Harmonic close-to-convex mappings associated with Salagean \(q\)-differential operator

Omendra Mishra, Asena Cetinkaya, Janusz Sokol

Abstract


In this paper, we define  a new subclass \(\mathcal{W}(n,\alpha ,q)\) of analytic functions and a new subclass \(\mathcal{W}_\mathcal{H}^{0}(n,\alpha ,q)\) of harmonic functions \(f=h+\overline{g}\in \mathcal{H}^{0}\) associated with Salagean \(q\)-differential operator. We prove that  a harmonic function \(f=h+\bar{g}\) belongs to the class \(\mathcal{W}_\mathcal{H}^{0}(n,\alpha ,q)\) if and only if the analytic functions \(h+\epsilon g\) belong to \(\mathcal{W}(n,\alpha ,q)\) for each \(\epsilon \ (|\epsilon| = 1)\), and using a method by Clunie and Sheil-Small, we determine a sufficient condition for the class \(\mathcal{W}_\mathcal{H}^{0}(n,\alpha ,q)\) to be close-to-convex. We provide sharp coefficient estimates, sufficient coefficient condition, and convolution properties for such functions classes.  We also determine several conditions of partial sums of \(f\in\mathcal{W}_\mathcal{H}^{0}(n,\alpha ,q)\).

Keywords


Salagean q-differential operator; analytic functions; harmonic functions; partial sums

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References


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DOI: http://dx.doi.org/10.24193/subbmath.2025.1.03

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