Bi-Starlike Function of Complex Order Involving Mathieu-Type Series in the Shell-Shaped Region

Bi-Starlike Function of Complex Order Involving Mathieu-Type Series in the Shell-Shaped Region

For functions of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>=</mo...

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Bibliographic Details
Main Authors: Ibrahim S. Elshazly, Gangadharan Murugusundaramoorthy, Borhen Halouani, Alaa H. El-Qadeem, Kaliappan Vijaya
Format: Article
Language:English
Published: MDPI AG 2024-10-01
Series:Axioms
Subjects:
bi-univalent
analytic
bi-convex
bi-starlike
coefficients bounds
Online Access:https://www.mdpi.com/2075-1680/13/11/747
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Summary:For functions of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>=</mo><mi>ξ</mi><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><mo>∞</mo></msubsup><msub><mi>c</mi><mi>n</mi></msub><msup><mi>ξ</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula>, we identified two new subclasses of bi-starlike functions and bi-convex functions by using Mathieu-type series defined in the disc <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mo>=</mo><mo>{</mo><mi>ξ</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>:</mo><mo>|</mo><mi>ξ</mi><mo>|</mo><mo><</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>. We derived constraints for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>c</mi><mn>2</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>c</mi><mn>3</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, and the subclasses are connected to the shell-shaped area. The Fekete–Szegö functional properties for the aforementioned function subclasses were also investigated. Additionally, a number of related corollaries are shown.
ISSN:2075-1680

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