Heinz Type Inequalities for Poisson Integrals

In 1958, E. Heinz obtained a lower bound for |∂xF|2+|∂yF|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\partial _x F|^2+|\partial _y F|^2$$\end{document}, where F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F$$\end{document} is a one-to-one harmonic mapping of the unit disk onto itself keeping the origin fixed. We show various variants of Heinz’s inequality in the case where F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F$$\end{document} is the Poisson integral of a function of bounded variation in the unit circle. In particular, we obtain such inequalities for F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F$$\end{document} when it is a locally injective quasiregular mapping or an injective mapping of the unit disk onto a bounded convex domain in the complex plane.

as well as (1.6) hold for every ζ ∈ D and all R 1 , R 2 > 0 satisfying D(0, R 1 ) ⊂ ⊂ D(0, R 2 ). Theorem 1.2 ([13,Thm. 2.2]) Given an injective harmonic mapping F of D onto a bounded convex domain including 0, assume that F(0) = 0 and |∂ F(0)|−|∂ F(0)| > 0. Then for all R 1 as well as (1.8) Note that if = D(0, R) for some R > 0, then Theorem 1.2 implies the inequality as well as If f is an integrable function on T, then we denote by P (1.12) Here and in what follows integrable means integrable in the sense of Lebesgue.
Our main goal is to improve this theorem by dropping the assumption that f is Dini smooth; cf. Theorem 4.4. We also present a number of related results dealing with estimates of Heinz type. In Sect. 2, we consider the general case of f being of bounded variation; cf. Theorem 2.3 and Corollary 2.4. Section 3 deals with the case where f is a regular mapping. Assuming that P[ f ](D) is a bounded convex domain we improve Theorems 1.3 and 1.1 by dropping the regularity of f . This is done in Sect. 4. The next section is devoted to study the subject under the assumption that P[ f ] is a locally injective quasiregular mapping, i.e. P[ f ] is locally injective and K -quasiregular for some K ≥ 1, which means that cf. e.g. [1, p. 25]. In the last section, we present a few applications of the earlier results.

The General Case
If f is a function of bounded variation, then we write P[d f ](ζ ) for the Poisson-Stieltjes integral of f at ζ ∈ D, i.e.
We recall that the harmonic conjugate operator A is defined for a function f : T → C integrable on T and z ∈ T as follows: as well as

Theorem 2.3
Given a function f : T → C of bounded variation assume that the for a certain p > 0. Then for every ζ ∈ D, Proof Fix a function f : T → C satisfying the assumption. From Lemmas 2.1 and 2.2 it follows that the radial limit lim r →1 − ∂ F(r z) exists for a.e. z ∈ T and by (2.6), If λ = 0 then the inequality (2.9) is obvious. Thus we may assume that λ > 0. By assumption, ∂ F is a holomorphic non-vanishing function in D. Therefore, we may define the function G : and we deduce from (2.11) and (2.10) that Hence and by (2.12), This implies the inequality (2.9), which completes the proof.
From Theorem 2.3, we can infer a number of lower estimates of |∂ F| in D. Let J[F] stand for the Jacobian of a differentiable mapping F : D → C, i.e. (2.14) Proof Fix a function f : T → C satisfying the assumption. Since F is a harmonic and locally injective mapping in D, it follows from Lewy's theorem that the Jacobian J[F] does not vanish on D; cf. [10]. Therefore, This shows the second inequality in (2.15). The last inequality in (2.15) is obvious, which proves the corollary.

The Smooth Case
In this section, we study the case where the function f in Theorem 2.3 is fairly regular.
Proof Fix a function f : T → C satisfying the assumption. Then f is absolutely continuous and thus the equality (2.4) holds. From (1.12), it follows that for every ζ ∈ D \ {0}, and consequently, Integrating by parts we conclude from (3.2) and (3.3) that Note that the equalities (3.4) hold for ζ = 0, too. Sinceḟ is a Dini continuous function on T, so isḟ . Applying now [15,Prop. 3.4] we conclude from the equalities (3.4) that both the functions ∂ F and∂ F have continuous extensions F 1 and F 2 to the closure cl(D), respectively. Since the limit lim r →1 − d dr P[ f ](r z) exists for every z ∈ T, and by Lemma 2.1,

Hence, A[ḟ ] is a continuous function in T.
Suppose that F 1 (z) = 0 for a certain z ∈ T. By (2.16), |∂ F(r z)| 2 < |∂ F(r z)| 2 for r ∈ [0; 1), and so Thus F 2 (z) = 0, and by (2.6) and (2.7) we obtain f (z) = 0 and A[ḟ ](z) = 0. This clearly forces the inequalities (3.1). Therefore, we may assume that |F 1 (z)| > 0 for every z ∈ T. From this and (2.17) we see that Since the function F 1 is continuous in the compact set cl(D), we conclude that 1/∂ F is a bounded function in D. Therefore, the condition (2.8) holds for p := 1. Since both the functions f and A[ḟ ] are continuous in T, we infer from Corollary 2.4 the inequalities (3.1), which completes the proof.
Proof Fix a function F : D → C satisfying the assumption. Given R ∈ (0; 1) we define the function T z → f (z) := F(Rz). Since the function D ζ → F(Rζ ) is harmonic in D and the function f is continuous in T, we see that Hence, By (3.7) and (2.5) we have (3.10) By the regularity of F we see thatḟ is a Dini continuous function in T. By the assumption and (3.7) it follows that P[ f ] is a locally injective mapping. Applying now Theorem 3.1 we infer from (3.8), (3.9) and (3.10) the inequalities in (3.6) for any ζ ∈ D(0, R), which proves the corollary.

The Case of a Mapping of Bounded Convex Image
We now focus our attention to Poisson integrals mapping the unit disk onto bounded convex domains. We will enhance Theorem 1.3. If ξ ∈ T satisfies the condition

2)
then the following limits exist and as well as and Re (4.8) where the constants a and b are given by (4.5). In particular, and Re Proof Fix f and z satisfying the assumption and assume that there exists ξ ∈ T satisfying the condition (4.2). From (2.5) and (4.3) it follows that which gives (4.7). From (2.6), (2.7) and (4.4) it follows that which leads to (4.8).
Since the function f is of bounded variation, there exists the derivative f (z) for a.e. z ∈ T. From Lemma 2.2 it follows that the limit lim r →1 − d dr P[ f ](r z) exists for a.e. z ∈ T. Moreover, for any point z ∈ T where f is differentiable, f is continuous at z, and so F(ζ ) → f (z) as D ζ → z. From this and the injectivity of F it follows that f (z) / ∈ . Since is a bounded convex domain, we see by Remark 4.2 that the point z satisfies the condition (4.2) for a certain ζ ∈ T. Therefore, the assumptions of Lemma 4.3 hold for a.e. z ∈ T. By Theorem 1.2, the inequality (1.7) holds for every ζ ∈ D. Thus 1/∂ F is a bounded function, and consequently the condition (2.8) is satisfied for any p > 0. Applying now Corollary 2.4 we deduce from (4.9) and (4.10) the first inequality in (4.13) for every ζ ∈ D. The second inequality in (4.13) follows directly from the inequality tan x ≥ x for x ∈ [0; π/2).
If  0 and F a (D) is a bounded convex domain. Therefore, the mapping F a satisfies the inequality in (4.13) for every ζ ∈ D with F and f replaced by F a and f a , respectively. Since ∂ F a = ∂ F in D and f = f a in T we obtain the estimate (4.13) without the assumption F(0) = 0, which completes the proof.

The Case of Quasiregularity
We are able to improve estimates which are obtained so far, provided the Poisson integral P[ f ] is a quasiregular mapping. It will be done by employing the following lemma.

Lemma 5.1 Given K ≥ 1 and a function f : T → C of bounded variation assume that F := P[ f ] is a K -quasiregular mapping. Then
for a.e. z ∈ T, to be specific, for z ∈ T where f is differentiable and the limit lim r →1 − d dr P[ f ](r z) exists.
Proof Fix K ≥ 1 and a function f : T → C satisfying the assumptions. Let z ∈ T be a point where f is differentiable and the limit lim r →1 − d dr P[ f ](r z) exists. Since F is a K -quasiregular mapping, we conclude from Lemma 2.1 that which implies the second inequality in (5.1). Likewise, which implies the first inequality in (5.1).
The following theorem corresponds to Corollary 2.4.
Proof Fix K ≥ 1 and a function f : T → C satisfying the assumptions. Since the mapping F is K -quasiregular, we see by (1.13) that Hence for every ζ ∈ D,

Applications
In this section, we provide a few applications of the results obtained in the previous sections.
Remark 6.1 All the estimates (2.9), (2.15), (3.1), (3.6), (4.13) and (5.2) are applicable under the assumption that F is a harmonic mapping of D onto a Jordan domain bounded by a rectifiable Jordan curve , which has a continuous and injective extension F * to the closure cl( ). Then the restriction f := F * |T is a function of bounded variation. Since F * is a continuous mapping in cl(D), it follows from [6,Thm. 2.11] and the maximal principle that the function F can be uniquely recovered from its boundary limiting valued function f by means of the Poisson integral, i.e. F = P[ f ]. Therefore, we can use the relevant results from the previous sections to get these estimates. Since the mapping F is locally injective and has continuous extension F * to the closure cl(D) such that F * |T = f , we conclude from the argument principle for topological mappings that F is an injective mapping; cf. also [2,Thm. 2.7]. Therefore, we can use respective results from the previous sections to get the estimates (2.9), (2.15), (3.1), (3.6), (4.13) and (5. for all R 1 , R 2 > 0 satisfying the condition (4.14) .  Since f = f a in T we obtain the inequalities (6.1) without the assumption F(0) = 0, which completes the proof. Theorem 6.4 Given K ≥ 1 and R 1 , R 2 > 0 let F be a K -quasiconformal and harmonic mapping of D onto a convex domain such that D(F(0), Proof Fix K , R 1 , R 2 and F satisfying the assumption. Since is a bounded convex domain, it is a Jordan domain bounded by a rectifiable Jordan curve ; cf. [4]. By assumption, F is a quasiconformal mapping of D onto . Therefore, it has a uniquely determined homeomorphic extension F * onto the closure cl(D) and F * (cl(D)) = cl( ) = ∪ ; cf. [9]. From Remark 6.1 it follows that the inequalities (4.13) holds. Then applying Lemma 6.3 we see that for every ζ ∈ D, which leads to (6.2). Proof Let F be a mapping satisfying the assumption. As in Corollary 2.4 we see that the condition (2.16) holds, and so the second dilatation of F is a well-defined holomorphic function as well as |ω(ζ )| < 1 , ζ ∈ D.
Remark 6.6 An easy computation shows that for every differentiable function F : D → C, Therefore, any lower estimate of |∂ F| in D leads to one of |∂ x F| 2 + |∂ y F| 2 in D. For example, under the assumptions of Theorem 4.4 the estimate holds for all R 1 , R 2 > 0 satisfying the condition (4.14). If additionally F is a Kquasiconformal mapping for a given K ≥ 1, we conclude from (6.4) and Theorem 6.4 that |∂ x F(ζ )| 2 + |∂ y F(ζ )| 2 ≥ 2 K + 1 K Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.