Abstract
In this paper, we study Schwarz lemma for solutions of the \(\alpha \)-harmonic equation. Applying the method given by Burgeth, we build a new form of Schwarz lemma for solutions of the \(\alpha \)-harmonic equation, which is suitable for \(\alpha >-1\). Moreover, we generalize the Schwarz–Pick inequality for solutions of the \(\alpha \)-harmonic equation from the case \(\alpha \ge 0\) to \(\alpha >-1\).
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1 Introduction
Let \(\mathbb {D}\) denote the unit disc of the complex plane \(\mathbb {C}\), and \(\mathbb {T}=\partial {\mathbb {D}}\), the boundary of \(\mathbb {D}\). Let \(\mathbb {R}^n\) be the standard Euclidean space, with the norm \(|x|=\sqrt{\sum _{i=1}^n{x_i}^2}\), where \(x=(x_1,\dots x_n)\). We write \(\mathbb {B}^n=\{x\in \mathbb {R}^n: |x|< 1\}\), and \(\mathbb {B}^2=\mathbb {D}\). Let \(S^n\) denote the boundary of \(\mathbb {B}^n\), and \(S^2=\mathbb {T}\). Denote by \(\mathcal {X}_A\) the characteristic function of a set A of \(S^{n}\), and \(\sigma \) the usual surface measure on \(S^{n}.\) Write
1.1 \(\alpha \)-harmonic Function and Related Definitions
Let \(\Delta _\alpha =\partial z(1-|z|^2)^{-\alpha }\bar{\partial }z\) denote the weighted Laplacian operator in \(\mathbb {D}\), and the standard weight by \(\omega _\alpha =(1-|z|^2)^{-\alpha }\), where \(\alpha >-1\). If \(\alpha =0\), then \(\Delta _\alpha \) reduces to the classical Laplacian operator. One can refer to [1] for the original study of weighted Laplacian operators. Let \(g \in \mathcal {C}(\mathbb {D})\) and \(f \in \mathcal {C}^2(\mathbb {D})\), we write an inhomogeneous \(\alpha \)-harmonic equation by
and the associated Dirichlet boundary value problem by
where the boundary data \(f^* \in L^1({\mathbb {T}})\) , and the boundary condition is understood as \(f_r \rightarrow f^*\) for \(r \rightarrow 1^-\), where \(f_r=f(re^{i\theta }), 0<r<1\). If \(g=0\), a Dirichlet solution of the Eq. (2) is called an \(\alpha \)-harmonic mapping.
Olofsson and Wittsten [2] proved that if the boundary data \(f^* \in L^1({\mathbb {T}})\), then an \(\alpha \)-harmonic mapping f had the form of a Poisson type integral
where
was called the \(\alpha \)-harmonic Poisson kernel. We note that \(P_{\alpha }\) is real if and only if \(\alpha =0\), for this special case, \(P_\alpha \) reduces to the classical Poisson kernel
and the convolution type integral
is the classical harmonic mapping in \({\mathbb {D}}\) with boundary data \(f^*\in L^1({\mathbb {T}})\). One can refer to [3] and the references therein for further study of harmonic mappings.
Behm [4] found that the weighted Green function \(G_\alpha \) of the weighted Laplacian operator \(\Delta _\alpha \) could be written by
where
and
The weighted potential of a function \(g\in \mathcal {C}(\mathbb {D})\) can be represented by
Where \(\mathrm {d}A(w)=\frac{1}{\pi }\mathrm {d}u\mathrm {d}v, w=u+iv\).
The combination of the results of Olofsson–Wittsten [2] and Behm [4] says that a solution of the \(\alpha \)-harmonic equation (2) has the form
For the latest research on \(\alpha \)-harmonic mappings one can see the papers [5,6,7,8,9,10,11,12,13,14,15].
1.2 Schwarz lemma and Schwarz–Pick lemma
The Schwarz lemma for analytic functions plays a vital role in complex analysis and has been generalized to various spaces of functions.
Heinz [16] generalized it to the class of complex-valued harmonic functions. That is, if f is a complex-valued harmonic function from \(\mathbb {D}\) into itself with \(f(0)=0\), then for \(z \in \mathbb {D}\),
Hethcote [17] and Pavlović [18] improved Heinz’s result, by removing the assumption \(f(0)=0\), i.e., let f be a harmonic function from \({\mathbb {D}}\) to \({\mathbb {D}}\), then
Colonna [19] obtained a Schwarz–Pick lemma for complex-valued harmonic functions which was stated that if f was a complex-valued harmonic function from \(\mathbb {D}\) into itself, then for \(z\in \mathbb {D}\),
where \(\Vert D_f(z)\Vert =\sup \{|D_f(z)\zeta |: |\zeta |=1\}=|f_z(z)|+|f_{\bar{z}}(z)|.\)
For \(\alpha \)-harmonic function, Li et al. [9] stated a Schwarz type lemma for solutions of the \(\alpha \)-harmonic equation with the condition \(\alpha \ge 0\).
Theorem A
Suppose that \(g\in \mathcal {C}(\overline{\mathbb {D}})\), and \(f^*\in \mathcal {C}^1(\mathbb {T})\). If \(f\in \mathcal {C}^2(\mathbb {D})\) satisfies (2) with \(\alpha \ge 0\) and \(\mathcal {P}_\alpha [f^*](0)=0\), then for \(z\in \mathbb {D},\)
where \(\Vert f^*\Vert _\infty =\sup _{z\in \mathbb {T}}\{|f^*(z)|\},\) and \(\Vert g\Vert _\infty =\sup _{z\in \mathbb {D}}\{|g(z)|\}.\)
Li et al. [9] also established a Schwarz–Pick type inequality for the solutions of the Eq. (2) for the case \(\alpha \ge 0\).
Theorem B
Suppose that \(g\in \mathcal {C}(\bar{\mathbb {D}})\), and \(f\in \mathcal {C}^2(\mathbb {D})\) satisfies (2) with \(f^*\in \mathcal {C}(\mathbb {T})\) and \(\alpha \ge 0\). Then for any \(z \in \mathbb {D},\)
where \(\Vert f^*\Vert _\infty =\sup _{z\in \mathbb {T}}\{|f^*(z)|\},\) and \(\Vert g\Vert _\infty =\sup _{z\in \mathbb {D}}\{|g(z)|\}.\)
One can refer to the papers [19,20,21,22,23,24,25,26,27,28] for recent progress on the Schwarz lemma and the Schwarz–Pick lemma.
1.3 Statement of main results
In this paper, we continue to study the Schwarz lemma and the Schwarz–Pick lemma for solutions of the \(\alpha \)-harmonic equation (2). The method to estimate an \(\alpha \)-Poisson kernel by the classical Poisson kernel is key to complete the proof of Theorem A. To reach their estimate, one needs to make sure that the condition \(\mathcal {P}_\alpha [f^*](0)=0\) imply that \(\mathcal {P}_\alpha [|f^*|](0)=0\). However, Example 2.1 shows that this is not necessarily true. Hethcote’s inequality may overcome this difficulty to reach an estimate for an \(\alpha \)-harmonic mapping when \(\alpha \ge 0\) as follows
We aim to give another form of the above estimate and obtain a Schwarz type lemma for solutions to the \(\alpha \)-harmonic equation (2) under a more general condition, which includes the case that \(-1<\alpha <0\).
Theorem 1.1
Suppose that \(g \in \mathcal {C}(\overline{{\mathbb {D}}})\). If non-constant \(f\in \mathcal {C}^2(\mathbb {D})\) with \(f^*\in \mathcal {C}^1(\mathbb {T})\) satisfies the Eq. (2), then for \(\alpha >-1 \) and \(z \in \mathbb {D}\),
where \(\Vert f^*\Vert _\infty =\sup _{z\in \mathbb {T}}\{|f^*(z)|\}\), \(\Vert g\Vert _\infty =\sup _{z\in \mathbb {D}}\{|g(z)|\}\), \(c=\frac{\mathcal {P}_{\alpha }[|f^*|](0)}{\Vert f^*\Vert _\infty }.\)
Note that the above theorem is an application of the growth estimate for positive harmonic functions, which is given by Burgeth at [29]. We also note that for a harmonic mapping, that is, \(\alpha =0\) and \(g=0\), the estimate at Theorem 1.1 attains equality if \(c=0\) or \(c=1\).
We next give the Schwarz–Pick inequality for the solution of the \(\alpha \)-harmonic equation when \(\alpha >-1\), which is a corresponding result given by Theorem B.
Theorem 1.2
Suppose that \(g \in \mathcal {C}(\overline{{\mathbb {D}}})\). If non-constant \(f\in \mathcal {C}^2(\mathbb {D})\) with \(f^*\in \mathcal {C}^1(\mathbb {T})\) satisfies the Eq. (2) with \(\alpha >-1\), then for \(z\in \mathbb {D},\)
where \(\Vert f^*\Vert _\infty =\sup _{z\in \mathbb {T}}\{|f^*(z)|\}\), \(\Vert g\Vert _\infty =\sup _{z\in \mathbb {D}}\{|g(z)|\}.\)
We note that both two estimates in Theorem 1.1 and Theorem 1.2 are continuous in \(\alpha \) for all \(\alpha >-1\).
2 Auxiliary Examples
In [9], Li et al. used the Heinz inequality of \(\mathcal {P}[|f^*|](z)\) to obtain an upper estimate of \(\mathcal {P}_\alpha [f^*](z)\). In fact, to reach their result, they need to require that the normalization that \(\mathcal {P}_{\alpha }[f^*](0)=0\) should imply that \(\mathcal {P}_{\alpha }[|f^*|](0)=0\). However, the following example shows that it will not be true in general.
Example 2.1
Let
Then
Thus the Poisson integral
implies that
but
In fact, the inequality
implies that
3 Schwarz Lemma for Real Harmonic Functions
Write
and
where \(L^\infty (S^{n})\) denotes the Lebesgue space of essentially bounded functions on \(S^{n}\), \(\sigma \) denotes the sphere measure. Let \(P_x\) denote the Poisson kernel of the harmonic function on \({\mathbb {B}}^n\) as follows
The Poisson kernel \(P_x\) satisfies a normalization that \(\int _{S^n}P_x(x,y)\mathrm {d}\sigma =1\).
For convenience, we set
and
where \(S_{(c,e_x)}\) denote the polar cap centered at \(e_x\) with \(\sigma \)-measure c.
After introducing spherical coordinates, we can write \(\sigma (S^n)\) as
So,
By the fact that
and
we have
So, we can rewrite (10) and (11) as
and
where \(\alpha (c)\) is the spherical angle of \(S_{(c,e_x)}\). Li and Chen [30] used the method of Theorem 1 at [4] to obtain the following theorem.
Theorem C
Let h be a harmonic function \(\mathbb {B}^n\), a, b are two real numbers with \(a<b\). If \(h(0)=d\) and \(a<h(x)<b\), then for \(\displaystyle c=\frac{d-a}{b-a}\) and any \(z\in \mathbb {B}^n\),
Since \(\mathcal {P}[|f^*|](z)\) is a real harmonic function on the unit disk \({\mathbb {D}}\), the above theorem will imply the following corollary.
Corollary 1
Let \(\mathcal {P}[|f^*|](z)\) be a positive harmonic function with \(\mathcal {P}[|f^*|](0)=\mathcal {P}_{\alpha }[|f^*|](0)=d\). If \(\mathcal {P}[|f^*|](z) \le \Vert f^*\Vert _{\infty }<\infty \) and \(\displaystyle c=\frac{d}{\Vert f^*\Vert _{\infty }}\), then for all \(z\in \mathbb {D}\)
Proof
By Theorem C, we obtain that
By the formula (12), we have
The fact that \(\displaystyle \alpha (c)=\pi \cdot c\), we have
So,
Thus, the proof of Corollary 1 is complete. \(\square \)
One can refer to [31, 32] for more knowledge of the growth estimate for real harmonic functions.
4 Some Basic Lemmas
In [9], Li et al. obtained an estimate for the function h(s) as follows.
Lemma A
For \(\alpha \ge 0\), the function h(s) satisfies the estimate
We note the above estimate of h(s) is not suitable for the case \(-1<\alpha <0\). In fact, in [4], Behm obtained an estimate of h(s) in the case that \(\alpha >-1\).
Lemma B
For \(\alpha >-1\), the function h(s) satisfies the estimate
Especially, for \(-1<\alpha <0\), it holds
For completeness, we give the proof of Lemma B.
Proof
By the definition of h(s) given by (5) and the fact that \(1+n+\alpha >n\) for all \(n\ge 1\) and \(\alpha >-1\), we obtain that
When \(-1<\alpha <0\), it follows that \(0<n+\alpha <n\) for all \(n\ge 1\) and hence, we have
Thus, the proof of Lemma B is complete. \(\square \)
Remark
-
(1)
Two estimates of the upper bound of h are two infinitesimals of the same order when s tends to 0; Furthermore,
-
(2)
For \(\alpha >0\), we have
$$\begin{aligned} \left\{ \begin{array}{rl} \displaystyle s^\alpha \log \frac{1}{1-s}\ge s^{1+\alpha }(\frac{1}{1+\alpha }+\log \frac{1}{1-s}),&{}0<s\le 1-e^{-\frac{\alpha }{\alpha +1}},\\ &{}\qquad \\ \displaystyle s^\alpha \log \frac{1}{1-s}<s^{1+\alpha }(\frac{1}{1+\alpha }+\log \frac{1}{1-s}),&{}1-e^{-\frac{\alpha }{\alpha +1}}<s<1. \end{array}\right. \end{aligned}$$
Proof
-
(1)
Since
$$\begin{aligned} \lim _{s\rightarrow 0}\frac{\log \frac{1}{1-s}}{s(\frac{1}{1+\alpha }+\log \frac{1}{1-s})}=1+\alpha , \end{aligned}$$the proof of Remarks (1) is complete.
-
(2)
By a direct calculation, we obtain that
$$\begin{aligned} s^\alpha \log \frac{1}{1-s}-s^{1+\alpha }(\frac{1}{1+\alpha }+\log \frac{1}{1-s})=s^\alpha \big ((1-s)\log \frac{1}{1-s}-\frac{s}{1+\alpha }\big ). \end{aligned}$$Let \(\displaystyle A_\alpha (s)=(1-s)\log \frac{1}{1-s}-\frac{s}{1+\alpha }\).
Since
we have the unique zero of the \(A'_\alpha (s)\), which is denoted by \(B(\alpha )=1-e^{-\frac{\alpha }{1+\alpha }}\).
Moreover,
By a change of variable \(\displaystyle x=\frac{1}{1+\alpha }\), \(A_\alpha (B(\alpha ))\) can be represented by \(e^{-1}e^x-x\).
Let \(H(x)=e^{-1}e^x-x\). The fact that
and
imply that
Thus, Remarks (2) is proved. \(\square \)
Li et al. used Lemma A to obtain an inequality of \(\Vert D_{{\mathcal {P}}_\alpha [f^*]}(z)\Vert \) for \(\alpha \ge 0\) in [9].
Lemma C
Assume that \(f^*\in \mathcal {C}(\mathbb {T})\). Then for \(\alpha \ge 0\)
Zhu and Kalaj used the hypergeometric function to obtain the maximum of the \(K_p(|z|)\) in [33] for \(z\in {\mathbb {D}}\), here \(\displaystyle K_p(|z|)=\int _{{\mathbb {D}}}\frac{1}{|w-z|^{q}}\mathrm {d}A(w).\)
Lemma D
For \(p>2\) and \(q\in {\mathbb {R}}\) such that \(\frac{1}{p}+\frac{1}{q}=1\), \(K_p(|z|)\) has its maximum
Next, we give Green’s formula of complex form.
Lemma 4.1
The complex form of Green’s formula can be expressed as
where \(z=x+iy\), \(u=\cos \alpha +i\cos \beta \), \(v=\cos \beta -i\cos \alpha \), and \((\cos \alpha , \cos \beta )\) is the unit tangent vector of the curve \(\partial {\mathbb {D}}\).
Proof
Let \((\cos \alpha , \cos \beta )\) be the unit tangent vector of the curve \(\partial {\mathbb {D}}\), then the unit exterior normal vector of \(\partial {\mathbb {D}}\) is \((\cos \beta , -\cos \alpha )\).
By the classic Green’s formula, it follows,
and
Write
Let \(f=P+iQ\), then we have
and
So
Therefore, the proof of Lemma 4.1 is complete. \(\square \)
Next, we will give an estimate of the weighted Green function \(G_\alpha (z,w)\).
Lemma 4.2
Assume that \(\alpha >-1\). Then the weighted Green function \(G_\alpha (z,w)\) satisfies the estimate
Proof
By the definition of \(G_\alpha (z,w)\) given by (4), we obtain that
where h(s)and \(\omega (z,w)\) are given by (5) and (6).
Moreover, we have from Lemma B that
Thus, the proof of Lemma 4.2 is complete. \(\square \)
The following Lemma will be used to give the form of the Schwarz–Pick inequality for \({\mathcal {P}}_\alpha [f^*]\) with \(\alpha >-1\).
Lemma 4.3
Assume that \(f^*\in {\mathcal {C}}(\mathbb {T})\), then
Two estimates coincides with each other when \(\alpha =0\).
Proof
For the case \(\alpha \ge 0\), one can see Lemma 3.3 of [9] for details. We next prove the case \(-1<\alpha <0\). By an elementary calculation of P(z) given by (3), we obtain that
and
Then
and
By the equality
and the fact that \(-1<\alpha <0\), we have
and
We obtain from (17) that
and from (18) that
Thus, the identities
imply
Therefore, the proof of Lemma 4.3 is complete. \(\square \)
We also need some inequalities of the weighted Green potential \({\mathcal {G}}[g](z)\) as the following lemma.
Lemma 4.4
For \(\alpha >-1\) and \(g\in {\mathcal {C}} ({\overline{\mathbb {D}} })\), the potential \({\mathcal {G}}[g](z)\) satisfies the following inequalities:
-
(a)
\(\displaystyle \big |{\mathcal {G}}[g](z)\big |\le 2^{|\alpha |}(1-|z|^2)^\alpha \bigg [\frac{2+\alpha }{1+\alpha }-|z|^2\bigg ]\Vert g\Vert _\infty ;\)
-
(b)
For fixed \(w\in \mathbb {D}\),
$$\begin{aligned} |\frac{\partial {\mathcal {G}}[g](z)}{\partial z}|\le 2^{\alpha +2} \bigg [\bigg |\frac{2\alpha }{1+\alpha }\bigg |+(2|\alpha |+1)(1-|z|^2)\bigg ](1-|z|^2)^{\alpha -1}\Vert g\Vert _\infty . \end{aligned}$$(20) -
(c)
For fixed \(w\in \mathbb {D}\),
$$\begin{aligned} |\frac{\partial {\mathcal {G}}[g](z)}{\partial \bar{z}}|\le 2^{\alpha +2}(1-|z|^2)^{\alpha }\Vert g\Vert _\infty . \end{aligned}$$(21)
Proof
We first give the proof of (a). By the Lemma 4.2, we have
Let \(\displaystyle I=\int _{\mathbb {D}} \log \bigg |\frac{1-\bar{z}w}{z-w}\bigg |^2\mathrm {d}A(w)\) and \(\displaystyle \eta =\varphi (w)=\frac{z-w}{1-w\bar{z}}=re^{iv}\), so,
Consequently,
Using the technique of power series expansion, we obtain that
which implies that
Next, we will give the proof of (b).
For \(\varepsilon >0\), let \(D_\varepsilon =\mathbb {D}(z,\varepsilon )\) and \(\varphi \in {\mathcal {C}}_0^\infty ({\mathbb {D}})\) be a test function. By Lebesgue’s dominated convergence theorem we get
where \(G_\alpha (z,w) \varphi _w (w)\) is the dominant function.
By Lemma 4.1, we have
where \(C_\varphi \) is a constant depending only on \(\Vert \varphi \Vert _\infty \). Therefore, let \(\varepsilon \rightarrow 0\), it follows
From the proof of (a), we obtain that
Hence, \( G_{\alpha }(z,w)\in L^1({\mathbb {D}}\times {\mathbb {D}}).\) Therefore, the assumption that \(\varphi (z)\in {\mathcal {C}}_0^\infty ({\mathbb {D}})\) and \(g(w)\in {\mathcal {C}}(\overline{{\mathbb {D}}})\) shows that the integral \(\int _{\mathbb {D}}\int _{\mathbb {D}}G_\alpha (z,w) g(w)\varphi _z(z)\mathrm {d}A(w)\mathrm {d}A(z)\) is absolutely convergent. Moreover, we also have that \({\mathcal {G}}[g](z)\in L^1(\mathbb {D})\) for \(g(w)\in {\mathcal {C}}(\overline{{\mathbb {D}}})\).
The definition of distributional partial derivatives says that for \(f\in L_{loc}^1(\mathbb {D})\) and \(g\in \mathcal {C}_0^\infty (\mathbb {D})\), it follows
Hence, the fact that \({\mathcal {G}}[g](z)\in L^1(\mathbb {D})\) and the absolute convergence of the integral \(\int _{\mathbb {D}}\int _{\mathbb {D}}G_\alpha (z,w) g(w)\varphi _z(z)\mathrm {d}A(w)\mathrm {d}A(z)\) shows that
From (a) and the proof of Proposition 4 of [9], for any fixed \(w\in \mathbb {D}\) we know that \(G_\alpha (z,w)\in L^1({\mathbb {D}})\). Thus, for a test function \(\varphi \in {\mathcal {C}}_0^\infty ({\mathbb {D}})\) it follows
To complete the proof of (b), we also need the absolute convergence of the integral \(\int _{{\mathbb {D}}}\int _{{\mathbb {D}}} \frac{\partial G_\alpha (z,w)}{\partial z}\mathrm {d}A(w)\mathrm {d}A(z)\). By some direct calculations, we obtain that for \(z\ne w\)
Hence,
Moreover, we have
So,
and
Therefore, we obtain that
By the Lemma D we can obtain
By the equality (22) we have
Using Parseval’s theorem, we can obtain that
Since \(\displaystyle \lim _{x\rightarrow 0}\frac{-\ln (1-x)}{x}=1,\) there exists a constant \(M_1\) such that
Thus, from the fact
we have that \(\int _{{\mathbb {D}}} I_1(z)\mathrm {d}A(z)\) is also finite. So we have completed the absolute convergence of the integral \(\int _{{\mathbb {D}}}\int _{{\mathbb {D}}} \frac{\partial G_\alpha (z,w)}{\partial z}\mathrm {d}A(w)\mathrm {d}A(z)\).
Therefore, we complete the proof of each steps of the following relation
Thus, we can conclude that
The following estimate
implies that
Hence, we conclude that
Finally, we will give the proof of (c). In fact, by (4), we obtain that for \(z\ne w\)
By a similar method as above, Lemma D implies that
Hence, we conclude that
Thus, the proof of the Lemma 4.4 is complete. \(\square \)
5 Proof of Theorem 1.1
Proof
By the formula (7) we have
For \(\alpha \ge 0\), we have
and \(\mathcal {P}[|f^*|](z)\) is a positive harmonic function in \(\mathbb {D}.\)
By the inequality (14), we know that, for \(z\in \mathbb {D}\)
On the other hand, for \(\alpha \ge 0\), by the inequality (23), we get
Hence, for \(\alpha \ge 0\),
For \(-1<\alpha <0\), we have
So,
By the formula (23), we obtain that
Hence, for \(-1<\alpha <0\),
Thus, the proof of Theorem 1.1 is complete. \(\square \)
6 Proof of Theorem 1.2
Proof
For \(-1<\alpha <0\), by the inequalities (19), (20) and (21) imply that
Hence, for \(\alpha \ge 0\), by the inequalities (20), (21) and Lemma C, we obtain that
Thus, the proof of the Theorem 1.2 is complete. \(\square \)
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Acknowledgements
The authors are deeply indebted to the referees for their very careful work and many helpful suggestions which improve the quality of this paper. The authors are also deeply indebted to Professor David Kalaj for his suggestion about the use of Lemma D to improve the estimate (b) and (c) at Lemma 4.4. This work is partially supported by the National Natural Science Foundation of China (11971182), the Natural Science Foundation of Fujian Province of China (2019J01066, 2021J01304).
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Communicated by Saminathan Ponnusamy.
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Li, M., Chen, X. Schwarz Lemma for Solutions of the \(\alpha \)-harmonic Equation. Bull. Malays. Math. Sci. Soc. 45, 2691–2713 (2022). https://doi.org/10.1007/s40840-022-01348-3
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DOI: https://doi.org/10.1007/s40840-022-01348-3
Keywords
- \(\alpha \)-harmonic equation
- Schwarz lemma
- Schwarz–Pick lemma
- Harmonic mapping
- Weighted Laplacian operator