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Some properties of a class of analytic functions involving a new generalized differential operator. (English) Zbl 1431.30009

Summary: In the present paper, we introduce a new generalized differential operator \(D^m_{\mu,\lambda,\sigma}(\alpha,\beta)\) defined on the open unit disc \(U=\{z \in \mathbb{C} :|z|< 1\}\). A novel subclass \(\Omega ^*_m(\delta,\lambda,\alpha,\beta, b)\) by means of the operator \(D^m_{\mu,\lambda,\sigma}(\alpha,\beta)\) is also introduced. Coefficient estimates, growth and distortion theorems, closure theorems, and class preserving integral operators for functions in the class \(\Omega^*_m(\delta,\lambda,\alpha,\beta,b)\) are discussed. Furthermore, sufficient conditions for close-to-convexity, starlikeness, and convexity for functions in the class \(\Omega ^*_m(\delta,\lambda,\alpha,\beta,b)\) are obtained.

MSC:

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

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