On Certain Bounds of Harmonic Univalent Functions
Harmonic functions are renowned for their application in the analysis of minimal surfaces. These functions are also very important in applied mathematics. Any harmonic function in the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inl...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/6/393 |
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| Summary: | Harmonic functions are renowned for their application in the analysis of minimal surfaces. These functions are also very important in applied mathematics. Any harmonic function in the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">U</mi><mo>=</mo><mfenced separators="" open="{" close="}"><mi>z</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>:</mo><mfenced open="|" close="|"><mi>z</mi></mfenced><mo><</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula> can be written as a sum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>=</mo><mi>h</mi><mo>+</mo><mover><mi>g</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula>, where <i>h</i> and <i>g</i> are analytic functions in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">U</mi></semantics></math></inline-formula> and are called the analytic part and the co-analytic part of <i>f</i>, respectively. In this paper, the harmonic shear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>=</mo><mi>h</mi><mo>+</mo><mover><mi>g</mi><mo>¯</mo></mover><mo>∈</mo><msub><mi>S</mi><mi mathvariant="script">H</mi></msub></mrow></semantics></math></inline-formula> and its rotation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>f</mi><mi>μ</mi></msup></semantics></math></inline-formula> by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mspace width="4pt"></mspace><mfenced separators="" open="(" close=")"><mi>μ</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>,</mo><mfenced open="|" close="|"><mi>μ</mi></mfenced><mo>=</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula> are considered. Bounds are established for this rotation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>f</mi><mi>μ</mi></msup></semantics></math></inline-formula>, specific inequalities that define the Jacobian of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>f</mi><mi>μ</mi></msup></semantics></math></inline-formula> are obtained, and the integral representation is determined. |
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| ISSN: | 2075-1680 |