Generalized \(k\)-uniformly close-to-convex functions associated with conic regions. (English) Zbl 1254.30018

Let \(\phi(z, \lambda, \mu)\) be the generalized Harwitz-Lerch Zeta function given as \[ \phi(z, \lambda, \mu)=\sum^{\infty}_{n=0} \frac{z^{n}}{(\mu+n)^{\lambda}}, \quad \lambda \in \mathbb{C},\, \mu \in \mathbb{C} \backslash\{-1, -2, \dots\}. \tag{1} \] Using (1), the following family of linear operators is defined in terms of the Hadamard product as \[ J_{\lambda, \mu}f(z)=H_{\lambda, \mu}(z)*f(z), \tag{2} \] where \(f \in A,\) \[ H_{\lambda, \mu}(z)=(1+\mu)^{\lambda}[\phi(z, \lambda, \mu)-\mu^{-\lambda}], \quad z \in E, \tag{3} \] and \(\phi(z, \lambda, \mu)\) is given by (1).
From (2) and (3), one can write \[ J_{\lambda, \mu}f(z)=z+\sum^{\infty}_{n=2} \left(\frac{1+\mu}{n+\mu} \right)^{\lambda} a_{n}z^{n}. \] The author consider the operator \(I_{\lambda, \mu}: A \rightarrow A\) as \[ I_{\lambda, \mu}f(z)*J_{\lambda, \mu}f(z)=\frac{z}{(1-z)}, \quad \lambda \text{ real}, \mu> -1. \] This means: \[ I_{\lambda, \mu}f(z)=z+ \sum^{\infty}_{n=2} \left(\frac{n+\mu}{1+\mu} \right)^{\lambda} a_{n}z^{n}, \quad \lambda \text{ real}, \mu> -1. \] By using this operator, the author defines and study some subclasses of analytic functions. These functions map the open unit disc onto the domains formed by parabolic and hyperbolic regions and extend the concept of uniform close-to-convexity. Some interesting properties of these classes, which include inclusion results, coefficient problems, and invariance under certain integral operators, are discussed. The results are shown to be the best possible.


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