Majorization Results Based upon the Bernardi Integral Operator
By making use of some families of integral and derivative operators, many distinct subclasses of analytic, starlike functions, and symmetric starlike functions have already been defined and investigated from numerous perspectives. In this article, with the help of the one-parameter Bernardi integral...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-07-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/7/1404 |
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| author | Isra Al-Shbeil Hari Mohan Srivastava Muhammad Arif Mirajul Haq Nazar Khan Bilal Khan |
| author_facet | Isra Al-Shbeil Hari Mohan Srivastava Muhammad Arif Mirajul Haq Nazar Khan Bilal Khan |
| author_sort | Isra Al-Shbeil |
| collection | DOAJ |
| description | By making use of some families of integral and derivative operators, many distinct subclasses of analytic, starlike functions, and symmetric starlike functions have already been defined and investigated from numerous perspectives. In this article, with the help of the one-parameter Bernardi integral operator, we investigate several majorization results for the class of normalized starlike functions, which are associated with the Janowski functions. We also give some particular cases of our main results. Finally, we direct the interested readers to the possibility of examining the fundamental or quantum (or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula>-) extensions of the findings provided in this work in the concluding section. However, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="fraktur">p</mi><mo>,</mo><mi mathvariant="fraktur">q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-variations of the suggested <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula>-results will provide relatively minor and inconsequential developments because the additional (rather forced-in) parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">p</mi></semantics></math></inline-formula> is obviously redundant. |
| first_indexed | 2024-03-09T12:56:12Z |
| format | Article |
| id | doaj.art-b52fb982ca414f258cbdb036052dd870 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T12:56:12Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-b52fb982ca414f258cbdb036052dd8702023-11-30T22:00:03ZengMDPI AGSymmetry2073-89942022-07-01147140410.3390/sym14071404Majorization Results Based upon the Bernardi Integral OperatorIsra Al-Shbeil0Hari Mohan Srivastava1Muhammad Arif2Mirajul Haq3Nazar Khan4Bilal Khan5Department of Mathematics, Faculty of Science, the University of Jordan, Amman 11942, JordanDepartment of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaDepartment of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, PakistanDepartment of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, PakistanDepartment of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, PakistanSchool of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, ChinaBy making use of some families of integral and derivative operators, many distinct subclasses of analytic, starlike functions, and symmetric starlike functions have already been defined and investigated from numerous perspectives. In this article, with the help of the one-parameter Bernardi integral operator, we investigate several majorization results for the class of normalized starlike functions, which are associated with the Janowski functions. We also give some particular cases of our main results. Finally, we direct the interested readers to the possibility of examining the fundamental or quantum (or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula>-) extensions of the findings provided in this work in the concluding section. However, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="fraktur">p</mi><mo>,</mo><mi mathvariant="fraktur">q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-variations of the suggested <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula>-results will provide relatively minor and inconsequential developments because the additional (rather forced-in) parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">p</mi></semantics></math></inline-formula> is obviously redundant.https://www.mdpi.com/2073-8994/14/7/1404analytic (holomorphic or regular) functionsunivalent functionsstarlike functionsmajorization problemsBernardi integral operatorcarathéodory class of functions |
| spellingShingle | Isra Al-Shbeil Hari Mohan Srivastava Muhammad Arif Mirajul Haq Nazar Khan Bilal Khan Majorization Results Based upon the Bernardi Integral Operator Symmetry analytic (holomorphic or regular) functions univalent functions starlike functions majorization problems Bernardi integral operator carathéodory class of functions |
| title | Majorization Results Based upon the Bernardi Integral Operator |
| title_full | Majorization Results Based upon the Bernardi Integral Operator |
| title_fullStr | Majorization Results Based upon the Bernardi Integral Operator |
| title_full_unstemmed | Majorization Results Based upon the Bernardi Integral Operator |
| title_short | Majorization Results Based upon the Bernardi Integral Operator |
| title_sort | majorization results based upon the bernardi integral operator |
| topic | analytic (holomorphic or regular) functions univalent functions starlike functions majorization problems Bernardi integral operator carathéodory class of functions |
| url | https://www.mdpi.com/2073-8994/14/7/1404 |
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