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Study of univalent holomorphic functions with negative coefficients by using generalized fractional derivative. (English) Zbl 1123.30005

Let \(\mathcal{A}\) be the class of analytic functions in the unit disc \(U=\{z:| z| <1\}\) normalized such that \(f(0)=f'(0)-1=0\) and let \(T\) be a subclass of \(\mathcal{A}\) of functions \(f(z)=z-\sum_{n=2}^\infty a_n z^n.\) Further let \(F(z)=z-\sum_{n=2}^\infty A_n^{(\alpha,\beta)} z^n\in T(\alpha,\beta)\subset T\) if and only if
\[ F(z)=(1-\alpha)f(z)+\alpha zf'(z)+\beta z^2f''(z) +(1-\beta)z^3 f'''(z) \]
and \(F(z)\in \Omega(\lambda,\xi,\theta,\nu)\subset T(\alpha,\beta)\subset T\) if and only if \[ \left| \frac{\frac{\Gamma(2-\eta)\Gamma(2-k+\gamma)}{\Gamma(2-\eta+\gamma)}z^{\eta-1} D_{0,z}^{k,\eta,\gamma}F(z)-1}{2\lambda \frac{\Gamma(2-\eta)\Gamma(2-k+\gamma)}{\Gamma(2-\eta+\gamma)} z^{\eta-1}D_{0,z}^{k,\eta,\gamma}F(z)-\xi(1+\theta)\lambda} \right| <\nu, \]
where \(D_{0,z}^{k,\eta,\gamma}\) is the generalization of a fractional derivative.
The authors give several properties for the last class such as coefficient estimates, Hadamard product, weighted and arithmetic mean and some integral operators acting on it are studied.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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