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Subclasses of starlike functions of complex order involving generalized hypergeometric functions. (English) Zbl 1242.30012

Let \(A\) denote the class of functions of the form \[ \displaystyle{f(z)=z+\sum^{\infty}_{n=2}a_{n}z^{n}} \] which are analytic and univalent in the open disc \(\mathbb{U}\). For functions \(\displaystyle{f(z)=z+\sum^{\infty}_{n=2}a_{n}z^{n}}\) and \(\displaystyle{g(z)=z+\sum^{\infty}_{n=2}b_{n}z^{n}}\) in \(A\), the Hadamard product (or convolution) of \(f\) and \(g\) is defined by \[ \displaystyle{(f*g)(z)=z+\sum^{\infty}_{n=2}a_{n}b_{n}z^{n},} \] where \(z\in \mathbb{U}\).
The author introduces the following subclass: for \(-1\leq \alpha<1\), \(\beta\geq 0\) and \(\gamma \in \mathbb{C}\setminus \{0\}\) let \(S^{\tau}_{\lambda}(\alpha, \beta, \gamma)\) be the subclass of \(A\) consisting of functions satisfying the criterion \[ \displaystyle{Re \left\{1+\frac{1}{\gamma}\left(\frac{z(L^{\tau,\alpha_{1}}_{\lambda,l,m}f(z))^{\prime}}{L^{\tau,\alpha_{1}}_{\lambda,l,m}f(z)}-\alpha \right) \right\}} \displaystyle{>\beta \left| 1+\frac{1}{\gamma}\left(\frac{z(L^{\tau,\alpha_{1}}_{\lambda,l,m}f(z))^{\prime}}{L^{\tau,\alpha_{1}}_{\lambda,l,m}f(z)}-1 \right) \right|, z \in U}. \] (\(L^{\tau,\alpha_{1}}_{\lambda,l,m}\) denotes a linear operator defined by Srivastava and coauthors.) Also, let \(T S^{\tau}_{\lambda}(\alpha,\beta,\gamma)=S^{\tau}_{\lambda}(\alpha,\beta,\gamma)\cap T\).
The main object of the paper is to study some quantities and properties that are usually of interest in geometric function theory such as the coefficient bound, extreme points, radii of close to convexity, starlikeness and convexity for the class \(T S^{\tau}_{\lambda}(\alpha,\beta,\gamma)\).
Reviewer: Mugur Acu (Sibiu)

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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