z : |z − a| < r}, the open disk with center a and radius r. For convenience, we use Dr to denote D(0, r) and D the open unit disk D1 . Let T be the unit circle, i.e., the boundary ∂D of D and D = D ∪ T. Also, we denote by C m (D) the set of all complex-valued m-times continuously differentiable functions from D into C, where D is a subset of C and m ∈ N0 := N ∪ {0}. In particular, let C(D) := C 0 (D), the set of all continuous functions in D. For a real 2 × 2 matrix A, we use the matrix norm A = sup{|Az| : |z| = 1} and the matrix function λ(A) = inf{|Az| : |z| = 1}. For z = x + iy ∈ C, the formal derivative of the complex-valued functions f = u + iv is given by   ux uy Df = . vx vy Then,

  Df  = |fz | + |fz | and λ(Df ) = |fz | − |fz |,

File: BULSCI2781_source.tex, printed:

28-1-2019, 12.20

2000 Mathematics Subject Classification. Primary: 31A30; Secondary: 31A05. Key words and phrases. Schwarz’s Lemma, Landau type theorem, biharmonic equation.

1

2

Shaolin Chen and Jian-feng Zhu

where fz =

  1 1 ∂f ∂f = fx − ify and fz = = fx + ify . ∂z 2 ∂z 2

Moreover, we use Jf := det Df = |fz |2 − |fz |2 to denote the Jacobian of f . For z, ζ ∈ D with z = ζ and |z| + |ζ| = 0, let   2  1 − zζ   and P (z, eit ) = 1 − |z| G(z, ζ) = log  z−ζ  |1 − ze−it |2 be the Green function and Poisson kernel, respectively, where t ∈ [0, 2π]. For g ∈ C(D), ϕ ∈ C(T) and f ∗ ∈ C(T), let BHϕ,g,f ∗ (D) be the class of all complexvalued functions f ∈ C 4 (D) satisfying Δ(Δf ) = g in D with Δf = ϕ in T and f = f ∗ in T, where ∂ 2f ∂ 2f + = 4fzz Δf := ∂x2 ∂y 2 stands for the Laplacian of f . In particular, if g ≡ 0, then f ∈ BHϕ,0,f ∗ (D) is a biharmonic function in D. By [15, Theorem 1] (or [2, Theorem 1]), we see that all f ∈ BHϕ,g,f ∗ (D) can be represented by f (z) = Pf ∗ (z) + G1 [ϕ](z) − G2 [g](z),

(1.1) where

1 Pf ∗ (z) = 2π

(1.2) 1 (1.3) G1 [ϕ](z) = 8π (1.4)



2π 0



2π

P (z, eiθ )f ∗ (eiθ )dθ,

0

log(1 − ze−iθ ) log(1 − zeiθ ) ϕ(eiθ )dθ, (1 − |z| ) 1 + + ze−iθ zeiθ 

2



1 2|ζ − z|2 G(z, ζ) + (1 − |z|2 )(1 − |ζ|2 ) G2 [g](z) = 16π D   log(1 − zζ) log(1 − zζ) g(ζ)dσ(ζ) + zζ zζ

and dσ denotes the Lebesgue area measure in D. We refer the reader to [16, 17, 27] for more discussions in this line. Heinz in his classical paper [18] proved the following result, which is called the Schwraz Lemma of complex-valued harmonic functions: If f is a complex-valued harmonic function from D into itself satisfying the condition f (0) = 0, then for z ∈ D, 4 |f (z)| ≤ arctan |z|. π

Schwarz type Lemmas and a Landau type theorem

3

Later, by removing the assumption f (0) = 0, Pavlovi´c [28, Theorem 3.6.1] got the following sharp form     4 1 − |z|2   (1.5) f (z) − 1 + |z|2 f (0) ≤ π arctan |z|, where f is a complex-valued harmonic function from D into itself. On the several dimensional case, see [21, 22]. Theorem 1. Let f ∈ BHϕ,g,f ∗ (D). Then, for z ∈ D, we have     1 − |z|2 4f ∗ ∞ (1 − |z|2 )ϕ∞  ∗ (0) P ≤ (1.6) f (z) − arctan |z| + τ (z) f  1 + |z|2 π 4 √ g∞ (1 − |z|4 ) g∞ 3π(1 − |z|2 ) + , + 64 96 where f ∗ ∞ = supζ∈T |f ∗ (ζ)|, ϕ∞ = supξ∈T |ϕ(ξ)|, g∞ = supz∈D |g(z)|,   12

12  |z|2 π2 2 log(1 − t) τ (z) = −1 − 2 ≤ dt −1 |z| 0 t 3 and Pf ∗ is defined in (1.2). Furthermore, if f ∗ ∈ C(T), then the inequality (1.6) is sharp in T. We remark that if ϕ∞ = g∞ = 0 in Theorem 1, then (1.6) coincides with (1.5). The classical Schwarz lemma at the boundary is as follows (cf. [13, p.42]). Theorem A. Suppose f : D → D is a holomorphic function with f (0) = 0, and, further, f is analytic at z = 1 with f (1) = 1. Then, the following two conclusions hold: (a) f  (1) ≥ 1; (b) f  (1) = 1 if and only if f (z) ≡ z. Recently, the study of Schwarz type lemmas at the boundary has been attracted much attention. For example, Burns and Krantz [6], Krantz [23], Liu and Tang [26] explored many versions of the Schwarz lemma at the boundary point of holomorphic functions, and reviewed several results of many authors. On the discussions of the boundary Schwarz type lemmas for harmonic functions, we refer the reader to [19, 21, 22]. In the following, by using Theorem 1, we establish a Schwarz type lemma at the boundary for f ∈ BHϕ,g,f ∗ (D). Theorem 2. Let f ∈ BHϕ,g,f ∗ (D) satisfying f (0) = 0 and f ∗ ∞ ≤ 1. (I) If for some ζ ∈ T such that limr→1− |f (rζ)| = 1, then    √

12  2 1 1 π2 3 |f (ζ) − f (rζ)| (3 + 3π) ≥ − + −1 + g∞ , ψ∞ − lim inf r→1− 1−r π 4 2 3 48 64

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Shaolin Chen and Jian-feng Zhu

where r ∈ [0, 1). In particular, if ψ∞ = g∞ = 0, then lim inf − r→1

2 |f (ζ) − f (rζ)| ≥ , 1−r π

and this estimate is sharp. (II) If f is differentiable at z = 1 with f (1) = 1, then    √

12   2  1 1 π2 3 (3 + 3π) + −1 + g∞ , ψ∞ − Re fx (1) ≥ − π 4 2 3 48 64 where “Re” denotes the real part. Especially, if ψ∞ = g∞ = 0, then  2  Re fx (1) ≥ , π and this estimate is sharp. Colonna [12] proved a sharp Schwraz-Pick type lemma for complex-valued harmonic functions, which is as follows: Let f be a complex-valued harmonic function from D into itself. Then, for z ∈ D, (1.7)

Df (z) ≤

1 4 . π 1 − |z|2

We extend (1.7) into the following form. Theorem 3. Let f ∈ BHϕ,g,f ∗ (D). Then, for z ∈ D, we have (1.8)

 2

12  1 π 4f ∗ ∞ ϕ∞ 1 + Df (z) ≤ + − 1 |z| π 1 − |z|2 2 2 3 ⎡   12 ⎤ 1 2 π 1 22 1 + 6 ⎥ ⎢ 1 (1 − |z|2 ) 2 + |z|⎦ . +g∞ ⎣ + 8 30 16

In particular, if ϕ∞ = g∞ = 0, then the extreme functions

1 + φ(z) 2αf ∞ arg f (z) = π 1 − φ(z) show that the estimate of (1.8) is sharp, where |α| = 1 is a constant and φ is a conformal automorphism of D. We remark that if ϕ∞ = g∞ = 0 in Theorem 3, then (1.8) coincides with (1.7). Denote by H(D) the set of all holomorphic functions f in D satisfying the standard normalization: f (0) = f  (0) − 1 = 0. In the early 20th century, Landau [24] showed that there is a constant r > 0, independent of elements in H(D), such that f (D) contains a disk of radius r. Later, the Landau theorem has become an important tool in geometric function theory (cf. [5, 30]). Unfortunately, for general classes of

Schwarz type Lemmas and a Landau type theorem

5

functions, there is no Landau type theorem (cf. [14, 29]). To establish analogs of the Landau type theorem for more general classes of functions, it is necessary to restrict our focus on certain subclasses (cf. [1, 3, 4, 7, 8, 9, 10, 29]). The following two results are the Landau type Theorems of complex-valued harmonic functions and complex-valued biharmonic functions, respectively. Theorem B. ([7, Theorem 1]) Let f be a complex-valued harmonic function of D such that f (0) = 0, Jf (0) = 1 and |f (z)| < M in D. Then, f is univalent on Dρ0 with ρ0 = π 3 /(64mM 2 ), and f (Dρ0 ) contains an univalent disk DR0 with R0 = πρ0 /(8M ) = π 4 /(512mM 3 ), where m ≈ 6.85 is the minimum of the function (3 − r2 )/[r(1 − r2 )] in (0, 1). Theorem C. ([1, Theorem 1]) Suppose that f (z) = |z|2 H(z) + K(z) is a complexvalued biharmonic fucntion, that is Δ(Δf ) = 0, in D such that f (0) = K(0) = Jf (0) − 1 = 0, where H and K are the complex-valued harmonic functions (i.e., ΔH = ΔK = 0) and max{|H(z)|, |K(z)|} < M with M being a positive constant. Then, there is a constant ρ2 ∈ (0, 1) such that f is univalent in Dρ2 , and f (Dρ2 ) contains a disk DR2 , where ρ2 satisfies the following equation: ρ2 π − 2M ρ2 − 4M = 0, 4M (1 − ρ2 )2 and R2 =

π ρ 4M 2

− 2M (ρ32 + ρ22 )/(1 − ρ2 ).

For convenience, we make a notational convention: For ϕ ∈ C(T), g ∈ C(D) and f ∗ ∈ C(T), let BH(M1 , M2 , M3 ) denote the class of all complex-valued functions f ∈ BHϕ,g,f ∗ (D) satisfying f (0) = Jf (0) − 1 = 0, f ∗ ∞ ≤ M1 , ϕ∞ ≤ M2 and g∞ ≤ M3 , where M1 > 0, M2 ≥ 0 and M3 ≥ 0 are constants. For x ∈ [0, 1), set (1.9)

M2 x ψ(x) = 2

 

12

π2 −1 3

+ 1

187 2 2 +M3 x + 240

(1 −  12 2 1 + π6



16



1+x 1

x2 ) 2 +

+

4M1 x(2 − x) π (1 − x)2



 

1+x

1

30(1 − x2 ) 2

.

It is easy to see that ψ is a strictly monotonically increasing function of [0, 1) onto [0, +∞). In the following, by Theorem 3, we establish a Landau type Theorem without the harmonicity assumption or biharmonicity assumption, which is an improvement of Theorems B and C. Theorem 4. Suppose that ψ is defined in (1.9). If f ∈ BH(M1 , M2 , M3 ), then f is r0 , where univalent in Dr0 , and f (Dr0 ) contains an univalent disk DR0 with R0 ≥ 2M 4 r0 satisfies the following equation:

4 M2 19 M1 + + M3 ψ(r0 ) = 1. π 4 120

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Shaolin Chen and Jian-feng Zhu

Remark 1.1. In general, the Landau type theorem is not true for the complexvalued functions in BHϕ,g,f ∗ (D) without any boundedness condition. For example, let fk (z) = kx + |z|4 /16 + iy/k, where z = x + iy ∈ D and k ∈ {1, 2, . . .}. Then, for all k ∈ {1, 2, . . .}, fk is univalent and Jfk (0) − 1 = fk (0) = 0. But, there is no absolute constant R0 > 0 such that DR0 belongs to fk (D) for each k. The proofs of Theorems 1, 2 and 3 will be presented in Section 2, and the proof of Theorem 4 will be given in Section 3. 2. The Schwarz type lemmas of f ∈ BHϕ,g,f ∗ (D) We begin this part with the following result which will play an important role in the proof of Theorem 1. Theorem D. (cf. [25]) For z ∈ D, we have

2  2π ∞  dθ Γ(n + α) 1 = |z|2n , 2π 0 |1 − zeiθ |2α n!Γ(α) n=0 where α > 0 and Γ denotes the Gamma function. Proof of Theorem 1. Let f ∈ BHϕ,g,f ∗ (D). Applying (1.5), we see that    4f ∗ ∞  1 − |z|2  ∗ (z) − ∗ (0) ≤ arctan |z|, P (2.1) P f f   1 + |z|2 π     where Pf ∗ is defined in (1.2). Next, we estimate G1 [ϕ] and G2 [g], where G1 [ϕ] and G2 [g] are defined in (1.3)  and (1.4), respectively. First, we estimate G1 [ϕ]. Let   2π   1 log(1 − ze−iθ ) log(1 − zeiθ )   + 1+ I1 (z) =  dθ. 2π 0  ze−iθ zeiθ By H¨older’s inequality, for z ∈ D, we get 

 12   −iθ iθ 2  log(1 − ze log(1 − ) ze ) 1 +  dθ + I1 (z) ≤   −iθ iθ ze ze 0  12  12    |z|2 ∞  |z|2(k−1) log(1 − t) 2 dt = −1 + 2 = −1 − 2 , 2 k |z| t 0 k=1 1 2π



2π

which implies that (2.2)

2   G1 [ϕ](z) ≤ (1 − |z| )ϕ∞ I1 (z) 4   12  |z|2 2 log(1 − t) (1 − |z|2 )ϕ∞ −1 − 2 dt . ≤ 4 |z| 0 t

Schwarz type Lemmas and a Landau type theorem

  Now we estimate G2 [g]. Let g∞ I2 (z) = 8π and g∞ (1 − |z|2 ) I3 (z) = 16π



 D

|ζ − z|2 |G(z, ζ)|dσ(ζ)

   log(1 − zζ) log(1 − zζ)    dσ(ζ). + (1 − |ζ| )   zζ zζ D 2

Set ξ=

z−ζ . 1 − zζ

Then (2.3)

|Jζ (ξ)| =

(1 − |z|2 )2 z−ξ ξ(|z|2 − 1) , ζ = and ζ − z = . |1 − zξ|4 1 − zξ 1 − zξ

By Theorem D, we obtain 1 2π



2π 0

 (n + 1)2 (n + 2)2 1 dθ = |z|2n ρ2n , |1 − zρeiθ |6 4 n=0 ∞

which, together with (2.3), yields that  (1 − |z|2 )4 2 g∞ 1 (2.4) I2 (z) = |ξ| log dσ(ξ) 6 8π D |1 − zξ| |ξ|  2π

2 4  1 1 dθ 1 g∞ (1 − |z| ) ρ3 log dρ = iθ 6 4 2π 0 |1 − zρe | ρ 0  ∞ g∞ (1 − |z|2 )4  (n + 1)2 (n + 2)2 2n 1 2n+3 1 |z| = ρ log dρ 4 4 ρ 0 n=0 ∞ g∞ (1 − |z|2 )4  (n + 1)2 (n + 2)2 2n = |z| 64 (n + 2)2 n=0 ∞ g∞ (1 − |z|2 )4  = (n + 1)2 |z|2n 64 n=0

g∞ (1 − |z|2 )4 1 + |z|2 64 (1 − |z|2 )3 g∞ (1 − |z|4 ) . = 64 =

Let 1 A(ρ) = 2π



2π 0

   ∞  (zρe−iθ )n−1  2Re dθ.  n n=1

7

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Shaolin Chen and Jian-feng Zhu

Then, by H¨older’s inequality, we get       12

1 ∞ ∞ 2π  −iθ n−1 2  (zρe ) |z|2(n−1) ρ2(n−1) 2 1   2Re = 2+2 , A(ρ) ≤  dθ 2π 0  n n2 n=1 n=2 which implies that (2.5)

g∞ (1 − |z|2 ) I3 (z) = 8



1 0

ρ(1 − ρ2 )A(ρ)dρ

1  ∞  g∞ (1 − |z|2 ) 1 1 2 2 ≤ ρ(1 − ρ ) 2 + 2 dρ 8 n2 0 n=2 √ g∞ 3π(1 − |z|2 ) . = 96

Hence, by (2.4) and (2.5), we obtain √ g∞ (1 − |z|4 ) g∞ 3π(1 − |z|2 ) + , |G2 [g](z)| ≤ I2 (z) + I3 (z) ≤ 64 96 which, together with (2.1) and (2.2), gives that     2 2       f (z) − 1 − |z| Pf ∗ (0) ≤ Pf ∗ (z) − 1 − |z| Pf ∗ (0) + G1 [ϕ](z) + |G2 [g](z)|     2 2 1 + |z| 1 + |z| ∗ 4f ∞ arctan |z| ≤ π   12  |z|2 2 log(1 − t) (1 − |z|2 )ϕ∞ −1 − 2 + dt 4 |z| 0 t √ g∞ (1 − |z|4 ) g∞ 3π(1 − |z|2 ) + . + 64 96 Now we prove the sharpness part. If f ∗ ∈ C(T), then Pf ∗ is harmonic in D and continuous in D. Hence there exists θ0 ∈ [0, 2π] such that |f ∗ (eiθ0 )| = f ∗ ∞ , which implies that   iθ0 2   ∗ iθ f (e 0 ) − 1 − |e | Pf ∗ (0) = f ∗ ∞ .   1 + |eiθ0 |2 The proof of this theorem is complete. Proof of Theorem 2. We first prove (I). It follows from Theorem 1 that (1 − |z|2 )ϕ∞ g∞ (1 − |z|4 ) 4 (2.6) |f (z)| ≤ c(|z|) := arctan |z| + τ (z) + 4 64 √π

1 − |z|2 g∞ 3π(1 − |z|2 ) + |Pf ∗ (0)|, + 96 1 + |z|2



Schwarz type Lemmas and a Landau type theorem

where

 τ (z) =

2 −1 − 2 |z|



|z|2 0

log(1 − t) dt t

 12

≤

9

12

π2 −1 3

and Pf ∗ is defined in (1.2). Since x2 log x1 − (1 − x2 ) ≤ 0 for x ∈ (0, 1], we see that  2π      ϕ∞ 1 iθ  G1 [ϕ](0) ≤ ϕ(e )dθ ≤  8π 0 4 and        2 G2 [g](0) ≤ g∞ |ζ| log 1 − (1 − |ζ|2 ) dσ(ζ)  8π D  |ζ|   g∞ 3g∞ 1 2 2 = dσ(ζ) = . (1 − |ζ| ) − |ζ| log 8π D |ζ| 64 It follows from f (0) = 0 and (1.1) that     ϕ∞ 3g∞ (2.7) |Pf ∗ (0)| ≤ G1 [ϕ](0) + G2 [g](0) ≤ + . 4 64 For r ∈ (0, 1), it follows from (2.6), (2.7) and lim−

r→1

1−

4 π

arctan r 2 = 1−r π

that

√

12 2 1 π2 (3 + 3π) 1 − c(r) ≥ − |Pf ∗ (0)| − − 1 ψ∞ − g∞ lim r→1− 1 − r π 2 3 48    √

12  2 (3 + 3π) 1 1 π2 3 g∞ ≥ − + −1 + ψ∞ − π 4 2 3 48 64

which yields that lim inf − r→1

|f (ζ) − f (rζ)| ≥ 1−r

1 − c(r) r→1 1−r    √

12  1 1 π2 3 (3 + 3π) 2 − + −1 + g∞ . ψ∞ − ≥ π 4 2 3 48 64 lim−

If ϕ∞ = g∞ = 0, then the sharpness part easily follows from [21, Theorem 2.6]. Next, we prove (II). Since f is differentiable at z = 1, we see that f (z) = 1 + fz (1)(z − 1) + fz (1)(z − 1) + o(|z − 1|), where o(x) denotes a function with lim o(x) = 0. This, together with (2.6), shows x→0 x that 2Re [fz (1)(1 − z) + fz (1)(1 − z)] ≥ 1 − c2 (|z|) + o(|z − 1|),

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Shaolin Chen and Jian-feng Zhu

which, together with z = r ∈ [0, 1) and (2.7), gives that   Re fx (1) ≥

1 − c(r) r→1 1−r    √

12  1 1 π2 3 (3 + 3π) 2 − + −1 + g∞ . ψ∞ − ≥ π 4 2 3 48 64 lim−

Now we prove the sharpness part for ϕ∞ = g∞ = 0. For z ∈ D, let f (z) =

2Re(z) 2 arctan , π 1 − |z|2

⎧ ⎨f ∗ (ξ) = lim f (rξ) = 1, ξ ∈ T and Re(ξ) = 0, − r→1

⎩f ∗ (ξ) = lim f (rξ) = 0, − r→1

ξ ∈ T and Re(ξ) = 0,

and ϕ = Δf ≡ 0 in T. It is not difficult to know that f is harmonic in D (i.e., Δf ≡ 0 in D) and f ∗ is a bounded integrable function in T satisfying f (0) = 0, g = Δ(Δf ) ≡ 0 in D and fx (1) =

2 . π

The proof of this theorem is complete.  The proof of Theorem 3 needs the following result (cf. [20, Proposition 2.4]). Theorem E. Suppose that X is an open subset of R, and (Ω, μ) denotes a measure space. Suppose, further, that a function F : X × Ω → R satisfies the following conditions: (1) F (x, w) is a measurable function of x and w jointly, and is integrable with respect to w for almost every x ∈ X. (2) For almost every w ∈ Ω, F (x, w) is an absolutely continuous function with respect to x. (This guarantees that ∂F/∂x exists almost everywhere.) (3) ∂F/∂x is locally integrable, that is, for all compact intervals [a, b] contained in X,   b  ∂   F (x, w) dμ(w)dx < ∞.   Ω ∂x a  Then, Ω F (x, w)dμ(w) is an absolutely continuous function with respect to x, and for almost every x ∈ X, its derivative exists, which is given by   ∂ d F (x, w)dμ(w). F (x, w)dμ(w) = dx Ω Ω ∂x

Schwarz type Lemmas and a Landau type theorem

11

Proof of Theorem 3. Let f ∈ BHϕ,g,f ∗ (D). By calculation, we have (2.8)

fz =

∂ ∂ ∂ Pf ∗ + G1 [ϕ] − G2 [g] ∂z ∂z ∂z

fz =

∂ ∂ ∂ Pf ∗ + G1 [ϕ] − G2 [g]. ∂z ∂z ∂z

and (2.9)

It follows from (2.8) and (2.9) that, for z ∈ D, Df (z) ≤ DPf ∗ (z) + DG1 [ϕ] (z) + DG2 [g] (z).

(2.10)

Step 2.1. Now we begin to estimate DG1 [ϕ] . Let (|z|2 − 1) L1 (z) = 8π and z L2 (z) = − 8π





2π 0

2π 0





eiθ log(1 − ze−iθ ) 1 + ϕ(eiθ )dθ z(1 − ze−iθ ) z2

log(1 − ze−iθ ) log(1 − zeiθ ) 1+ ϕ(eiθ )dθ. + ze−iθ zeiθ

First, we estimate |L1 (z)|. Since (1 − |z|2 )ϕ∞ |L1 (z)| ≤ 8π



2π 0

 ∞  (n − 1)(ze−iθ )n−2 e−iθ     dθ,    n n=2

and since H¨older’s inequality implies 1 2π



2π 0

⎛  2 ⎞ 12 ∞  2π  ∞  (n − 1)(ze−iθ )n−2 e−iθ  −iθ n−2  1 (n − 1)(ze )      dθ ≤ ⎝  dθ⎠ ,      n=2 n 2π 0  n=2 n

we see that

∞  12 (1 − |z|2 )ϕ∞  (n − 1)2 2(n−2) |z| . |L1 (z)| ≤ 4 n2 n=2

Let

 h(x) = (1 − x)

∞  (n − 1)2 n=2

n2

 12 xn−2

For x ∈ [0, 1), let k(x) = h2 (x). Then k(x) =

∞  n=0

d n xn ,

in [0, 1).

12

Shaolin Chen and Jian-feng Zhu

where

⎧ 1 ⎪ ⎪ , ⎪ ⎪ 4 ⎪ ⎪ 1 ⎪ ⎪ ⎨− , 18 dn = 11 ⎪ , − ⎪ ⎪ ⎪ 144 ⎪ ⎪ 4 + 4n − 6n2 − 4n3 ⎪ ⎪ , ⎩ 2 n (n + 1)2 (n + 2)2 √ and thus h0 = maxx∈[0,1) {h(x)} = d0 = 1/2. Then, (2.11)

|L1 (z)| ≤

n = 0, n = 1, n = 2, n>2 we obtain that for z ∈ D,

ϕ∞ . 8

Next, we estimate |L2 (z)|. Since    |z|ϕ∞ 2π  log(1 − ze−iθ ) log(1 − zeiθ )  + |L2 (z)| ≤ 1 +  dθ, 8π ze−iθ zeiθ 0 we obtain from H¨older’s inequality that "   # 12  2π  −iθ iθ 2 log(1 − ze 1 |z|ϕ∞ log(1 − ) ze ) 1 +  dθ |L2 (z)| ≤ + .  4 2π 0  ze−iθ zeiθ Then, it follows from "   # 12  ∞  12  2π  −iθ iθ 2  1 1 ) ze ) log(1 − log(1 − ze 1 +  dθ + ≤ 2 −1  2π 0  ze−iθ zeiθ n2 n=1 that

(2.12)

ϕ∞ |L2 (z)| ≤ 4



12

π2 −1 3

|z|.

Now, (2.11), (2.12) and Theorem E guarantee that

(2.13)

 2    2    ∂     G1 [ϕ](z) =  L (z) ≤ |Lj (z)|  j  ∂z    j=1

j=1

ϕ∞ ϕ∞ + ≤ 8 4



12

π2 −1 3

|z|.

From symmetry, we can get  

12  ϕ∞ ϕ∞ π 2 ∂  G1 [ϕ](z) ≤ + − 1 |z|. (2.14)   ∂z 8 4 3 It follows from (2.13) and (2.14) that

Schwarz type Lemmas and a Landau type theorem

ϕ∞ ϕ∞ DG1 [ϕ]  ≤ + 4 2

(2.15)



12

π2 −1 3

13

|z|.

Step 2.2. Next, we estimate DG2 [g] . For this, let

 1 (z − ζ)G(z, ζ)g(ζ)dσ(ζ), L3 (z) = 8π D  1 ∂ |ζ − z|2 G(z, ζ)g(ζ)dσ(ζ), L4 (z) = 8π D ∂z   1 log(1 − zζ) log(1 − zζ) 2 L5 (z) = − g(ζ)dσ(ζ), z(1 − |ζ| ) + 16π D zζ zζ

and 1 L6 (z) = − 16π



log(1 − zζ) 1 g(ζ)dσ(ζ). + (1 − |ζ| )(1 − |z| ) z(1 − zζ) z2ζ D 

2

2

We are going to estimate the norms of Lj (z), where j ∈ {3, 4, 5, 6}. Before these estimates, we need some preparation. Set z−ζ . w= 1 − zζ Then, |Jζ (w)| =

(2.16)

(1 − |z|2 )2 w(|z|2 − 1) , , ζ − z = |1 − zw|4 1 − zw

and 1 − |ζ|2 =

(2.17)

(1 − |w|2 )(1 − |z|2 ) . |1 − zw|2

Firstly, we estimate |L3 (z)|. Since (2.16) and (2.17) guarantee that

 dσ(w) g∞ 1 2 3 , (1 − |z| ) |w| log |L3 (z)| ≤ 8π |w| |1 − zw|5 D by letting w = reiθ , we obtain g∞ |L3 (z)| ≤ (1 − |z|2 )3 4



1 0

1 r log r 2



1 2π



2π 0

dθ |1 − zreiθ |5

dr.

Moreover, H¨older’s inequality and Theorem D show that

12  2π

12  2π  2π dθ dθ dθ 1 1 1 ≤ . 2π 0 |1 − zreiθ |5 2π 0 |1 − zreiθ |4 2π 0 |1 − zreiθ |6 Since



1 2π



2π 0

dθ |1 − zreiθ |4



12 =

∞  n=0

 12 2

(n + 1) |zr|

2n

14

Shaolin Chen and Jian-feng Zhu

and



1 2π



2π

dθ |1 − zreiθ |6

0

we see that



1 2π

2π 0

12 =

∞ 

(n + 1)2

n

n=0

2

 12

2

+ 1 |zr|2n

,

dθ 1 ≤ (n + 1)2 (n + 2)2 |zr|2n , |1 − zreiθ |5 4 n=0 ∞

which implies (2.18)

|L3 (z)| ≤

∞  g∞ g∞ (1 − |z|2 )3 . (n + 1)(n + 2)|z|2n = 64 32 n=0

Secondly, we estimate |L4 (z)|. By Theorem D, we obtain that 1 2π



2π 0

 (n + 1)2 (n + 2)2 1 |z|2n r2n , dθ = |1 − zreiθ |6 4 n=0 ∞

which implies 

1



(2.19) 0

1 2π



2π 0

r2 (1 − r2 ) 1 dθ dr ≤ . iθ 6 |1 − zre | 4(1 − |z|2 )3

Moreover, by (2.16), (2.17), and by applying w = reiθ , we have

  1  2π 2 1 |z − ζ|(1 − |ζ|2 ) r (1 − r2 ) 1 2 3 dσ(ζ) = (1 − |z| ) dθ dr, 2π D 2π 0 |1 − zreiθ |6 |1 − zζ| 0 which, together with (2.19), yields  |z − ζ|(1 − |ζ|2 ) g∞ g∞ |L4 (z)| ≤ (2.20) . dσ(ζ) = 16π D 32 |1 − zζ| Next, we estimate |L5 (z)|. Let   2π  ∞ ∞ −it n−1 it n−1   (zρe ) (zρe ) 1   + B1 (z, ρ) =  dt.   2π 0  n=1 n n n=1 It follows from H¨older’s inequality that ⎛ B1 (z, ρ) ≤ ⎝

1 2π



2π 0

2 ⎞ 12 ∞

1 ∞  (zρe−it )n−1  it n−1  1 π2 2 (zρe )  ⎠  + ≤ 22 1 + .  dt    n n 6 n=1 n=1

By letting ζ = ρeit , we get

(2.21) |L5 (z)| ≤

|z|g∞ 8



1 0

ρ(1 − ρ2 )B1 (z, ρ)dρ ≤

2

1 2

 1+

π2 6

 12

32

g∞

|z|.

Schwarz type Lemmas and a Landau type theorem

15

Finally, we estimate |L6 (z)|. Let     ∞  1 2π  (n − 1) n−2 n−1 −it(n−1)  1  2 ρ(1 − ρ ) z ρ e B2 (z) =   dt dρ.  2π 0  n=2 n 0 By using H¨older’s inequality, we obtain that ⎡ 2 ⎤ 12  2π   1 ∞ 1 (n − 1) n−2 n−1 −it(n−1)  ⎦  B2 (z) ≤ z ρ e ρ(1 − ρ2 ) ⎣   dt dρ.  2π 0  n=2 n 0 Since 1 2π



2π 0

∞ 2 ∞  (n − 1)   (n − 1)2   z n−2 ρn−1 e−it(n−1)  dt = ρ2 (|z|ρ)2(n−2)  2   n n n=2 n=2 ≤ ρ2

∞ 

|z|2(n−2) ,

n=2

we see that  B2 (z) ≤

1 0

 ρ2 (1 − ρ2 )

∞ 

 12 |z|2n

dρ =

n=0

2 1

15(1 − |z|2 ) 2

,

which imples 1

(1 − |z|2 )g∞ g∞ (1 − |z|2 ) 2 B2 (z) ≤ . 8 60 Therefore, by (2.18), (2.20), (2.21), (2.22) and Theorem E, we conclude that (2.22)

|L6 (z)| ≤

⎡   12 ⎤ 1 2   π 1 6 22 1 + 6 ∂   (1 − |z|2 ) 2 ⎢1 ⎥  G2 [g](z) ≤ |L (z)| ≤ g + + |z|⎦ . ⎣ j ∞  ∂z  16 60 32 j=3 From symmetry, we can prove

⎡  1 ⎤ 1   π2 2 1 2 2 1+ 6 ∂  1 (1 − |z|2 ) 2 ⎥  G2 [g](z) ≤ g∞ ⎢ + |z|⎦ , ⎣ +  ∂z  16 60 32

which implies that ⎡ 2

(2.23)

1 2

⎢ 1 (1 − |z| ) + DG2 [g]  ≤ g∞ ⎣ + 8 30

 1 22 1 + 16

π2 6

 12

⎤ ⎥ |z|⎦ .

Hence (1.8) follows from (1.7), (2.10), (2.15) and (2.23). The proof of this theorem is complete. 

16

Shaolin Chen and Jian-feng Zhu

3. The Landau type theorem for f ∈ BH(M1 , M2 , M3 ) The following result will be used in the proof of Theorem 4. Lemma F. ([9, Lemma 1]) Suppose f is a harmonic mapping of D into C such that |f (z)| ≤ M and ∞ ∞   n an z + bn z n , f (z) = n=0

n=1

where M is a positive constant. Then, |a0 | ≤ M and for all n ≥ 1, 4M . π This estimate is sharp, and the extreme function is

⎧ n ⎨ 2M α1 arg 1 + β1 z , |α1 | = |β1 | = 1, π 1 − β1 z n fn (z) = ⎩ M, |an | + |bn | ≤

if n ≥ 1, if n = 0.

Proof of Theorem 4. We will prove this theorem in six steps. Step 3.1. We first estimate     ∂  ∂  ∂ ∂ I4 (z) :=  G1 [ϕ](z) − G1 [ϕ](0) and I5 (z) :=  G1 [ϕ](z) − G1 [ϕ](0) . ∂z ∂z ∂z ∂z Let −z A1 (z) = 8π



2π

1−

0

A2 (z) =

∞  (ze−iθ )n−1

n

n=1

|z|2 8π

and 1 A3 (z) = 8π



∞ 2π  0



n=2

∞ 2π  0

n=3

∞  (zeiθ )n−1 − ϕ(eiθ )dθ, n n=1

(n − 1) −iθ n−2 −iθ (ze ) e ϕ(eiθ )dθ n

(n − 1) −iθ n−2 −iθ (ze ) e ϕ(eiθ )dθ. n

In order to estimate I4 (z), we first estimate |Aj (z)|, where j ∈ {1, 2, 3}. By H¨older’s inequality, for z ∈ D, we have  1  2π  ∞ ∞ −iθ n−1 iθ n−1 2   2  (ze ) (ze ) |z|ϕ∞ 1 1 −  dθ |A1 (z)| ≤ −   4 2π 0 n n n=1 n=1   12 ∞  |z|2(k−1) |z|ϕ∞ = −1 + 2 4 k2 k=1

1 |z|ϕ∞ π2 2 ≤ , −1 + 4 3

Schwarz type Lemmas and a Landau type theorem

17

 1  2π   2  ∞ (n − 1) −iθ n−2 2 |z|2 ϕ∞ 1   dθ |A2 (z)| ≤ ) (ze  4 2π 0  n=2 n ∞ ∞  12  12 |z|2 ϕ∞  (n − 1)2 2(n−2) |z|2 ϕ∞  2(n−2) = |z| ≤ |z| 4 n2 4 n=2 n=2 =

|z|2 ϕ∞ 1

4(1 − |z|2 ) 2

and  1  2π   2  ∞ (n − 1) −iθ n−3 2 |z|ϕ∞ 1  (ze )  dθ |A3 (z)| ≤  4 2π 0 n n=3 ∞  12 |z|ϕ∞  (n − 1)2 2(n−3) = |z| 4 n2 n=3 ≤

|z|ϕ∞ 1

4(1 − |z|2 ) 2

,

which imply that

(3.1)

ϕ∞ |z| I4 (z) = |A1 (z) + A2 (z) − A3 (z)| ≤ 4



12

π2 −1 3

+



1 + |z| 1

(1 − |z|2 ) 2 By using the similar proof process of the above inequality, we get ϕ∞ |z| I5 (z) ≤ 4

(3.2)



12

π2 −1 3

+

.



1 + |z| 1

(1 − |z|2 ) 2

.

Step 3.2. Now, we estimate     ∂  ∂  ∂ ∂ I6 (z) :=  G2 [g](z) − G2 [g](0) and I7 (z) :=  G2 [g](z) − G2 [g](0) . ∂z ∂z ∂z ∂z For this, let

 1 (z − ζ)G(z, ζ)g(ζ)dσ(ζ), A4 (z) = 8π D  ∂ 1 A5 (z) = |ζ − z|2 G(z, ζ)g(ζ)dσ(ζ), 8π D ∂z   log(1 − zζ) log(1 − zζ) 1 2 + z(1 − |ζ| ) A6 (z) = − g(ζ)dσ(ζ), 16π D zζ zζ

and 1 A7 (z) = − 16π



1 log(1 − zζ) g(ζ)dσ(ζ). (1 − |ζ| )(1 − |z| ) + z(1 − zζ) z2ζ D 

2

2

18

Shaolin Chen and Jian-feng Zhu

By [20, Lemma 2.7], we obtain    1 2 (3.3) |Gw (w, ζ)| + |Gw (w, ζ)| dσ(ζ) ≤ . 2π D 3 By direct calculation, we get   |G(z, ζ) − G(0, ζ)| = 

[0,z]

   dG(w, ζ) ≤

[0,z]

(|Gw (w, ζ)| + |Gw (w, ζ)|)|dw|,

which, together with Fubini’s Theorem and (3.3), gives that    1 G(z, ζ) − G(0, ζ)dσ(ζ) A8 (z) := 2π D

    1 |Gw (w, ζ)| + |Gw (w, ζ)| |dw| dσ(ζ) ≤ 2π D [0,z]

    dσ(ζ) 2 |Gw (w, ζ)| + |Gw (w, ζ)| |dw| ≤ |z|. = 2π 3 [0,z] D

(3.4)

Since Theorem D implies 1 2π by letting w = 1 2π



z−ζ 1−zζ



2π 0

 dt = (n + 1)2 |z|2n r2n , |1 − zreit |4 n=0 ∞

= reit , we obtain

1 G(z, ζ)dσ(ζ) = 2π D

 D

1 log |w|

(1 − |z|2 )2 (1 − |z|2 ) , dσ(w) = |1 − zw|4 4

which gives that (3.5)

   g∞ |z|(1 − |z|2 ) 1  ≤ . zG(z, ζ)g(ζ)dσ(ζ)  8π  D 16

It follows from (3.4) and (3.5) that

(3.6)

   g∞ 1  zG(z, ζ)g(ζ)dσ(ζ) + |A4 (z) − A4 (0)| ≤ A8 (z)  8π D 4  g∞ |z|(1 − |z|2 ) ≤ + A8 (z) 4 4   11 − 3|z|2 |z| . ≤ g∞ 48

By computation, we obtain

Schwarz type Lemmas and a Landau type theorem

19

  2   − zζ ) (z   g(ζ)dσ(ζ)  (|ζ|2 − 1)  D  1 − zζ  (1 − |ζ|2 )(1 + |ζ|2 ) g∞ |z| dσ(ζ) ≤ 16π 1 − |ζ| D 77|z| g∞ . = 480   2π  ∞ ∞ (zρe−it )n−1  (zρeit )n−1  1  A9 (z, ρ) = +  dt.   2π 0  n=1 n n n=1

1 |A5 (z) − A5 (0)| = 16π

(3.7)

Let

It follows from H¨older’s inequality that ⎛ A9 (z, ρ) ≤ ⎝



1 2π

2π 0

 2 ⎞ 12

1 ∞ ∞  −it n−1 it n−1 2 2  (zρe )  1 π (zρe )  . +   dt⎠ ≤ 2 2 1 +  n=1  n n 6 n=1

By letting ζ = ρeit , we get |z|g∞ |A6 (z) − A6 (0)| ≤ 8

(3.8)

g∞ 2

≤ Let

 A10 (z, ρ) =

and

1 0

 A11 (z, ρ) =

1 0

1 0







1 ρ(1 − ρ ) 2π 1 ρ(1 − ρ2 ) 2π

2π 0



2π 0

ρ(1 − ρ2 )A9 (z, ρ)dρ

1+ 32

2



1 2



π2 6

 12 |z|.

   ∞  (n − 1) n−2 n−1 −it(n−1)   z ρ e   dt dρ   n n=2    ∞  (n − 1) n−2 n−1 −it(n−1)   z ρ e   dt dρ.   n n=3

By using H¨older’s inequality, we obtain that ⎡ 2 ⎤ 12  2π   1 ∞  (n − 1) 1   A10 (z, ρ) ≤ z n−2 ρn−1 e−it(n−1)  dt⎦ dρ ρ(1 − ρ2 ) ⎣    2π n 0 0 n=2 and  A11 (z, ρ) ≤ Since

1 0

⎡ ρ(1 − ρ2 ) ⎣

1 2π



2π 0

∞ 2 ⎤ 12  (n − 1)   n−2 n−1 −it(n−1)  ρ e z   dt⎦ dρ.  n=3  n

20

Shaolin Chen and Jian-feng Zhu

1 2π



2π 0

∞ 2 ∞  (n − 1)   (n − 1)2  n−2 n−1 −it(n−1)  2 z ρ e dt = ρ (|z|ρ)2(n−2)   2   n n n=2 n=2 2

≤ ρ

∞ 

|z|2(n−2)

n=2

and 1 2π



2π 0

∞ 2 ∞  (n − 1)   (n − 1)2  n−2 n−1 −it(n−1)  2 (|z|ρ)2(n−2) z ρ e   dt = ρ 2  n=3  n n n=3 2

≤ ρ

∞ 

|z|2(n−2) ,

n=3

we see that  A10 (z, ρ) ≤ and

1 0

 A11 (z, ρ) ≤

1 0

 2

2

ρ (1 − ρ )

∞ 

 12 |z|

2n

dρ =

n=0

 ρ2 (1 − ρ2 )

∞ 

2 1

15(1 − |z|2 ) 2

 12 |z|2n

dρ =

n=1

2|z| 1

15(1 − |z|2 ) 2

,

which yield that    ∞  (n − 1) n−2 n−1  g∞ |z|2 2  (3.9) |A7 (z) − A7 (0)| ≤ (1 − |ζ| )  z ζ  dσ(ζ)   16π n D n=2    ∞  g∞ (n − 1) n−2 n−1  2  + (1 − |ζ| )  z ζ  dσ(ζ)  n=3  16π D n g∞ |z|2 g∞ A10 (z, ρ) + A11 (z, ρ) 8 8  g∞ |z| + |z|2 ≤ . 1 60(1 − |z|2 ) 2 =

Hence, by (3.6), (3.7), (3.8), (3.9) and Theorem E, we get   7     (3.10) I6 (z) =  Aj (z) − Aj (0)    j=4



2

(187 − 30|z| ) ≤ g∞ |z| + 480

 1 22 1 +

By the similar proof process, we can prove

32

π2 6

 12

 +

1 + |z|

  1

60(1 − |z|2 ) 2

.

Schwarz type Lemmas and a Landau type theorem



(3.11)

2

(187 − 30|z| ) I7 (z) ≤ g∞ |z| + 480

 1 22 1 +

π2 6

 12

32

21

 +

1 + |z|

  1

60(1 − |z|2 ) 2

.

Step 3.3. Next, we estimate     I8 (z) := [Pf ∗ (z)]z − [Pf ∗ (0)]z  and I9 (z) := [Pf ∗ (z)]z − [Pf ∗ (0)]z . Since Pf ∗ is harmonic in D, we see that it can be written as the following form: P (z) =

∞ 

f∗

n

an z +

n=0

∞ 

bn z n .

n=1

By applying Lemma F, we have

(3.12)

 ∞  ∞       n−1  n−1  I8 (z) + I9 (z) =  nan z  +  nbn z      n=2 n=2 ∞ ∞    n−1 4f ∗ ∞  ≤ n |an | + |bn | |z| ≤ n|z|n−1 π n=2 n=2

=

4f ∗ ∞ |z|(2 − |z|) . π (1 − |z|)2

Step 3.4. In this step, we begin to estimate I10 (z) := |fz (z) − fz (0)| and I11 (z) := |fz (z) − fz (0)|. Since f ∗ ∞ ≤ M1 , ϕ∞ ≤ M2 and g∞ ≤ M3 , by (1.1), (3.1), (3.2), (3.10), (3.11) and (3.12), we see that

(3.13)



12 π2 1 + |z| + −1 1 3 (1 − |z|2 ) 2  1  1    π2 2 2 1 + 2 1 + |z| 6 187 + + +M3 |z| 1 240 16 30(1 − |z|2 ) 2

9 

M2 |z| I10 (z) + I11 (z) ≤ Ij (z) ≤ 2 j=4

+



4M1 |z|(2 − |z|) , π (1 − |z|)2

where M1 > 0, M2 ≥ 0 and M3 ≥ 0 are constants. Step 3.5. We will prove f is univalent in Dr0 , where r0 satisfies the following equation: 1 ψ(r0 ) − 4M1 M2 = 0, 19 + 4 + 120 M3 π where ψ is defined in (1.9).

22

Shaolin Chen and Jian-feng Zhu

It follows from Theorem 3 that λ(Df (0)) =

(3.14)

1 Jf (0) = ≥ Df (0) Df (0)

1 4M1 π

+

M2 4

+

19 M3 120

.

To prove the univalence of f in Dr0 , we choose two points z1 = z2 ∈ Dr0 . Then, by (3.13), we have    |fz (z) − fz (0)| + |fz (z) − fz (0)| |dz| ≤ |z2 − z1 |ψ(r0 ) [z1 ,z2 ]

and, by (3.14), we see that    

[z1 ,z2 ]

  fz (0)dz + fz (0)dz  ≥ λ(Df (0))|z2 − z1 | ≥

1 4M1 π

+

M2 4

+

19 M3 120

|z2 − z1 |,

which give that   |f (z2 ) − f (z1 )| ≥ 

[z1 ,z2 ]

  fz (0)dz + fz (0)dz 

  fz (z) − fz (0) dz + fz (z) − fz (0) dz  [z1 ,z2 ]   1 ≥ |z2 − z1 | 4M1 M2 − ψ(r0 ) > 0. 19 + 4 + 120 M3 π   − 









Thus, from the arbitrariness of z1 and z2 , the univalence of f follows. Step 3.6. At last, we will prove f (Dr0 ) contains an univalent disk DR0 with the radius R0 satisfying r0 R0 ≥ 8M1 M2 19 . + 2 + 60 M3 π To reach this goal, let ζ = r0 eiθ ∈ ∂Dr0 . Then, by (3.13), we have  I12 (ζ) :=

[0,ζ]

 |fz (z) − fz (0)| + |fz (z) − fz (0)| |dz|



  2

12 π M2 1 + r0 −1 |z||dz| + + |z||dz| ≤ 1 2 3 (1 − r02 ) 2 [0,ζ] [0,ζ]  1  1    π2 2 2 1 + 2 1 + r0 6 187 r0 ψ(r0 ) +M3 |z||dz| = + + 1 240 16 2 30(1 − r02 ) 2 [0,ζ] 4M1 (2 − r0 ) π (1 − r0 )2



and, by (3.14), we see that      fz (0)dz + fz (0)dz  ≥ λ(Df (0))r0 ≥  [0,ζ]

4M1 π

r0 + + M2 4

19 M3 120

,

Schwarz type Lemmas and a Landau type theorem

23

which give that   |f (ζ) − f (0)| ≥ 

[0,ζ]

  fz (0)dz + fz (0)dz  − I12 (ζ) ≥

The proof of this theorem is complete.

8M1 π

+

r0 . M2 + 19 M3 2 60 

Acknowledgements: We thank the referee for providing constructive comments and help in improving this paper. This research was partly supported by the Science and Technology Plan Project of Hengyang City (No. 2018KJ125), the National Natural Science Foundation of China (No. 11571216), the Science and Technology Plan Project of Hunan Province (No. 2016TP1020), the Science and Technology Plan Project of Hengyang City (No. 2017KJ183), and the Application-Oriented Characterized Disciplines, Double First-Class University Project of Hunan Province (Xiangjiaotong [2018]469). References 1. Z. Abdulhadi and Y. Abu Muhanna, Landau’s theorem for biharmonic mappings, J. Math. Anal. Appl., 338 (2008), 705–709. 2. H. Begehr, Dirichlet problems for the biharmonic equation, Gen. Math., 13 (2005), 65–72. 3. S. Bochner, Bloch’s theorem for real variables, Bull. Amer. Math. Soc., 52 (1946), 715–719. 4. M. Bonk and A. Eremenko, Covering properties of meromorphic functions, negative curvature and spherical geometry, Ann. Math., 152 (2000), 551–592. 5. R. Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc., 235 (1978), 213– 219. 6. D. M. Burns and S. G. Krantz, Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer. Math. Soc., 7 (1994), 661–676. 7. H. Chen, P.M. Gauthier, and W. Hengartner, Bloch constants for planar harmonic mappings, Proc. Amer. Math. Soc., 128 (2000), 3231–3240. 8. H. Chen and P. M. Gauthier, Bloch constants in several variables, Trans. Amer. Math. Soc., 353 (2001), 1371–1386. 9. Sh. Chen, S. Ponnusamy and X. Wang, Bloch constant and Landau’s theorems for planar p-harmonic mappings, J. Math. Anal. Appl., 373 (2011), 102–110. 10. Sh. Chen and M. Vuorinen, Some properties of a class of elliptic partial differential operators, J. Math. Anal. Appl., 431 (2015), 1124–1137. 11. Sh. Chen and X. Wang, Bi-Lipschitz characteristic of quasiconformal self-mappings of the unit disk satisfying bi-harmonic equation, submitted. 12. F. Colonna, The Bloch constant of bounded harmonic mappings, Indiana Univ. Math. J., 38 (1989), 829–840. 13. J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. 14. P. M. Gauthier and M. R. Pouryayevali, Failure of Landau’s theorem for quasiconformal mappings of the disc, Contemporay Math., 355 (2004), 265–268. 15. W. K. Hayman and B. Horenblum, Representation and uniqueness theorems for polyharmonic functions, J. Anal. Math., 60 (1993), 113–133. 16. H. Hedenmalm, A computation of Green functions for the weighted biharmonic Green functions Δ|z|−2α Δ, Duke Math. J., 75 (1994), 51–78. 17. H. Hedenmalm, S. Jakobsson and S. Shimorin, A biharmonic maximum principle for hyperbolic surfaces, J. Reine Angew. Math., 550 (2002), 25–75. 18. E. Heinz, On one-to-one harmonic mappings, Pacific J. Math., 9 (1959), 101–105.

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Shaolin Chen and Jian-feng Zhu

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