On the Third Hankel Determinant of a Certain Subclass of Bi-Univalent Functions Defined by (<i>p</i>,<i>q</i>)-Derivative Operator
In this study, the generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow><...
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2025-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/8/1269 |
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| author | Mohammad El-Ityan Qasim Ali Shakir Tariq Al-Hawary Rafid Buti Daniel Breaz Luminita-Ioana Cotîrlă |
| author_facet | Mohammad El-Ityan Qasim Ali Shakir Tariq Al-Hawary Rafid Buti Daniel Breaz Luminita-Ioana Cotîrlă |
| author_sort | Mohammad El-Ityan |
| collection | DOAJ |
| description | In this study, the generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-derivative operator is used to define a novel class of bi-univalent functions. For this class, we define constraints on the coefficients up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">|</mo></mrow><msub><mo>ℓ</mo><mn>5</mn></msub><mrow><mo stretchy="false">|</mo></mrow></mrow></semantics></math></inline-formula>. The functions are analyzed using a suitable operational method, which enables us to derive new bounds for the Fekete–Szegö functional, as well as explicit estimates for important coefficients like <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">|</mo></mrow><msub><mo>ℓ</mo><mn>2</mn></msub><mrow><mo stretchy="false">|</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 stretchy="false">|</mo></mrow><msub><mo>ℓ</mo><mn>3</mn></msub><mrow><mo stretchy="false">|</mo></mrow></mrow></semantics></math></inline-formula>. In addition, we establish the upper bounds of the second and third Hankel determinants, providing insights into the geometrical and analytical properties of this class of functions. |
| format | Article |
| id | doaj-art-a3cae6f49c3e4e07a676883701001926 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-a3cae6f49c3e4e07a6768837010019262025-08-20T02:18:00ZengMDPI AGMathematics2227-73902025-04-01138126910.3390/math13081269On the Third Hankel Determinant of a Certain Subclass of Bi-Univalent Functions Defined by (<i>p</i>,<i>q</i>)-Derivative OperatorMohammad El-Ityan0Qasim Ali Shakir1Tariq Al-Hawary2Rafid Buti3Daniel Breaz4Luminita-Ioana Cotîrlă5Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Al-Salt 19117, JordanDepartment of Mathematics, College of Computer Science and Information Technology, University of Al-Qadisiyah, Diwaniyah 58006, IraqDepartment of Applied Science, Ajloun College, Al Balqa Applied University, Ajloun 26816, JordanDepartment of Mathematics, College of Education for Pure Science, Al Muthanna University, Al Muthanna 66002, IraqDepartment of Mathematics, University of Alba Iulia, 510009 Alba Iulia, RomaniaDepartment of Mathematics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, RomaniaIn this study, the generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-derivative operator is used to define a novel class of bi-univalent functions. For this class, we define constraints on the coefficients up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">|</mo></mrow><msub><mo>ℓ</mo><mn>5</mn></msub><mrow><mo stretchy="false">|</mo></mrow></mrow></semantics></math></inline-formula>. The functions are analyzed using a suitable operational method, which enables us to derive new bounds for the Fekete–Szegö functional, as well as explicit estimates for important coefficients like <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">|</mo></mrow><msub><mo>ℓ</mo><mn>2</mn></msub><mrow><mo stretchy="false">|</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 stretchy="false">|</mo></mrow><msub><mo>ℓ</mo><mn>3</mn></msub><mrow><mo stretchy="false">|</mo></mrow></mrow></semantics></math></inline-formula>. In addition, we establish the upper bounds of the second and third Hankel determinants, providing insights into the geometrical and analytical properties of this class of functions.https://www.mdpi.com/2227-7390/13/8/1269(<i>p</i>,<i>q</i>)-derivative operatorFekete–SzegöHankel determinantsunivalentbi-univalent functions |
| spellingShingle | Mohammad El-Ityan Qasim Ali Shakir Tariq Al-Hawary Rafid Buti Daniel Breaz Luminita-Ioana Cotîrlă On the Third Hankel Determinant of a Certain Subclass of Bi-Univalent Functions Defined by (<i>p</i>,<i>q</i>)-Derivative Operator Mathematics (<i>p</i>,<i>q</i>)-derivative operator Fekete–Szegö Hankel determinants univalent bi-univalent functions |
| title | On the Third Hankel Determinant of a Certain Subclass of Bi-Univalent Functions Defined by (<i>p</i>,<i>q</i>)-Derivative Operator |
| title_full | On the Third Hankel Determinant of a Certain Subclass of Bi-Univalent Functions Defined by (<i>p</i>,<i>q</i>)-Derivative Operator |
| title_fullStr | On the Third Hankel Determinant of a Certain Subclass of Bi-Univalent Functions Defined by (<i>p</i>,<i>q</i>)-Derivative Operator |
| title_full_unstemmed | On the Third Hankel Determinant of a Certain Subclass of Bi-Univalent Functions Defined by (<i>p</i>,<i>q</i>)-Derivative Operator |
| title_short | On the Third Hankel Determinant of a Certain Subclass of Bi-Univalent Functions Defined by (<i>p</i>,<i>q</i>)-Derivative Operator |
| title_sort | on the third hankel determinant of a certain subclass of bi univalent functions defined by i p i i q i derivative operator |
| topic | (<i>p</i>,<i>q</i>)-derivative operator Fekete–Szegö Hankel determinants univalent bi-univalent functions |
| url | https://www.mdpi.com/2227-7390/13/8/1269 |
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