Certain Subclasses of Analytic and Bi-Univalent Functions Governed by the Gegenbauer Polynomials Linked with <i>q</i>-Derivative
In this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions using the <i>q</i>-derivative operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><...
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| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/6/1192 |
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| author | Sercan Kazımoğlu Erhan Deniz Luminiţa-Ioana Cotîrlă |
| author_facet | Sercan Kazımoğlu Erhan Deniz Luminiţa-Ioana Cotîrlă |
| author_sort | Sercan Kazımoğlu |
| collection | DOAJ |
| description | In this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions using the <i>q</i>-derivative operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>q</mi></msub><mspace width="3.33333pt"></mspace><mfenced separators="" open="(" close=")"><mn>0</mn><mo><</mo><mi>q</mi><mo><</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula> and the Gegenbauer polynomials in a symmetric domain, which is the open unit disc <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">Λ</mi><mo>=</mo><mfenced separators="" open="{" close="}"><mo>℘</mo><mo>:</mo><mo>℘</mo><mo>∈</mo><mi mathvariant="double-struck">C</mi><mspace width="3.33333pt"></mspace><mi>and</mi><mspace width="3.33333pt"></mspace><mfenced open="|" close="|"><mo>℘</mo></mfenced><mo><</mo><mn>1</mn></mfenced><mo>.</mo></mrow></semantics></math></inline-formula> For these subclasses of analytic and bi-univalent functions, the coefficient estimates and Fekete–Szegö inequalities are solved. Some special cases of the main results are also linked to those in several previous studies. The symmetric nature of quantum calculus itself motivates our investigation of the applications of such quantum (or <i>q</i>-) extensions in this paper. |
| first_indexed | 2024-03-11T01:52:58Z |
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| id | doaj.art-4865760fd7f142419077fc0c13cd7d1e |
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| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T01:52:58Z |
| publishDate | 2023-06-01 |
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| series | Symmetry |
| spelling | doaj.art-4865760fd7f142419077fc0c13cd7d1e2023-11-18T12:50:44ZengMDPI AGSymmetry2073-89942023-06-01156119210.3390/sym15061192Certain Subclasses of Analytic and Bi-Univalent Functions Governed by the Gegenbauer Polynomials Linked with <i>q</i>-DerivativeSercan Kazımoğlu0Erhan Deniz1Luminiţa-Ioana Cotîrlă2Department of Mathematics, Kafkas University, Kars 36100, TurkeyDepartment of Mathematics, Kafkas University, Kars 36100, TurkeyDepartment of Mathematics, Technical University of Cluj-Napoca, 400020 Cluj-Napoca, RomaniaIn this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions using the <i>q</i>-derivative operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>q</mi></msub><mspace width="3.33333pt"></mspace><mfenced separators="" open="(" close=")"><mn>0</mn><mo><</mo><mi>q</mi><mo><</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula> and the Gegenbauer polynomials in a symmetric domain, which is the open unit disc <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">Λ</mi><mo>=</mo><mfenced separators="" open="{" close="}"><mo>℘</mo><mo>:</mo><mo>℘</mo><mo>∈</mo><mi mathvariant="double-struck">C</mi><mspace width="3.33333pt"></mspace><mi>and</mi><mspace width="3.33333pt"></mspace><mfenced open="|" close="|"><mo>℘</mo></mfenced><mo><</mo><mn>1</mn></mfenced><mo>.</mo></mrow></semantics></math></inline-formula> For these subclasses of analytic and bi-univalent functions, the coefficient estimates and Fekete–Szegö inequalities are solved. Some special cases of the main results are also linked to those in several previous studies. The symmetric nature of quantum calculus itself motivates our investigation of the applications of such quantum (or <i>q</i>-) extensions in this paper.https://www.mdpi.com/2073-8994/15/6/1192analytic functions<i>q</i>-derivative operatorbi-univalent functionssubordinationFekete–Szegö inequalityGegenbauer polynomials |
| spellingShingle | Sercan Kazımoğlu Erhan Deniz Luminiţa-Ioana Cotîrlă Certain Subclasses of Analytic and Bi-Univalent Functions Governed by the Gegenbauer Polynomials Linked with <i>q</i>-Derivative Symmetry analytic functions <i>q</i>-derivative operator bi-univalent functions subordination Fekete–Szegö inequality Gegenbauer polynomials |
| title | Certain Subclasses of Analytic and Bi-Univalent Functions Governed by the Gegenbauer Polynomials Linked with <i>q</i>-Derivative |
| title_full | Certain Subclasses of Analytic and Bi-Univalent Functions Governed by the Gegenbauer Polynomials Linked with <i>q</i>-Derivative |
| title_fullStr | Certain Subclasses of Analytic and Bi-Univalent Functions Governed by the Gegenbauer Polynomials Linked with <i>q</i>-Derivative |
| title_full_unstemmed | Certain Subclasses of Analytic and Bi-Univalent Functions Governed by the Gegenbauer Polynomials Linked with <i>q</i>-Derivative |
| title_short | Certain Subclasses of Analytic and Bi-Univalent Functions Governed by the Gegenbauer Polynomials Linked with <i>q</i>-Derivative |
| title_sort | certain subclasses of analytic and bi univalent functions governed by the gegenbauer polynomials linked with i q i derivative |
| topic | analytic functions <i>q</i>-derivative operator bi-univalent functions subordination Fekete–Szegö inequality Gegenbauer polynomials |
| url | https://www.mdpi.com/2073-8994/15/6/1192 |
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