Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions
In this paper, we demonstrate a relationship between a generalized distribution series and a comprehensive subclass of analytic functions. The primary aim of this study is to determine a necessary and sufficient condition for the generalized distribution series <inline-formula><math xmlns=&...
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MDPI AG
2025-06-01
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| author | Tariq Al-Hawary Basem Frasin Ibtisam Aldawish |
| author_facet | Tariq Al-Hawary Basem Frasin Ibtisam Aldawish |
| author_sort | Tariq Al-Hawary |
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| description | In this paper, we demonstrate a relationship between a generalized distribution series and a comprehensive subclass of analytic functions. The primary aim of this study is to determine a necessary and sufficient condition for the generalized distribution series <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>E</mi><mrow><mi>ϕ</mi></mrow><mo>∗</mo></msubsup><mrow><mo>(</mo><mi>ς</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> to belong to the inclusive subclass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Π</mi><mi>η</mi></msub><mrow><mo>(</mo><msub><mi>Q</mi><mn>3</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>2</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>1</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Necessary and sufficient conditions are also given for the generalized distribution series <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>E</mi><mrow><mi>ϕ</mi></mrow><mo>∗</mo></msubsup><mrow><mo>(</mo><mi>ς</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>ℏ</mo></mrow></semantics></math></inline-formula> and the integral operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>J</mi><mrow><mi>ς</mi></mrow><mi>ϕ</mi></msubsup><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> to be in the inclusive subclass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Π</mi><mi>η</mi></msub><mrow><mo>(</mo><msub><mi>Q</mi><mn>3</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>2</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>1</mn></msub><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Further, we provide a number of corollaries, which improve the existing ones that are available in some recent studies. The results presented here not only improve the earlier studies, but also give rise to a number of new results for particular choices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mn>3</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>2</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mn>0</mn></msub></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-63e3c8cc19f54c07b3c5123e288a0101 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-63e3c8cc19f54c07b3c5123e288a01012025-08-20T03:27:18ZengMDPI AGMathematics2227-73902025-06-011312202910.3390/math13122029Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic FunctionsTariq Al-Hawary0Basem Frasin1Ibtisam Aldawish2Department of Applied Science, Ajloun College, Al Balqa Applied University, Ajloun 26816, JordanFaculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq 25113, JordanMathematics and Statistics Department, College of Science, IMSIU (Imam Mohammad Ibn Saud Islamic University), Riyadh 13327, Saudi ArabiaIn this paper, we demonstrate a relationship between a generalized distribution series and a comprehensive subclass of analytic functions. The primary aim of this study is to determine a necessary and sufficient condition for the generalized distribution series <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>E</mi><mrow><mi>ϕ</mi></mrow><mo>∗</mo></msubsup><mrow><mo>(</mo><mi>ς</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> to belong to the inclusive subclass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Π</mi><mi>η</mi></msub><mrow><mo>(</mo><msub><mi>Q</mi><mn>3</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>2</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>1</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Necessary and sufficient conditions are also given for the generalized distribution series <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>E</mi><mrow><mi>ϕ</mi></mrow><mo>∗</mo></msubsup><mrow><mo>(</mo><mi>ς</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>ℏ</mo></mrow></semantics></math></inline-formula> and the integral operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>J</mi><mrow><mi>ς</mi></mrow><mi>ϕ</mi></msubsup><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> to be in the inclusive subclass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Π</mi><mi>η</mi></msub><mrow><mo>(</mo><msub><mi>Q</mi><mn>3</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>2</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>1</mn></msub><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Further, we provide a number of corollaries, which improve the existing ones that are available in some recent studies. The results presented here not only improve the earlier studies, but also give rise to a number of new results for particular choices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mn>3</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>2</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mn>0</mn></msub></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/13/12/2029analyticunivalentgeneralized distributionintegral operator |
| spellingShingle | Tariq Al-Hawary Basem Frasin Ibtisam Aldawish Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions Mathematics analytic univalent generalized distribution integral operator |
| title | Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions |
| title_full | Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions |
| title_fullStr | Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions |
| title_full_unstemmed | Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions |
| title_short | Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions |
| title_sort | sufficient and necessary conditions for generalized distribution series on comprehensive subclass of analytic functions |
| topic | analytic univalent generalized distribution integral operator |
| url | https://www.mdpi.com/2227-7390/13/12/2029 |
| work_keys_str_mv | AT tariqalhawary sufficientandnecessaryconditionsforgeneralizeddistributionseriesoncomprehensivesubclassofanalyticfunctions AT basemfrasin sufficientandnecessaryconditionsforgeneralizeddistributionseriesoncomprehensivesubclassofanalyticfunctions AT ibtisamaldawish sufficientandnecessaryconditionsforgeneralizeddistributionseriesoncomprehensivesubclassofanalyticfunctions |