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...
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/6/393 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849418189958545408 |
|---|---|
| author | Fethiye Müge Sakar Omendra Mishra Georgia Irina Oros Basem Aref Frasin |
| author_facet | Fethiye Müge Sakar Omendra Mishra Georgia Irina Oros Basem Aref Frasin |
| author_sort | Fethiye Müge Sakar |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-463a13da4c2846f2beb8b13a0cb02e12 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-463a13da4c2846f2beb8b13a0cb02e122025-08-20T03:32:31ZengMDPI AGAxioms2075-16802025-05-0114639310.3390/axioms14060393On Certain Bounds of Harmonic Univalent FunctionsFethiye Müge Sakar0Omendra Mishra1Georgia Irina Oros2Basem Aref Frasin3Department of Management, Dicle University, Diyarbakir 21280, TurkeyDepartment of Mathematical and Statistical Sciences, Institute of Natural Sciences and Humanities, Shri Ramswaroop Memorial University, Lucknow 225003, IndiaDepartment of Mathematics and Computer Science, University of Oradea, 410087 Oradea, RomaniaDepartment of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 25113, JordanHarmonic 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.https://www.mdpi.com/2075-1680/14/6/393univalent harmonic functionshear constructionconvex functionconvexity in a directiongrowth and distortion bounds |
| spellingShingle | Fethiye Müge Sakar Omendra Mishra Georgia Irina Oros Basem Aref Frasin On Certain Bounds of Harmonic Univalent Functions Axioms univalent harmonic function shear construction convex function convexity in a direction growth and distortion bounds |
| title | On Certain Bounds of Harmonic Univalent Functions |
| title_full | On Certain Bounds of Harmonic Univalent Functions |
| title_fullStr | On Certain Bounds of Harmonic Univalent Functions |
| title_full_unstemmed | On Certain Bounds of Harmonic Univalent Functions |
| title_short | On Certain Bounds of Harmonic Univalent Functions |
| title_sort | on certain bounds of harmonic univalent functions |
| topic | univalent harmonic function shear construction convex function convexity in a direction growth and distortion bounds |
| url | https://www.mdpi.com/2075-1680/14/6/393 |
| work_keys_str_mv | AT fethiyemugesakar oncertainboundsofharmonicunivalentfunctions AT omendramishra oncertainboundsofharmonicunivalentfunctions AT georgiairinaoros oncertainboundsofharmonicunivalentfunctions AT basemareffrasin oncertainboundsofharmonicunivalentfunctions |