Bell Distribution Series Defined on Subclasses of Bi-Univalent Functions That Are Subordinate to Horadam Polynomials
Several different subclasses of the bi-univalent function class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Σ</mi></semantics></math></inline-formu...
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MDPI AG
2023-04-01
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| Online Access: | https://www.mdpi.com/2075-1680/12/4/362 |
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| author | Ibtisam Aldawish Basem Frasin Ala Amourah |
| author_facet | Ibtisam Aldawish Basem Frasin Ala Amourah |
| author_sort | Ibtisam Aldawish |
| collection | DOAJ |
| description | Several different subclasses of the bi-univalent function class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Σ</mi></semantics></math></inline-formula> were introduced and studied by many authors using distribution series like Pascal distribution, Poisson distribution, Borel distribution, the Mittag-Leffler-type Borel distribution, Miller–Ross-Type Poisson Distribution. In the present paper, by making use of the Bell distribution, we introduce and investigate a new family <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">G</mi><mrow><mi mathvariant="sans-serif">Σ</mi></mrow><mi>t</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of normalized bi-univalent functions in the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">U</mi></semantics></math></inline-formula>, which are associated with the Horadam polynomials and estimate the second and the third coefficients in the Taylor-Maclaurin expansions of functions belonging to this class. Furthermore, we establish the Fekete–Szegö inequality for functions in the family <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">G</mi><mrow><mi mathvariant="sans-serif">Σ</mi></mrow><mi>t</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> After specializing the parameters used in our main results, a number of new results are demonstrated to follow. |
| first_indexed | 2024-03-11T05:15:22Z |
| format | Article |
| id | doaj.art-be4ad6d54dd848e59dfe56b6176a5581 |
| institution | Directory Open Access Journal |
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| language | English |
| last_indexed | 2024-03-11T05:15:22Z |
| publishDate | 2023-04-01 |
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| spelling | doaj.art-be4ad6d54dd848e59dfe56b6176a55812023-11-17T18:19:19ZengMDPI AGAxioms2075-16802023-04-0112436210.3390/axioms12040362Bell Distribution Series Defined on Subclasses of Bi-Univalent Functions That Are Subordinate to Horadam PolynomialsIbtisam Aldawish0Basem Frasin1Ala Amourah2Department of Mathematics and Statistics, College of Science, IMSIU (Imam Mohammad Ibn Saud Islamic University), Riyadh 11564, Saudi ArabiaFaculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq 25113, JordanDepartment of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 21110, JordanSeveral different subclasses of the bi-univalent function class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Σ</mi></semantics></math></inline-formula> were introduced and studied by many authors using distribution series like Pascal distribution, Poisson distribution, Borel distribution, the Mittag-Leffler-type Borel distribution, Miller–Ross-Type Poisson Distribution. In the present paper, by making use of the Bell distribution, we introduce and investigate a new family <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">G</mi><mrow><mi mathvariant="sans-serif">Σ</mi></mrow><mi>t</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of normalized bi-univalent functions in the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">U</mi></semantics></math></inline-formula>, which are associated with the Horadam polynomials and estimate the second and the third coefficients in the Taylor-Maclaurin expansions of functions belonging to this class. Furthermore, we establish the Fekete–Szegö inequality for functions in the family <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">G</mi><mrow><mi mathvariant="sans-serif">Σ</mi></mrow><mi>t</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> After specializing the parameters used in our main results, a number of new results are demonstrated to follow.https://www.mdpi.com/2075-1680/12/4/362fekete-Szegö problemhoradam polynomialsbi-univalent functionsbell distributionanalytic functions |
| spellingShingle | Ibtisam Aldawish Basem Frasin Ala Amourah Bell Distribution Series Defined on Subclasses of Bi-Univalent Functions That Are Subordinate to Horadam Polynomials Axioms fekete-Szegö problem horadam polynomials bi-univalent functions bell distribution analytic functions |
| title | Bell Distribution Series Defined on Subclasses of Bi-Univalent Functions That Are Subordinate to Horadam Polynomials |
| title_full | Bell Distribution Series Defined on Subclasses of Bi-Univalent Functions That Are Subordinate to Horadam Polynomials |
| title_fullStr | Bell Distribution Series Defined on Subclasses of Bi-Univalent Functions That Are Subordinate to Horadam Polynomials |
| title_full_unstemmed | Bell Distribution Series Defined on Subclasses of Bi-Univalent Functions That Are Subordinate to Horadam Polynomials |
| title_short | Bell Distribution Series Defined on Subclasses of Bi-Univalent Functions That Are Subordinate to Horadam Polynomials |
| title_sort | bell distribution series defined on subclasses of bi univalent functions that are subordinate to horadam polynomials |
| topic | fekete-Szegö problem horadam polynomials bi-univalent functions bell distribution analytic functions |
| url | https://www.mdpi.com/2075-1680/12/4/362 |
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