Certain Extremal Problems on a Classical Family of Univalent Functions

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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Bibliographic Details
Main Authors: Lateef Ahmad Wani, Saiful R. Mondal
Format: Article
Language:English
Published: MDPI AG 2025-04-01
Series:Mathematics
Subjects:
univalent functions
starlike and convex functions
extremal problems
subordination
arc length
partial sums
Online Access:https://www.mdpi.com/2227-7390/13/8/1216
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Summary: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.
ISSN:2227-7390

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