Abstract
We consider the convolution operator dζf(z)=1zf(z)∗z(1-ζz)(1-z)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathrm{d}_\\zeta f(z) = \\frac{1}{z}\\left\\{ f(z)*\\frac{z}{(1-\\zeta z)(1-z)}\\right\\} \\end{aligned}$$\\end{document}on the class of analytic functions f(z)=z+a_2z^2+cdots , |z|<1, in the complex plane, where zeta is complex, |zeta |le 1. For zeta =1, the operator becomes the derivative f'(z), while for real zeta =q, 0<q<1 we obtain the Jackson’s q-derivative mathrm{d}_qf(z).
Highlights
We consider the convolution operator dζ fz) on the class of analytic functions f (z) = z + a2z2 + · · · , |z| < 1, in the complex plane, where ζ is complex, |ζ | ≤ 1. For ζ = 1, the operator becomes the derivative f (z), while for real ζ = q, 0 < q < 1 we obtain the Jackson’s q-derivative dq f (z). Keywords Analytic functions · Convex functions · Starlike functions · q-calculus · q-starlike · q-derivative
Let H denote the class of analytic functions in the unit disc D = {z : |z| < 1} on the complex plane C
Let the function f ∈ H be univalent in the unit disc D with the normalization f (0) = 0
Summary
z) on the class of analytic functions f (z) = z + a2z2 + · · · , |z| < 1, in the complex plane, where ζ is complex, |ζ | ≤ 1. For ζ = 1, the operator becomes the derivative f (z), while for real ζ = q, 0 < q < 1 we obtain the Jackson’s q-derivative dq f (z). Keywords Analytic functions · Convex functions · Starlike functions · q-calculus · q-starlike · q-derivative
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