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An asymptotically sharp coefficients estimate for harmonic K-quasiconformal mappings
Journal of Inequalities and Applications volume 2016, Article number: 84 (2016)
Abstract
By using the improved Hübner inequalities, in this paper we obtain an asymptotically sharp lower bound estimate for the coefficients of harmonic K-quasiconformal self-mappings of the unit disk \({\mathbb{D}}\) which keep the origin fixed. The result partly improves the former results given by (Partyka and Sakan in Ann. Acad. Sci. Fenn., Math. 30:167-182, 2005) and (Zhu and Zeng in J. Comput. Anal. Appl. 13:1081-1087, 2011). Furthermore, using some estimate for the derivative of the boundary function of a harmonic K-quasiconformal self-mapping w of \({\mathbb{D}}\) which keeps the origin fixed, we obtain an upper bound estimate for the coefficients of w.
1 Introduction
Let \({\mathbb{D}}=\{z: |z|<1\}\) denote the unit disk, \(w(z)\) be a harmonic mapping defined in \({\mathbb{D}}\). Then \(w(z)\) can be presented as \(w(z)=h(z)+\overline{g(z)}\), where
are both analytic in \({\mathbb{D}}\). By Lewy’s theorem [3], we know that \(w(z)\) is locally univalent and sense-preserving in \({\mathbb{D}}\) if and only if its Jacobian satisfies the following inequality:
for all \(z\in{\mathbb{D}}\). One of the basic properties for harmonic self-mappings of \({\mathbb {D}}\) is the Heinz inequality [4].
Lemma A
Let w map the unit disk harmonically onto itself with \(w(0)=0\). Then
for some absolute constant \(c>0\).
Subsequently, in 1982, Hall [5] obtained the sharp lower bound of c.
Theorem B
Let \(w(z)=h(z)+\overline{g(z)}=\sum_{n=1}^{\infty }a_{n}z^{n}+\overline{\sum_{n=1}^{\infty}b_{n}z^{n}}\) be a univalent harmonic mapping of the unit disk onto itself, then its coefficients satisfy the inequality
The lower bound \(\frac{27}{4\pi^{2}}\) is the best possible.
Let
denote the Poisson kernel, then every bounded harmonic mapping w defined in \({\mathbb{D}}\) has the following representation:
where \(z=re^{i\varphi}\in{\mathbb{D}}\) and f is a bounded integrable function defined on the unit circle \(\mathbf{T}:=\partial{\mathbb{D}}\).
Suppose that \(w(z)\) is a sense-preserving univalent harmonic mapping of \({\mathbb{D}}\) onto a domain \(\Omega\subseteq\mathbb{C}\). Then \(w(z)\) is a harmonic K-quasiconformal mapping if and only if
Under the additional assumption that \(w(z)\) is a K-quasiconformal mapping, in 2005 Partyka and Sakan [1] obtained an asymptotically sharp variant of Heinz’s inequality as follows (see also [2]).
Theorem C
Let \(w(z)\) be a harmonic K-quasiconformal mapping of \({\mathbb{D}}\) onto itself satisfying \(w(0)=0\). Then the inequality
holds for every \(z\in{\mathbb{D}}\), where
is a strictly decreasing function of K. For \(L>0\), \(\Phi_{L}(s)\) is the Hersch-Pfluger distortion function defined by the equalities \(\Phi_{L}(s):=\mu^{-1}(\mu(s)/L) \), \(0< s<1\); \(\Phi_{L}(0):=0 \), \(\Phi_{L}(1):=1\), where \(\mu(s)\) stands for the module of Grötzsch’s extremal domain \({\mathbb{D}}\backslash[0,s]\).
In 2010, Qiu and Ren [6] improved the Hübner inequalities as follows.
Theorem D
For all \(s\in(0,1)\) and \(K\in(1, \infty)\), we have
and
where \(D(s)=(1-s)(1+s)^{1/\ln4}\).
A sense-preserving harmonic mapping of \({\mathbb{D}}\) onto itself can be represented as the Poisson extension of the boundary function \(f(e^{it})=e^{i\gamma(t)}\), where \(\gamma(t)\) is a continuous non-decreasing function with \(\gamma(2\pi )-\gamma(0)=2\pi\) and \(\gamma(t+2\pi)=\gamma(t)+2\pi\) (cf. [7, 8]). The coefficients \(a_{n}\) and \(b_{n}\) have an alternative interpretation as Fourier coefficients of the periodic function \(e^{i\gamma(t)}\), and so Heinz’s lemma can be viewed as a statement about Fourier series.
In this paper, assuming that \(w(z)\) is a harmonic K-quasiconformal mapping of \({\mathbb{D}}\) onto itself satisfying \(w(0)=0\), by using Theorem D we obtain a sharp lower bound for its coefficients as follows:
which satisfies \(\lim_{K\rightarrow1^{+}}B_{1}(K)=1\), where Γ is the gamma function.
For \(n\geq2\) we have
where
and
is a decreasing function of K with \(\chi(1)=0\).
Assume that \(w(z)=P[f](z)\) is a harmonic K-quasiconformal mapping of \({\mathbb{D}}\) onto itself with the boundary function \(f(e^{it})=e^{i\gamma(t)}\), satisfying \(w(0)=0\). In Theorem 3.2 of [9], Partyka and Sakan proved that the following inequalities:
hold for a.e. \(z=e^{it}\in\mathbf{T}\). Applying the above inequalities we obtain an upper bound for the coefficients of a harmonic K-quasiconformal self-mapping \(w(z)\) of \({\mathbb{D}}\) satisfying \(w(0)=0\) as follows:
Furthermore we show that (9) and (10) are sharp as \(K\rightarrow1\).
2 Auxiliary results
Lemma 1
Let \(K>1\) be a constant. Then the equality
holds for all nonnegative integer numbers \(n=0,1,2,\ldots \) .
Lemma 2
Let \(\varphi(t):=\vert \cos\frac{t}{2}\vert ^{\frac{3}{2}}+ \vert \sin\frac{t}{2}\vert ^{\frac{3}{2}}\), for any \(t\in[0, 2\pi]\). Then
Lemma 3
Let \(w=P[f](z)\) be a harmonic K-quasiconformal self-mapping of \({\mathbb{D}}\) with the boundary function \(f(e^{it})=e^{i\gamma(t)}\). For every \(z_{1}=e^{i(s+t)}, z_{2}=e^{i(s-t)}\in\mathbf{T}\), let \(\theta=\gamma(s+t)-\gamma(s-t)\). Then \(f(z_{1})=e^{i\theta}f(z_{2})\) and the inequalities
hold for every \(0\leq s<2\pi\), \(0\leq t\leq\pi\).
Proof
According to the quasi-invariance of the harmonic measure (see (1.9) in [1]), we have
for every \(0\leq s<2\pi\), \(0\leq t\leq\pi\), and \(\theta=\gamma (s+t)-\gamma(s-t)\). Since \(\Phi^{2}_{K}(x)+\Phi^{2}_{1/K}(\sqrt{1-x^{2}})=1\) holds for every \(0\leq x\leq1\), this shows that
Using the Hübner inequalities, (7) and (8), we see that \(4^{1-K}s^{K}\leq\Phi_{1/K}(s)<4^{D(s)(1-K)}s^{K}\) and \(s^{1/K}\leq\Phi_{K}(s)<4^{(1-s^{2})^{\frac{3}{4}}(1-1/K)}s^{1/K}\). Applying (18), (19), and the above two inequalities, we have
and
By using Lemma 2 we see that
This implies that
hold for every \(0\leq s<2\pi\), \(0\leq t\leq\pi\), and \(\theta=\gamma (s+t)-\gamma(s-t)\).
This completes the proof. □
3 Main results
Theorem 1
Given \(K> 1\), let \(w(z)=P[f](z)=h(z)+\overline{g(z)}\) be a harmonic K-quasiconformal self-mapping of \({\mathbb{D}}\) satisfying \(w(0)=0\) with the boundary function \(f(e^{it})=e^{i\gamma (t)}\), where
are both analytic in \({\mathbb{D}}\). Then
where \(B_{1}(K)\) is given by (9) and satisfies \(\lim_{K\rightarrow1^{+}}B_{1}(K)=1\). For \(n\geq2\),
where \(B_{n}(K)\) is given by (11) and satisfies \(\lim_{n\rightarrow\infty}\lim_{K\rightarrow1^{+}}B_{n}(K)=0\).
Proof
Since \(w(z)=P[f](z)=\sum_{n=1}^{\infty}a_{n}z^{n}+\overline{\sum_{n=1}^{\infty}b_{n}z^{n}}\), using Parseval’s relation (cf. [7]) we have
for arbitrary \(t\in R\). Taking real parts, we arrive at the formula
where
Since \(w(z)\) is a harmonic K-quasiconformal mapping, by Lemma 3 we have
Hence \((|a_{n}|^{2}+|b_{n}|^{2})\int_{0}^{\pi}\cos(2nt)(1+\cos(2nt))\, dt= \int _{0}^{\pi}(1-2J(t))(1+\cos(2nt))\, dt\). Using (15) we also obtain
For \(n=1\), using the formula \(\Gamma(z+1)=z\Gamma(z)\) and simplifying the above result we obtain the following inequality:
By computation we know that \(B_{1}(K)\) is a decreasing function of K and satisfies
The above estimate is sharp. Consider the conformal mapping \(w(z)=e^{ix}z\), where \(x\in\mathbb{R}\) is a real number. Then we have \(|a_{1}|+|b_{1}|=1\).
For \(n\geq2\), we have
then
By calculating we see that \(\chi(K)\) is a decreasing function of K with \(\chi(1)=0\). The function \(B_{n}(K)\) is a continuous function of K with \(\lim_{K\rightarrow1^{+}}B_{n}(K)=\frac{2}{(n+1)n(n-1)}\). This implies that \(B_{n}(K)>0\) holds for all \(n\geq2\) and some \(K>1\).
The proof is completed. □
Remark 1
By computation we obtain
for all \(1\leq K\leq1.05174\). This shows that under the additional assumption that w is a K-quasiconformal mapping, the lower bound of the inequality (3) can be improved.
By the definition of the Gamma function we see that \(\Gamma(-n)=\infty \) holds for all nonnegative integer numbers n. According to the proof of Theorem 1 we know that for all \(n\geq2\), \(\lim_{K\rightarrow1^{+}}\Gamma(1+\frac{1}{K}-n)=\infty\). Therefore
holds for all \(n\geq2\).
Let \(t=0\) in equation (21). Then we have \(\sum_{n=1}^{\infty}(|a_{n}|^{2}+|b_{n}|^{2})=1\). The sharp coefficient estimate of \(a_{1}\) and \(b_{1}\) shows that if \(K\rightarrow1^{+}\) then \(|a_{1}|^{2}+|b_{1}|^{2}\geq B_{1}(K)\rightarrow1\). This shows that under the assumptions of Theorem 1 if additionally \(w(z)\) is a conformal self-mapping of \({\mathbb{D}}\) satisfying \(w(0)=0\), then all the coefficients \(b_{n}\) for \(n\geq1\) and \(a_{n}\) for \(n\geq2\) are zeros and \(|a_{1}|=1\), that is, \(w(z)=e^{i\theta}z\) for some \(\theta\in \mathbb{R}\).
Remark 2
In [1] the authors showed that an asymptotically sharp inequality holds for all z in \({\mathbb{D}}\). Our Theorem 1, however, gives an estimate at \(z=0\) only. In this sense, Theorem 1 partly improves the former results.
Theorem 2 shows that \(n^{2}(|a_{n}|^{2}+|b_{n}|^{2})\) is less than or equal to a positive number determined by K.
Theorem 2
Under the assumption of Theorem 1, the coefficients of \(w(z)\) satisfy the following inequality:
Proof
For every \(z=re^{i\theta}\in{\mathbb{D}}\),
hence
For every n we set \(a_{n}=|a_{n}|e^{i\alpha_{n}}\), \(b_{n}=|b_{n}|e^{i\beta _{n}}\), and \(\theta_{n}=\frac{\alpha_{n}+\beta_{n}}{2n}\). Then
Integrating by parts we have
In Theorem 2.8 of [10], Kalaj proved that the radial limits of \(w_{\theta}\) and \(w_{r}\) exist almost everywhere and
for almost every \(z=e^{i\theta}\in\mathbf{T}\). Here f is the boundary function of w. Hence, letting \(r\rightarrow1^{-}\) and using (13), (23) we see that
It shows that \(|a_{n}|^{2}+|b_{n}|^{2}\leq(|a_{n}|+|b_{n}|)^{2}\leq\frac {16K^{6K}2^{5(K-1/K)}}{n^{2}\pi^{2}}:=A_{n}(K)\).
The proof is completed. □
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Acknowledgements
The author of this work is supported by NNSF of China (Nos. 11301195, 11501220) and the China Scholarship Council and a research foundation of Huaqiao University (Project No. 2014KJTD14). The author would like to express her appreciation to Professor Jian-Feng Zhu and the referee for their helpful advice.
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Li, HP. An asymptotically sharp coefficients estimate for harmonic K-quasiconformal mappings. J Inequal Appl 2016, 84 (2016). https://doi.org/10.1186/s13660-016-1033-0
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DOI: https://doi.org/10.1186/s13660-016-1033-0