Two Families of Bi-Univalent Functions Associating the (<i>p</i>, <i>q</i>)-Derivative with Generalized Bivariate Fibonacci Polynomials
Making use of generalized bivariate Fibonacci polynomials, we propose two families of regular functions of the type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mrow><mo&...
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| author | Sondekola Rudra Swamy Basem Aref Frasin Daniel Breaz Luminita-Ioana Cotîrlă |
| author_facet | Sondekola Rudra Swamy Basem Aref Frasin Daniel Breaz Luminita-Ioana Cotîrlă |
| author_sort | Sondekola Rudra Swamy |
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| description | Making use of generalized bivariate Fibonacci polynomials, we propose two families of regular functions of the type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>ζ</mi><mo>)</mo></mrow><mo>=</mo><mi>ζ</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>2</mn></mrow><mo>∞</mo></munderover></mstyle><msub><mi>d</mi><mi>j</mi></msub><msup><mi>ζ</mi><mi>j</mi></msup></mrow></semantics></math></inline-formula>, which are bi-univalent in the disc <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>ζ</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>:</mo><mo>|</mo><mi>ζ</mi><mo>|</mo><mo><</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula> involving the (<i>p</i>, <i>q</i>)-derivative operator. We find estimates on the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>d</mi><mn>2</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>d</mi><mn>3</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula> and the of Fekete–Szegö functional for members of these families. Relevant connections to the existing results and new consequences of the main result are presented. |
| format | Article |
| id | doaj-art-466b7becf6f34340a64a0e8f1ee11528 |
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| language | English |
| publishDate | 2024-12-01 |
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| spelling | doaj-art-466b7becf6f34340a64a0e8f1ee115282025-08-20T02:00:39ZengMDPI AGMathematics2227-73902024-12-011224393310.3390/math12243933Two Families of Bi-Univalent Functions Associating the (<i>p</i>, <i>q</i>)-Derivative with Generalized Bivariate Fibonacci PolynomialsSondekola Rudra Swamy0Basem Aref Frasin1Daniel Breaz2Luminita-Ioana Cotîrlă3Department of Infomation Science and Engineering, Acharya Institute of Technology, Bengaluru 560 107, Karnataka, IndiaDepartment of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq, JordanDepartment of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, RomaniaDepartment of Mathematics, Tehnical University of Cluj-Napoca, 400114 Cluj-Napoca, RomaniaMaking use of generalized bivariate Fibonacci polynomials, we propose two families of regular functions of the type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>ζ</mi><mo>)</mo></mrow><mo>=</mo><mi>ζ</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>2</mn></mrow><mo>∞</mo></munderover></mstyle><msub><mi>d</mi><mi>j</mi></msub><msup><mi>ζ</mi><mi>j</mi></msup></mrow></semantics></math></inline-formula>, which are bi-univalent in the disc <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>ζ</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>:</mo><mo>|</mo><mi>ζ</mi><mo>|</mo><mo><</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula> involving the (<i>p</i>, <i>q</i>)-derivative operator. We find estimates on the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>d</mi><mn>2</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>d</mi><mn>3</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula> and the of Fekete–Szegö functional for members of these families. Relevant connections to the existing results and new consequences of the main result are presented.https://www.mdpi.com/2227-7390/12/24/3933bi-univalent functions(p, q)-derivative operatorsubordinationHorodam polynomialsFekete–Szegö functional |
| spellingShingle | Sondekola Rudra Swamy Basem Aref Frasin Daniel Breaz Luminita-Ioana Cotîrlă Two Families of Bi-Univalent Functions Associating the (<i>p</i>, <i>q</i>)-Derivative with Generalized Bivariate Fibonacci Polynomials Mathematics bi-univalent functions (p, q)-derivative operator subordination Horodam polynomials Fekete–Szegö functional |
| title | Two Families of Bi-Univalent Functions Associating the (<i>p</i>, <i>q</i>)-Derivative with Generalized Bivariate Fibonacci Polynomials |
| title_full | Two Families of Bi-Univalent Functions Associating the (<i>p</i>, <i>q</i>)-Derivative with Generalized Bivariate Fibonacci Polynomials |
| title_fullStr | Two Families of Bi-Univalent Functions Associating the (<i>p</i>, <i>q</i>)-Derivative with Generalized Bivariate Fibonacci Polynomials |
| title_full_unstemmed | Two Families of Bi-Univalent Functions Associating the (<i>p</i>, <i>q</i>)-Derivative with Generalized Bivariate Fibonacci Polynomials |
| title_short | Two Families of Bi-Univalent Functions Associating the (<i>p</i>, <i>q</i>)-Derivative with Generalized Bivariate Fibonacci Polynomials |
| title_sort | two families of bi univalent functions associating the i p i i q i derivative with generalized bivariate fibonacci polynomials |
| topic | bi-univalent functions (p, q)-derivative operator subordination Horodam polynomials Fekete–Szegö functional |
| url | https://www.mdpi.com/2227-7390/12/24/3933 |
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