Certain Extremal Problems on a Classical Family of Univalent Functions
Consider the collection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> of analytic functions <i>f</...
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| author | Lateef Ahmad Wani Saiful R. Mondal |
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| description | Consider the collection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> of analytic functions <i>f</i> defined within the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">D</mi></semantics></math></inline-formula>, subject to the conditions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. For the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, define the subclass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">R</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> as follows:<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">R</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mfenced separators="" open="{" close="}"><mi>f</mi><mo>∈</mo><mi mathvariant="script">A</mi><mo>:</mo><mi>Re</mi><mfenced separators="" open="(" close=")"><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mfenced><mo>></mo><mi>λ</mi><mo>,</mo><mspace width="0.277778em"></mspace><mi>z</mi><mo>∈</mo><mi mathvariant="double-struck">D</mi></mfenced><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, we derive sharp bounds on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="|" close="|"><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>/</mo><mrow><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mfenced></semantics></math></inline-formula> for <i>f</i> in the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">R</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> and compute the boundary length of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Additionally, we investigate the inclusion properties of the sequences of partial sums <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>z</mi><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>k</mi></msub><msup><mi>z</mi><mi>k</mi></msup></mrow></semantics></math></inline-formula> for functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>z</mi><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><mo>∞</mo></msubsup><msub><mi>a</mi><mi>n</mi></msub><msup><mi>z</mi><mi>n</mi></msup><mo>∈</mo><mi mathvariant="script">R</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Our results extend and refine several classical results in the theory of univalent functions. |
| format | Article |
| id | doaj-art-3785aa0dbe1c4afd9586cf0bc12bb38a |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-3785aa0dbe1c4afd9586cf0bc12bb38a2025-08-20T02:18:19ZengMDPI AGMathematics2227-73902025-04-01138121610.3390/math13081216Certain Extremal Problems on a Classical Family of Univalent FunctionsLateef Ahmad Wani0Saiful R. Mondal1Department of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147001, Punjab, IndiaDepartment of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi ArabiaConsider the collection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> of analytic functions <i>f</i> defined within the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">D</mi></semantics></math></inline-formula>, subject to the conditions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. For the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, define the subclass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">R</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> as follows:<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">R</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mfenced separators="" open="{" close="}"><mi>f</mi><mo>∈</mo><mi mathvariant="script">A</mi><mo>:</mo><mi>Re</mi><mfenced separators="" open="(" close=")"><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mfenced><mo>></mo><mi>λ</mi><mo>,</mo><mspace width="0.277778em"></mspace><mi>z</mi><mo>∈</mo><mi mathvariant="double-struck">D</mi></mfenced><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, we derive sharp bounds on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="|" close="|"><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>/</mo><mrow><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mfenced></semantics></math></inline-formula> for <i>f</i> in the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">R</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> and compute the boundary length of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Additionally, we investigate the inclusion properties of the sequences of partial sums <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>z</mi><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>k</mi></msub><msup><mi>z</mi><mi>k</mi></msup></mrow></semantics></math></inline-formula> for functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>z</mi><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><mo>∞</mo></msubsup><msub><mi>a</mi><mi>n</mi></msub><msup><mi>z</mi><mi>n</mi></msup><mo>∈</mo><mi mathvariant="script">R</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Our results extend and refine several classical results in the theory of univalent functions.https://www.mdpi.com/2227-7390/13/8/1216univalent functionsstarlike and convex functionsextremal problemssubordinationarc lengthpartial sums |
| spellingShingle | Lateef Ahmad Wani Saiful R. Mondal Certain Extremal Problems on a Classical Family of Univalent Functions Mathematics univalent functions starlike and convex functions extremal problems subordination arc length partial sums |
| title | Certain Extremal Problems on a Classical Family of Univalent Functions |
| title_full | Certain Extremal Problems on a Classical Family of Univalent Functions |
| title_fullStr | Certain Extremal Problems on a Classical Family of Univalent Functions |
| title_full_unstemmed | Certain Extremal Problems on a Classical Family of Univalent Functions |
| title_short | Certain Extremal Problems on a Classical Family of Univalent Functions |
| title_sort | certain extremal problems on a classical family of univalent functions |
| topic | univalent functions starlike and convex functions extremal problems subordination arc length partial sums |
| url | https://www.mdpi.com/2227-7390/13/8/1216 |
| work_keys_str_mv | AT lateefahmadwani certainextremalproblemsonaclassicalfamilyofunivalentfunctions AT saifulrmondal certainextremalproblemsonaclassicalfamilyofunivalentfunctions |