Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions
In the past few years, many scholars gave much attention to the use of <i>q</i>-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric <i>q</i>-calculus in geometric function theory, very little wo...
| Main Authors: | , , , , |
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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/2073-8994/14/10/2188 |
| _version_ | 1797469870415151104 |
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| author | Mohammad Faisal Khan Isra Al-Shbeil Najla Aloraini Nazar Khan Shahid Khan |
| author_facet | Mohammad Faisal Khan Isra Al-Shbeil Najla Aloraini Nazar Khan Shahid Khan |
| author_sort | Mohammad Faisal Khan |
| collection | DOAJ |
| description | In the past few years, many scholars gave much attention to the use of <i>q</i>-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric <i>q</i>-calculus in geometric function theory, very little work has been published so far. In this research, with the help of fundamental concepts of symmetric <i>q</i>-calculus and the symmetric <i>q</i>-Salagean differential operator for harmonic functions, we define a new class of harmonic functions connected with Janowski functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><msubsup><mi mathvariant="script">S</mi><mrow><mi mathvariant="script">H</mi></mrow><mn>0</mn></msubsup><mo>˜</mo></mover><mfenced separators="" open="(" close=")"><mi>m</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi mathvariant="script">A</mi><mo>,</mo><mi mathvariant="script">B</mi></mfenced></mrow></semantics></math></inline-formula>. First, we illustrate the necessary and sufficient convolution condition for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><msubsup><mi mathvariant="script">S</mi><mrow><mi mathvariant="script">H</mi></mrow><mn>0</mn></msubsup><mo>˜</mo></mover><mfenced separators="" open="(" close=")"><mi>m</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi mathvariant="script">A</mi><mo>,</mo><mi mathvariant="script">B</mi></mfenced></mrow></semantics></math></inline-formula> and then prove that this sufficient condition is a sense preserving and univalent, and it is necessary for its subclass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><msubsup><mi mathvariant="script">TS</mi><mrow><mi mathvariant="script">H</mi></mrow><mn>0</mn></msubsup><mo>˜</mo></mover><mfenced separators="" open="(" close=")"><mi>m</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi mathvariant="script">A</mi><mo>,</mo><mi mathvariant="script">B</mi></mfenced></mrow></semantics></math></inline-formula>. Furthermore, by using this necessary and sufficient coefficient condition, we establish some novel results, particularly convexity, compactness, radii of <i>q</i>-starlike and <i>q</i>-convex functions of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, and extreme points for this newly defined class of harmonic functions. Our results are the generalizations of some previous known results. |
| first_indexed | 2024-03-09T19:26:12Z |
| format | Article |
| id | doaj.art-c27829c05b0a4ccd8638457306d63dcb |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T19:26:12Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-c27829c05b0a4ccd8638457306d63dcb2023-11-24T02:53:56ZengMDPI AGSymmetry2073-89942022-10-011410218810.3390/sym14102188Applications of Symmetric Quantum Calculus to the Class of Harmonic FunctionsMohammad Faisal Khan0Isra Al-Shbeil1Najla Aloraini2Nazar Khan3Shahid Khan4Department of Basic Science, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi ArabiaDepartment of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, JordanDepartment of Mathematics, College of Arts and Sciences Onaizah, Qassim University, Buraidah 51452, Saudi ArabiaDepartment of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22500, PakistanDepartment of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22500, PakistanIn the past few years, many scholars gave much attention to the use of <i>q</i>-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric <i>q</i>-calculus in geometric function theory, very little work has been published so far. In this research, with the help of fundamental concepts of symmetric <i>q</i>-calculus and the symmetric <i>q</i>-Salagean differential operator for harmonic functions, we define a new class of harmonic functions connected with Janowski functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><msubsup><mi mathvariant="script">S</mi><mrow><mi mathvariant="script">H</mi></mrow><mn>0</mn></msubsup><mo>˜</mo></mover><mfenced separators="" open="(" close=")"><mi>m</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi mathvariant="script">A</mi><mo>,</mo><mi mathvariant="script">B</mi></mfenced></mrow></semantics></math></inline-formula>. First, we illustrate the necessary and sufficient convolution condition for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><msubsup><mi mathvariant="script">S</mi><mrow><mi mathvariant="script">H</mi></mrow><mn>0</mn></msubsup><mo>˜</mo></mover><mfenced separators="" open="(" close=")"><mi>m</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi mathvariant="script">A</mi><mo>,</mo><mi mathvariant="script">B</mi></mfenced></mrow></semantics></math></inline-formula> and then prove that this sufficient condition is a sense preserving and univalent, and it is necessary for its subclass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><msubsup><mi mathvariant="script">TS</mi><mrow><mi mathvariant="script">H</mi></mrow><mn>0</mn></msubsup><mo>˜</mo></mover><mfenced separators="" open="(" close=")"><mi>m</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi mathvariant="script">A</mi><mo>,</mo><mi mathvariant="script">B</mi></mfenced></mrow></semantics></math></inline-formula>. Furthermore, by using this necessary and sufficient coefficient condition, we establish some novel results, particularly convexity, compactness, radii of <i>q</i>-starlike and <i>q</i>-convex functions of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, and extreme points for this newly defined class of harmonic functions. Our results are the generalizations of some previous known results.https://www.mdpi.com/2073-8994/14/10/2188analytic functionssymmetric <i>q</i>-calculussymmetric <i>q</i>-derivative operatorharmonic functionsJanowski functionssymmetric Salagean <i>q</i>-differential operator |
| spellingShingle | Mohammad Faisal Khan Isra Al-Shbeil Najla Aloraini Nazar Khan Shahid Khan Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions Symmetry analytic functions symmetric <i>q</i>-calculus symmetric <i>q</i>-derivative operator harmonic functions Janowski functions symmetric Salagean <i>q</i>-differential operator |
| title | Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions |
| title_full | Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions |
| title_fullStr | Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions |
| title_full_unstemmed | Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions |
| title_short | Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions |
| title_sort | applications of symmetric quantum calculus to the class of harmonic functions |
| topic | analytic functions symmetric <i>q</i>-calculus symmetric <i>q</i>-derivative operator harmonic functions Janowski functions symmetric Salagean <i>q</i>-differential operator |
| url | https://www.mdpi.com/2073-8994/14/10/2188 |
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