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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/12/2029 |
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| Summary: | 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>. |
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| ISSN: | 2227-7390 |