Heinz type inequalities for mappings satisfying Poisson’s equation

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Heinz type inequalities for mappings satisfying Poisson’s equation Deguang Zhong ∗, Hongqiang Tu School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, PR China Received 28 December 2017; received in revised form 26 August 2018; accepted 29 August 2018 Communicated by J.J.O.O. Wiegerinck

Abstract In this paper, we study the Heinz type inequalities for mappings satisfying Poisson’s equation. Some results generalize the ones obtained by Partyka and Sakan. c 2018 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝ Keywords: Heinz type inequality; Poisson’s equation; Quasiconformality

1. Introduction

1

Let D be an open unit disk in the complex plane C. Let D and D ′ be subdomains of C and C(D) be the set of all continuous functions defined in D. A real-valued function u, defined in an open subset D of the complex plane C, is real harmonic if it is twice continuously differentiable in D and satisfies Laplace’s equation: ∂ u ∂ u (z) + 2 (z) = 0, z ∈ D. ∂x2 ∂y A complex-valued function w = u + iv is harmonic if both u and v are real harmonic. We say that a function u : D ↦→ R is absolutely continuous on line in the region D if for every closed rectangle R ⊂ D with sides parallel to the x and y-axes, u is absolutely continuous on a.e. horizontal line and a.e. vertical line in R. Such a function has, of course, partial derivatives 2

2 3 4 5

2

∆u(z) =

∗ Corresponding author.

E-mail addresses: [email protected] (D. Zhong), [email protected] (H. Tu). https://doi.org/10.1016/j.indag.2018.08.006 c 2018 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝

Please cite this article in press as: D. Zhong, H. Tu, Heinz type inequalities for mappings satisfying Poisson’s equation, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.08.006.

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u x and u y a.e. in D. A sense-preserving topological mapping ω = u + iv : D → D ′ , between open regions D, D ′ ⊂ C, is said to be K -quasiconformal (K ≥ 1) if it satisfies : (a) ω is absolutely continuous on lines in D; (b) there is constant K ≥ 1, such that

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11 12 13

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17 18 19 20

21

22 23

24

(1)

where k = ∂ω = − iω y ) and ∂ω = Let P be the Poisson kernel, i.e. the function 1 (ωx 2

K −1 , K +1

P(z, eiθ ) =

8

9

for a.e. z ∈ D,

|∂ω(z)| ≤ k|∂ω(z)|

1 − |z|2 |z − eiθ |2

1 (ωx 2

+ iω y ).

, z ∈ D, θ ∈ R,

and let G be the Green function of the unit disk, i.e. the function ⏐ ⏐ ⏐ 1 − zw ⏐ 1 ⏐ ⏐ , z, w ∈ D, z ̸= w. log ⏐ G(z, w) = 2π z−w ⏐ Let f : T → C be a bounded integrable function on the unit circle T = {z ∈ C : |z| = 1} and let g : D → C be continuous. The solution of the Poisson’s equation ∆ω = g in D satisfying the boundary condition ω|T = f ∈ L 1 (T) is given by ω(z) = P[ f ](z) − G[g](z), z ∈ D,

(2)

where P[ f ](z) =

1 2π

2π

∫

P(z, eiϕ ) f (eiϕ )dϕ, G[g](z) = 0

∫

G(z, w)g(w)dm(w),

(3)

D

where m denotes the Lebesgue measure in the plane. It is well known that if f and g are continuous in T and in D respectively, then the mapping ω = P[ f ] − G[g] has a continuous extension ω˜ to the boundary, and ω˜ = f on T, see [4]. It was proved in [1] that for any g ∈ C(D) the function G := G[g] satisfies the following inequalities ∥g∥∞ ∥g∥∞ ∥g∥∞ · (1 − |z|2 ), |∂G(z)| ≤ and |∂G(z)| ≤ , (4) 4 3 3 where ∥g∥∞ = supz∈D |g(z)|. Write Hom+ (T) for the class of all sense-preserving homeomorphic self-mappings of T. Given f ∈ Hom+ (T), set |G(z)| ≤

d f := essinf| f ′ (z)|, z∈T

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where for each z ∈ T, f (u) − f (z) f ′ (z) = lim u→z u−z provided the limit exists and f (z) = 0 otherwise. If f ∈ Hom+ (T), from [7, Lemma 2.1] it follows that for a.e. z ∈ T both the functions ∂ P[ f ] and ∂ P[ f ] have radial limiting values at z and the following equalities hold ⎧ [ ] f (z) − P[ f ](r z) ⎪ ′ ⎪ ⎪ 2z lim ∂ P[ f ](r z) = lim + z f (z) , ⎨ r →1− 1−r r →1− (5) [ ] ⎪ f (z) − P[ f ](r z) ⎪ ′ ⎪ − z f (z) . ⎩ 2z lim− ∂ P[ f ](r z) = lim 1−r r →1 r →1− Please cite this article in press as: D. Zhong, H. Tu, Heinz type inequalities for mappings satisfying Poisson’s equation, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.08.006.

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Thus we may define ⏐ ⏐ ⏐ ⏐ ∗ ⏐ d f := essinf ⏐ lim ∂ P[ f ](r z)⏐⏐ . z∈T

3 1

(6)

r →1−

In 1958, E. Heinz proved that the inequality

3

2 (7) π2 holds for every z = x + i y ∈ D, if F is an univalent harmonic mapping of D onto itself satisfying F(0) = 0; cf. [3]. Recent papers [8–11,14] improved the above Heinz inequality from various ways. In particular, Partyka and Sakan improved the Heinz inequality for two cases: In the first one F = P[ f ] for some f ∈ Hom+ (T); see [8, Theorem 2.2]. In the second one F is a quasiconformal mapping; see [8, Theorem 3.2]. In this paper, we consider the above two theorems [8, Theorem 2.2] and [8, Theorem 3.2] for the case of the mapping ω of D onto itself satisfying Poisson’s equation ∆ω = g for a given g ∈ C(D). The organization at the rest of this paper is as follows: In section 2 we prove a lemma which is a counterpart of [8, Theorem 2.2] with removed the assumption F(0) = 0. Then we use this lemma to derive a version of [8, Theorem 2.2] for mapping satisfying Poisson’s equation. Next we will give counterparts of [8, Lemma 3.1 and Theorem 3.2] with removed the assumption F(0) = 0 and use them to obtain a version of [8, Theorem 3.2] for mappings satisfying Poisson’s equation. |∂x F(z)|2 + |∂ y F(z)|2 ≥

2. Main results

4

5 6 7 8 9 10 11 12 13 14 15 16 17

18

By virtue of [12, Theorem 3.6.1], the assumption of F(0) = 0 in [8, Theorem 2.2] can be removed as presented by the following lemma. For the completeness, we will give the proof which is a small modification of the proof of [8, Theorem 2.2] given by Partyka and Sakan. Lemma 2.1. If f ∈ Hom+ (T) and if F := P[ f ] satisfies |F(0)| < ⏐ ⏐ ⏐1 |F(0)| ⏐⏐2 1 2 1 3 2 ⏐ inf |∂ F(z)| ≥ ⏐ − + df + df z∈D π 2 ⏐ 4 2

2

2 , π

then

19 20 21

22

(8)

and

23

24

⏐ ⏐ ⏐1 |F(0)| ⏐⏐2 1 2 2 2 ⏐ inf {|∂x F(z)| + |∂ y F(z)| } ≥ 2⏐ − + d f + d 3f . z∈D π 2 ⏐ 2 Proof. Know from the proof of [8, Theorem 2.2], the following equality ⏐ ⏐ ⏐ f (z) − F(r z) ⏐2 1 1 ′ 2 1 2 ⏐ ⏐ + lim J [F](r z) lim |∂ F(r z)| = | f (z)| + lim ⏐ ⏐ 4 4 r →1− 1−r 2 r →1− r →1−

(9)

25

26

(10)

hold for a.e. z ∈ T. Since F is harmonic on D and F(D) = D, so from [12, Theorem 3.6.1] we have ⏐ ⏐ 2 ⏐ ⏐ ⏐ F(z) − 1 − |z| F(0)⏐ ≤ 4 arctan|z|, z ∈ D. (11) ⏐ ⏐ π 2 1 + |z| Please cite this article in press as: D. Zhong, H. Tu, Heinz type inequalities for mappings satisfying Poisson’s equation, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.08.006.

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Hence for every z ∈ T and r ∈ (0, 1), we have ⏐ ⏐ ⏐ f (z) − F(r z) ⏐ ⏐ ⏐ ⏐ ⏐ 1−r ⏐ ⏐ 2 ⏐ ⏐ 1−r 2 1 − 1−r z) − |F(0)| − F(0) ⏐F(r ⏐ 2 2 1+r 1+r ≥ 1−r 2 4 arctan r |F(0)| − 1 − 1−r 2 π 1+r 2 ≥ → − |F(0)| as r → 1− . 1−r π By [8, Theorem 1.2], we have

(12)

lim J [F](r z) ≥ d 3f

4

r →1− 5

6

7

for a.e. z ∈ T. Combining this with (10) and (12), we obtain ⏐2 ⏐ ⏐ 1 1 1⏐ 2 (d ∗f )2 ≥ ⏐⏐ − |F(0)|⏐⏐ + d 2f + d 3f . 4 π 4 2 So from [8, Lemma 2.1], we can derive (8). In addition, since 2

|∂x F(z)|2 + |∂ y F(z)|2 = 2(|∂ F(z)|2 + |∂ F(z)| ), z ∈ D,

8

9

10

11 12

13

thus (9) holds. This completes the proof. □ From Lemma 2.1, we can derive the following theorem. Theorem 2.2. For given g ∈ C(D) and f ∈ Hom+ (T) let ω ∈ C(D) be a twice continuously differentiable function from D onto itself √⏐satisfying ⏐∆ω = g in D, ω|T = f and |ω(0)| < ⏐ ⏐2 1 2 1 3 2 − ∥g∥4 ∞ . Let F := P[ f ]. If ∥g∥∞ < 3 ⏐ π1 − |F(0)| + 4 d f + 2 d f , then π 2 ⏐

14

15

(13)

⎞2 ⎛√ ⏐ ⏐2 ⏐ ⏐ |F(0)| ⏐ 1 ∥g∥∞ ⎠ 1 1 inf |∂ω(z)|2 ≥ ⎝ ⏐⏐ − + d 2f + d 3f − z∈D π 2 ⏐ 4 2 3

(14)

inf (|∂x ω(z)|2 + |∂ y ω(z)|2 ) ⎛√ ⎞2 ⏐ ⏐2 ⏐1 ⏐ |F(0)| 1 1 ∥g∥ ∞⎠ ⏐ + d2 + d3 − ≥ 2⎝ ⏐⏐ − . π 2 ⏐ 4 f 2 f 3

(15)

and z∈D

16

Proof. Let G = G[g]. As |ω(0)| < π2 − ∥g∥4 ∞ , |F(0)| ≤ |ω(0)| + |G(0)| < Hom+ (T), by Lemma 2.1 and (4), we have √⏐ ⏐2 ⏐1 ⏐ ⏐ − |F(0)| ⏐ + 1 d 2 + 1 d 3 ⏐π 2 ⏐ 4 f 2 f ∥g∥∞ ≤ |∂ F(z)| ≤ |∂ω(z)|+|∂G(z)| ≤ |∂ω(z)| + . 3

2 . π

Since ω|T = f ∈

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As ∥g∥∞

√⏐ ⏐ < 3 ⏐ π1 −

⏐2

|F(0)| ⏐ 2 ⏐

5

+ 14 d 2f + 21 d 3f , we obtain

1

⎛√ ⎞2 ⏐ ⏐2 ⏐ ⏐ 1 1 1 ∥g∥ |F(0)| ∞ ⏐ + d2 + d3 − ⎠ . |∂ω(z)|2 ≥ ⎝ ⏐⏐ − π 2 ⏐ 4 f 2 f 3

2

Hence, (14) holds, and (15) follows immediately from (14). This completes the proof. □

3

It is well known that a quasiconformal self-mapping F of D has a homeomorphic extension F ∗ to the closure D; cf. [6]. We call the restriction f := F|∗T the boundary limiting valued function of F. Using (11), we also can remove the assumption of F(0) = 0 in [8, Lemma 3.1], and consequently the assumption of F(0) = 0 in [8, Theorem 3.2] can be removed. Generalized forms of these results are given in the following two lemmas. Their proofs are straightforward modifications of the proofs of [8, Lemma 3.1 and Theorem 3.2]. Lemma 2.3. Given K ≥ 1, let F be a K -quasiconformal and harmonic self-mapping of D satisfying |F(0)| < π2 . If f is the boundary limiting valued function of F, then ( ) 1 2 − |F(0)| . (16) df ≥ K π Proof. From (5) it follows that for a.e. z ∈ T, ⎧ f (z) − F(r z) ⎪ ⎨ lim [z∂ F(r z) + z∂ F(r z)] = lim , − 1−r r →1− r →1 ⎪ ⎩ lim [z∂ F(r z) − z∂ F(r z)] = z f ′ (z).

4 5 6 7 8 9

10 11

12

13

(17)

14

r →1−

Since F is K -quasiconformal mapping, we see from (12) and (17) that for a.e. z ∈ T, | f ′ (z)| = lim |z∂ F(r z) − z∂ F(r z)| r →1−

≥ lim (|∂ F(r z)| − |∂ F(r z)|) r →1−

1 K 1 ≥ K 1 = K 1 ≥ K ≥

lim (|∂ F(r z)| + |∂ F(r z)|)

r →1−

lim (|z∂ F(r z) + z∂ F(r z)|) ⏐ ⏐ ⏐ f (z) − F(r z) ⏐ ⏐ lim ⏐⏐ ⏐ 1−r r →1− ( ) 2 − |F(0)| . π

r →1−

This finishes the proof. □ Lemma 2.4. Given K ≥ 1, let F be a K -quasiconformal and harmonic self-mapping of D satisfying |F(0)| < π2 . If f is the boundary limiting valued function of F, then the inequalities ( ) 1+K 2 |∂ F(z)| ≥ − |F(0)| (18) 2K π Please cite this article in press as: D. Zhong, H. Tu, Heinz type inequalities for mappings satisfying Poisson’s equation, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.08.006.

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and |∂x F(z)|2 + |∂ y F(z)|2 ≥

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( )2 (1 + K )2 2 − |F(0)| 2K 2 π

(19)

hold for every z ∈ D. Proof. By (2.7) in [8, Theorem 2.2], we have for a.e. z ∈ T, ⏐ ⏐ ⏐ f (z) − F(r z) ⏐2 2 2 ′ 2 ⏐ . ⏐ 2 lim (|∂ F(r z)| + |∂ F(r z)| ) = | f (z)| + lim ⏐ ⏐ 1−r r →1− r →1−

(20)

Since F is K -quasiconformal mapping, we have 2

2(K 2 + 1)|∂ F(w)|2 ≥ (K + 1)2 (|∂ F(w)|2 + |∂ F(w)| ), w ∈ D.

7

8

9

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13 14

15 16

17 18

19

20

By (12), (16) and (20), we get for a.e. z ∈ T, ( ) ) ( 1 |F(0)| 2 1 2 2 lim (|∂ F(r z)| + |∂ F(r z)| ) ≥ 2 − 1+ 2 . π 2 K r →1−

22

23 24 25 26

27

(22)

From (21) and (22) it follows that for a.e. z ∈ T, ( ) |F(0)| K +1 1 − lim |∂ F(r z)| ≥ . K π 2 r →1− By virtue of [8, Lemma 2.1], we get (18). Thus (19) follows directly from (13) and (18). □ By virtue of Lemma 2.4, we can obtain the Heinz type inequality for K -quasiconformal mapping ω from D onto itself satisfying Poisson’s equation ∆ω = g for a certain g ∈ C(D). Theorem 2.5. Given K ≥ 1 and g ∈ C(D), let ω be a K -quasiconformal twice continuously differentiable self-mapping of D satisfying ∆ω = g and |ω(0)| < π2 − ∥g∥4 ∞ . Let f be the boundary limiting valued function of ω, and let F := P[ f ]. If ∥g∥∞ < inequalities [ ]2 |F(0)| 2 1 3( − ) ∥g∥ 1 + K ∞ π 2 |∂ω(z)|2 ≥ · − √ K 3 3( π1 − |F(0)| ) + 2∥g∥∞ 2

) 3( π1 − |F(0)| √ 2 , 2K

then the

(23)

and [

21

(21)

|∂x ω(z)|2 + |∂ y ω(z)|2 ≥ 2

3( π1 − 3( π1 −

|F(0)| ) 2

|F(0)| 2 ) 2

+

√ 2∥g∥∞

1+K ∥g∥∞ · − K 3

]2 (24)

hold for every z ∈ D. Proof. Let G := G[g], then by (4) we have |F(0)| < π2 . Since ω is a K -quasiconformal mapping, f is a homeomorphism. Hence by Choquet–Rad´o–Kneser theorem F is a diffeomorphism (See [2,5,13]). Thus F is a harmonic diffeomorphism, and by Heinz type inequality (See Remark 1), we have for an arbitrarily fixed z ∈ D, ⏐ ⏐ ⏐ ⏐2 ⏐ 1 |F(0)| ⏐⏐2 2 ⏐ ⏐ ⏐ |∂ F(z)| + ∂ F(z) ≥ ⏐ − , π 2 ⏐ Please cite this article in press as: D. Zhong, H. Tu, Heinz type inequalities for mappings satisfying Poisson’s equation, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.08.006.

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which, in view of the fact that |∂ F(z)| ≥ |∂ F(z)|, implies that √ 2 · |∂ F(z)| ≥ 1. |F(0)| 1 − 2 π

7 1

(25)

In addition, by the quasiconformality of ω we have ⏐ ⏐ ⏐ ∂ F(z) − ∂G(z) ⏐ K −1 ⏐ ⏐ . ⏐ ≤ k := ⏐ ⏐ ∂ F(z) − ∂G(z) ⏐ K +1

2

3

(26)

4

< 1. This shows that F is a

5

· K -quasiconformal mapping of the unit disk onto itself. Thus by Lemma 2.4

6

Combining (25) with (4) and (26), we obtain ⏐ ⏐ ⏐∂ F(z)⏐ ≤ k · |∂ F(z)| + (1 + k)∥g∥∞ √3 2 (1 + k)∥g∥∞ ≤ k · |∂ F(z)| + 1 · · |∂ F(z)| |F(0)| 3 − 2 π ] [ √ 2∥g∥∞ (1 + k) · |∂ F(z)| . = k+ 3( π1 − |F(0)| ) 2 Since ∥g∥∞ < √

3( π1 − |F(0)| ) √ 2 , 2K

3( π1 − |F(0)| 2∥g∥∞ 2 )+ √ 3( π1 − |F(0)| )− 2∥g∥ ∞K 2

√

we have 0 ≤ k +

2∥g∥∞ (1+k) 3( π1 − |F(0)| 2 )

we have

7

)2 3( π1 − |F(0)| 2 √ 3( π1 − |F(0)| ) + 2∥g∥∞ 2 Again, since ∥g∥∞ < 3( π1 − 3( π1

−

3( π1 − |F(0)| ) √ 2 , 2K

|F(0)| 2 ) 2

|F(0)| ) 2

√

+

·

2∥g∥∞

·

∥g∥∞ 1+K ≤ |∂ω(z)| + . K 3

8

thus

9

1+K ∥g∥∞ > . K 3

This implies the inequalities (23) and (24). □ Remark 1. Using (11), we can also remove the assumption of F(0) = 0 in (7) by the following: If F is a one-to-one harmonic mapping of D onto itself and |F(0)| < π2 , then for every z ∈ D, ( ) 1 |F(0)| 2 2 |∂ F(z)|2 + |∂ F(z)| ≥ − . (27) π 2 Acknowledgments This work was completed with the support of the Innovation Research for the Postgraduates of Guangzhou University under Grant No. 2017GDJC-D06. The authors would like to express their sincere thanks to the referees for their great efforts to improve this paper. References [1] S. Chen, S. Ponnusamy, Schwarz’s lemmas for mappings satisfying Poisson’s equation. arXiv:1708.03924v2 [math.CV]. [2] G. Choquet, Sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques, Bull. Sci. Math. 69 (1945) 156–165. Please cite this article in press as: D. Zhong, H. Tu, Heinz type inequalities for mappings satisfying Poisson’s equation, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.08.006.

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[3] E. Heinz, On one-to-one harmonic mappings, Pacific J. Math. 9 (1959) 101–105. [4] L. Hörmander, Notions of Convexity, in: Progress in Mathematics, vol. 127, Birkhäuser Boston Inc., Boston, MA, 1994. [5] H. Kneser, Lösung der Aufgabe 41, Jahresber, Deutsch. Math.-Verein. 35 (1926) 123–124. [6] O. Lehto, K.I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, New York, 1973. [7] D. Partyka, K. Sakan, Quasiconformality of harmonic extensions, J. Comput. Appl. Math. 105 (1999) 425–436. [8] D. Partyka, K. Sakan, On Heinz’s inequality, in: Bull. Soc. Sci. Lett. Łód´z, 52, in: Sér. Rech. Déform., vol. 36, 2002, pp. 27–34. [9] D. Partyka, K. Sakan, On an asymptotically sharp variant of Heinz’s inequality, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 30 (2005) 167–182. [10] D. Partyka, K. Sakan, On a variant of Heinz’s inequality for harmonic mappings of the unit disk onto bounded convex domains, in: Bull. Soc. Sci. Lett. Łód´z, 59, in: Série: Recherches sur les déformations, vol. 59 (2), 2009, pp. 25–36. [11] D. Partyka, K. Sakan, Heinz type inequalities for Poisson integrals, Comput. Methods Funct. Theory 14 (2–3) (2014) 219–236. [12] M. Pavlovi´c, Introduction to function spaces on the disk, in: Posebna Izdanja [Special Editions], Vol. 20, Matematiki Institut SANU, Belgrade, 2004. [13] T. Radó, Aufgabe 41. (Gestellt in Jahresbericht D. M. V. 35, 49) Lösung von H. Kneser, Jahresbericht D. M. V. 35, 1926, pp. 123–124 (in German). [14] J. Zhu, X. Huang, Estimate for Heinz’s inequality in harmonic quasiconformal mappings, J. Math. Anal. Approx. Theory 2 (2007) 52–59.

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