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A Class of k -Symmetric Harmonic Functions Involving a Certain q -Derivative Operator

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  • Hari M. Srivastava

    (Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
    Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan
    Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy)

  • Nazar Khan

    (Department of Mathematics Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan)

  • Shahid Khan

    (Department of Mathematics, Riphah International University, Islamabad 44000, Pakistan)

  • Qazi Zahoor Ahmad

    (Government Akhtar Nawaz Khan (Shaheed) Degree College KTS, Haripur 22620, Pakistan)

  • Bilal Khan

    (School of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China)

Abstract

In this paper, we introduce a new class of harmonic univalent functions with respect to k -symmetric points by using a newly-defined q -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 q -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 ( p , q ) -variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter p is obviously unnecessary.

Suggested Citation

  • Hari M. Srivastava & Nazar Khan & Shahid Khan & Qazi Zahoor Ahmad & Bilal Khan, 2021. "A Class of k -Symmetric Harmonic Functions Involving a Certain q -Derivative Operator," Mathematics, MDPI, vol. 9(15), pages 1-14, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1812-:d:605611
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    References listed on IDEAS

    as
    1. Hari M. Srivastava & Qazi Zahoor Ahmad & Nasir Khan & Nazar Khan & Bilal Khan, 2019. "Hankel and Toeplitz Determinants for a Subclass of q -Starlike Functions Associated with a General Conic Domain," Mathematics, MDPI, vol. 7(2), pages 1-15, February.
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