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On some \(\alpha\)-convex functions. (English) Zbl 1151.30307

Summary: We define a general class of \(\alpha\)-convex functions, denoted by \(ML_{\beta,\alpha}(q)\), with respect to a convex domain \(D(q(z) \in \mathcal{H}_{u}(U), \, q(0) = 1, \, q(U) = D)\) contained in the right half plane by using the linear operator \(D_{\lambda}^{\beta}\) defined by \[ D_{\lambda}^{\beta} \, : \, A \to A, \]
\[ D_{\lambda}^{\beta}f(z)=z+\sum \limits_{j=2}^{\infty} {\left(1+\left(j-1\right) \lambda \right)}^{\beta} a_{j}z^{j}, \] where \(\beta,\lambda \in \mathbb{R}, \, \beta \geq 0, \, \lambda \geq 0\) and \(f(z)=z + \sum \limits_{j=2}^{\infty} a_{j}z^{j}\). Regarding the class \(ML_{\beta,\alpha}(q)\), we give a inclusion theorem and a transforming theorem, from which we may obtain many particular results.

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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