Duality on <i>q</i>-Starlike Functions Associated with Fractional <i>q</i>-Integral Operators and Applications
In this paper, we make use of the Riemann–Liouville fractional <i>q</i>-integral operator to discuss the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>S</mi&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/10/2076 |
| _version_ | 1797469873603870720 |
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| author | Ebrahim Amini Shrideh Al-Omari Mojtaba Fardi Kamsing Nonlaopon |
| author_facet | Ebrahim Amini Shrideh Al-Omari Mojtaba Fardi Kamsing Nonlaopon |
| author_sort | Ebrahim Amini |
| collection | DOAJ |
| description | In this paper, we make use of the Riemann–Liouville fractional <i>q</i>-integral operator to discuss the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>S</mi><mrow><mi>q</mi><mo>,</mo><mi>δ</mi></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of univalent functions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>α</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>−</mo><mo>{</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mo>|</mo><mi>q</mi><mo>|</mo><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Then, we develop convolution results for the given class of univalent functions by utilizing a concept of the fractional <i>q</i>-difference operator. Moreover, we derive the normalized classes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">P</mi><mrow><mi>δ</mi><mo>,</mo><mi>q</mi></mrow><mi>ζ</mi></msubsup><mrow><mo>(</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">P</mi><mrow><mi>δ</mi><mo>,</mo><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>β</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mo>|</mo><mi>q</mi><mo>|</mo><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>≤</mo><mi>β</mi><mo>≤</mo><mn>1</mn><mo>,</mo><mi>ζ</mi><mo>></mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula> of analytic functions on a unit disc and provide conditions for the parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>ζ</mi><mo>,</mo><mi>β</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> so that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">P</mi><mrow><mi>δ</mi><mo>,</mo><mi>q</mi></mrow><mi>ζ</mi></msubsup><mrow><mo>(</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo></mrow><mo>⊂</mo><msubsup><mi>S</mi><mrow><mi>q</mi><mo>,</mo><mi>δ</mi></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">P</mi><mrow><mi>δ</mi><mo>,</mo><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>β</mi><mo>)</mo></mrow><mo>⊂</mo><msubsup><mi>S</mi><mrow><mi>q</mi><mo>,</mo><mi>δ</mi></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>−</mo><mo>{</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula>. Finally, we also propose an application to symmetric <i>q</i>-analogues and Ruscheweh’s duality theory. |
| first_indexed | 2024-03-09T19:27:13Z |
| format | Article |
| id | doaj.art-d1149197eef7403cb6ff7b43119173bd |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T19:27:13Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-d1149197eef7403cb6ff7b43119173bd2023-11-24T02:51:57ZengMDPI AGSymmetry2073-89942022-10-011410207610.3390/sym14102076Duality on <i>q</i>-Starlike Functions Associated with Fractional <i>q</i>-Integral Operators and ApplicationsEbrahim Amini0Shrideh Al-Omari1Mojtaba Fardi2Kamsing Nonlaopon3Department of Mathematics, Payme Noor University, Tehran P.O. Box 19395-4697, IranDepartment of Scientific Basic Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, JordanDepartment of Applied Mathematics, Faculty of Mathematical Science, Shahrekord University, Shahrekord P.O. Box 115, IranDepartment of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, ThailandIn this paper, we make use of the Riemann–Liouville fractional <i>q</i>-integral operator to discuss the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>S</mi><mrow><mi>q</mi><mo>,</mo><mi>δ</mi></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of univalent functions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>α</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>−</mo><mo>{</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mo>|</mo><mi>q</mi><mo>|</mo><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Then, we develop convolution results for the given class of univalent functions by utilizing a concept of the fractional <i>q</i>-difference operator. Moreover, we derive the normalized classes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">P</mi><mrow><mi>δ</mi><mo>,</mo><mi>q</mi></mrow><mi>ζ</mi></msubsup><mrow><mo>(</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">P</mi><mrow><mi>δ</mi><mo>,</mo><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>β</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mo>|</mo><mi>q</mi><mo>|</mo><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>≤</mo><mi>β</mi><mo>≤</mo><mn>1</mn><mo>,</mo><mi>ζ</mi><mo>></mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula> of analytic functions on a unit disc and provide conditions for the parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>ζ</mi><mo>,</mo><mi>β</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> so that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">P</mi><mrow><mi>δ</mi><mo>,</mo><mi>q</mi></mrow><mi>ζ</mi></msubsup><mrow><mo>(</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo></mrow><mo>⊂</mo><msubsup><mi>S</mi><mrow><mi>q</mi><mo>,</mo><mi>δ</mi></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">P</mi><mrow><mi>δ</mi><mo>,</mo><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>β</mi><mo>)</mo></mrow><mo>⊂</mo><msubsup><mi>S</mi><mrow><mi>q</mi><mo>,</mo><mi>δ</mi></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>−</mo><mo>{</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula>. Finally, we also propose an application to symmetric <i>q</i>-analogues and Ruscheweh’s duality theory.https://www.mdpi.com/2073-8994/14/10/2076Riemann–Liouville<i>q</i>-analoguedifference operator<i>q</i>-starlike functionsduality principledual set |
| spellingShingle | Ebrahim Amini Shrideh Al-Omari Mojtaba Fardi Kamsing Nonlaopon Duality on <i>q</i>-Starlike Functions Associated with Fractional <i>q</i>-Integral Operators and Applications Symmetry Riemann–Liouville <i>q</i>-analogue difference operator <i>q</i>-starlike functions duality principle dual set |
| title | Duality on <i>q</i>-Starlike Functions Associated with Fractional <i>q</i>-Integral Operators and Applications |
| title_full | Duality on <i>q</i>-Starlike Functions Associated with Fractional <i>q</i>-Integral Operators and Applications |
| title_fullStr | Duality on <i>q</i>-Starlike Functions Associated with Fractional <i>q</i>-Integral Operators and Applications |
| title_full_unstemmed | Duality on <i>q</i>-Starlike Functions Associated with Fractional <i>q</i>-Integral Operators and Applications |
| title_short | Duality on <i>q</i>-Starlike Functions Associated with Fractional <i>q</i>-Integral Operators and Applications |
| title_sort | duality on i q i starlike functions associated with fractional i q i integral operators and applications |
| topic | Riemann–Liouville <i>q</i>-analogue difference operator <i>q</i>-starlike functions duality principle dual set |
| url | https://www.mdpi.com/2073-8994/14/10/2076 |
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