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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2024-10-01
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| author | Ibrahim S. Elshazly Gangadharan Murugusundaramoorthy Borhen Halouani Alaa H. El-Qadeem Kaliappan Vijaya |
| author_facet | Ibrahim S. Elshazly Gangadharan Murugusundaramoorthy Borhen Halouani Alaa H. El-Qadeem Kaliappan Vijaya |
| author_sort | Ibrahim S. Elshazly |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-8a7739dcea2a438e80b74a9f4ef14a20 |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-10-01 |
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| series | Axioms |
| spelling | doaj-art-8a7739dcea2a438e80b74a9f4ef14a202025-08-20T01:53:40ZengMDPI AGAxioms2075-16802024-10-01131174710.3390/axioms13110747Bi-Starlike Function of Complex Order Involving Mathieu-Type Series in the Shell-Shaped RegionIbrahim S. Elshazly0Gangadharan Murugusundaramoorthy1Borhen Halouani2Alaa H. El-Qadeem3Kaliappan Vijaya4Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 11451, Saudi ArabiaSchool of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, IndiaDepartment of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, EgyptSchool of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, IndiaFor 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.https://www.mdpi.com/2075-1680/13/11/747bi-univalentanalyticbi-convexbi-starlikecoefficients bounds |
| spellingShingle | Ibrahim S. Elshazly Gangadharan Murugusundaramoorthy Borhen Halouani Alaa H. El-Qadeem Kaliappan Vijaya Bi-Starlike Function of Complex Order Involving Mathieu-Type Series in the Shell-Shaped Region Axioms bi-univalent analytic bi-convex bi-starlike coefficients bounds |
| title | Bi-Starlike Function of Complex Order Involving Mathieu-Type Series in the Shell-Shaped Region |
| title_full | Bi-Starlike Function of Complex Order Involving Mathieu-Type Series in the Shell-Shaped Region |
| title_fullStr | Bi-Starlike Function of Complex Order Involving Mathieu-Type Series in the Shell-Shaped Region |
| title_full_unstemmed | Bi-Starlike Function of Complex Order Involving Mathieu-Type Series in the Shell-Shaped Region |
| title_short | Bi-Starlike Function of Complex Order Involving Mathieu-Type Series in the Shell-Shaped Region |
| title_sort | bi starlike function of complex order involving mathieu type series in the shell shaped region |
| topic | bi-univalent analytic bi-convex bi-starlike coefficients bounds |
| url | https://www.mdpi.com/2075-1680/13/11/747 |
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