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

Hari M. Srivastava, Nazar Khan, Shahid Khan, Qazi Zahoor Ahmad and Bilal Khan
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Hari M. Srivastava: Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
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

Mathematics, 2021, vol. 9, issue 15, 1-14

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.

Keywords: univalent functions; harmonic functions; q -derivative (or q -difference) operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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