A Class of <i>k</i>-Symmetric Harmonic Functions Involving a Certain <i>q</i>-Derivative Operator
In this paper, we introduce a new class of harmonic univalent functions with respect to <i>k</i>-symmetric points by using a newly-defined <i>q</i>-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a suffic...
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-07-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/9/15/1812 |
| _version_ | 1797525311857885184 |
|---|---|
| author | Hari M. Srivastava Nazar Khan Shahid Khan Qazi Zahoor Ahmad Bilal Khan |
| author_facet | Hari M. Srivastava Nazar Khan Shahid Khan Qazi Zahoor Ahmad Bilal Khan |
| author_sort | Hari M. Srivastava |
| collection | DOAJ |
| description | In this paper, we introduce a new class of harmonic univalent functions with respect to <i>k</i>-symmetric points by using a newly-defined <i>q</i>-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized <i>q</i>-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter <i>p</i> is obviously unnecessary. |
| first_indexed | 2024-03-10T09:11:58Z |
| format | Article |
| id | doaj.art-58dc6cd937ee4d8c9cd4802561f3e74f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T09:11:58Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-58dc6cd937ee4d8c9cd4802561f3e74f2023-11-22T05:57:05ZengMDPI AGMathematics2227-73902021-07-01915181210.3390/math9151812A Class of <i>k</i>-Symmetric Harmonic Functions Involving a Certain <i>q</i>-Derivative OperatorHari M. Srivastava0Nazar Khan1Shahid Khan2Qazi Zahoor Ahmad3Bilal Khan4Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaDepartment of Mathematics Abbottabad University of Science and Technology, Abbottabad 22010, PakistanDepartment of Mathematics, Riphah International University, Islamabad 44000, PakistanGovernment Akhtar Nawaz Khan (Shaheed) Degree College KTS, Haripur 22620, PakistanSchool of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai 200241, ChinaIn this paper, we introduce a new class of harmonic univalent functions with respect to <i>k</i>-symmetric points by using a newly-defined <i>q</i>-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized <i>q</i>-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter <i>p</i> is obviously unnecessary.https://www.mdpi.com/2227-7390/9/15/1812univalent functionsharmonic functions<i>q</i>-derivative (or <i>q</i>-difference) operator |
| spellingShingle | Hari M. Srivastava Nazar Khan Shahid Khan Qazi Zahoor Ahmad Bilal Khan A Class of <i>k</i>-Symmetric Harmonic Functions Involving a Certain <i>q</i>-Derivative Operator Mathematics univalent functions harmonic functions <i>q</i>-derivative (or <i>q</i>-difference) operator |
| title | A Class of <i>k</i>-Symmetric Harmonic Functions Involving a Certain <i>q</i>-Derivative Operator |
| title_full | A Class of <i>k</i>-Symmetric Harmonic Functions Involving a Certain <i>q</i>-Derivative Operator |
| title_fullStr | A Class of <i>k</i>-Symmetric Harmonic Functions Involving a Certain <i>q</i>-Derivative Operator |
| title_full_unstemmed | A Class of <i>k</i>-Symmetric Harmonic Functions Involving a Certain <i>q</i>-Derivative Operator |
| title_short | A Class of <i>k</i>-Symmetric Harmonic Functions Involving a Certain <i>q</i>-Derivative Operator |
| title_sort | class of i k i symmetric harmonic functions involving a certain i q i derivative operator |
| topic | univalent functions harmonic functions <i>q</i>-derivative (or <i>q</i>-difference) operator |
| url | https://www.mdpi.com/2227-7390/9/15/1812 |
| work_keys_str_mv | AT harimsrivastava aclassofikisymmetricharmonicfunctionsinvolvingacertainiqiderivativeoperator AT nazarkhan aclassofikisymmetricharmonicfunctionsinvolvingacertainiqiderivativeoperator AT shahidkhan aclassofikisymmetricharmonicfunctionsinvolvingacertainiqiderivativeoperator AT qazizahoorahmad aclassofikisymmetricharmonicfunctionsinvolvingacertainiqiderivativeoperator AT bilalkhan aclassofikisymmetricharmonicfunctionsinvolvingacertainiqiderivativeoperator AT harimsrivastava classofikisymmetricharmonicfunctionsinvolvingacertainiqiderivativeoperator AT nazarkhan classofikisymmetricharmonicfunctionsinvolvingacertainiqiderivativeoperator AT shahidkhan classofikisymmetricharmonicfunctionsinvolvingacertainiqiderivativeoperator AT qazizahoorahmad classofikisymmetricharmonicfunctionsinvolvingacertainiqiderivativeoperator AT bilalkhan classofikisymmetricharmonicfunctionsinvolvingacertainiqiderivativeoperator |