<p>The purpose of this paper is to study Phragmén-Lindelöf theorems for harmonic mappings. As generalizations of the maximum principle, we consider the boundedness of harmonic mappings on an unbounded domain, typically half-planes and angular sectors, from the hypotheses that the harmonic mapping is bounded on the boundary and not too rapid growth inside, which are called Phragmén-Lindelöf theorems for harmonic mappings. As applications, we consider asymptotic values of harmonic mappings in an angular sector, the growth of a harmonic mapping in the right half-plane under certain growth condition on the boundary, and Montel’s Theorem for harmonic mappings. Next, we investigate the Phragmén-Lindelöf indicator function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$h_f(\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a harmonic mapping <i>f</i> in an angular sector. We establish a (unique) sinusoid which dominates <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$h_f(\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and using this result, we give properties of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$h_f(\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and consider when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$h_f(\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a supporting function. Finally, we consider the effect on the general behaviour of harmonic mappings of exponential type by imposing a condition of boundedness along a line.</p>

Phragmén-Lindelöf Theorems for Harmonic Mappings

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发表日期：

2026年4月29日

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Yule Bai,
Saminathan Ponnusamy,
Jinjing Qiao,
Jing Wang

摘要

The purpose of this paper is to study Phragmén-Lindelöf theorems for harmonic mappings. As generalizations of the maximum principle, we consider the boundedness of harmonic mappings on an unbounded domain, typically half-planes and angular sectors, from the hypotheses that the harmonic mapping is bounded on the boundary and not too rapid growth inside, which are called Phragmén-Lindelöf theorems for harmonic mappings. As applications, we consider asymptotic values of harmonic mappings in an angular sector, the growth of a harmonic mapping in the right half-plane under certain growth condition on the boundary, and Montel’s Theorem for harmonic mappings. Next, we investigate the Phragmén-Lindelöf indicator function $h_f(\theta )$ $h_{f} (θ)$ of a harmonic mapping f in an angular sector. We establish a (unique) sinusoid which dominates $h_f(\theta )$ $h_{f} (θ)$ , and using this result, we give properties of $h_f(\theta )$ $h_{f} (θ)$ and consider when $h_f(\theta )$ $h_{f} (θ)$ is a supporting function. Finally, we consider the effect on the general behaviour of harmonic mappings of exponential type by imposing a condition of boundedness along a line.