Acu, Mugur; Al-Oboudi, Fatima; Darus, Maslina; Owa, Shigeyoshi; Polatoğlu, Yaşar; Yavuz, Emel On some \(\alpha\)-convex functions. (English) Zbl 1151.30307 Int. J. Open Probl. Comput. Sci. Math., IJOPCM 1, No. 1, 1-10 (2008). 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 Keywords:\(\alpha\)-convex functions; generalized Libera integral operator; Briot–Bouquet differential subordination; modified Sălăgean operator PDFBibTeX XMLCite \textit{M. Acu} et al., Int. J. Open Probl. Comput. Sci. Math., IJOPCM 1, No. 1, 1--10 (2008; Zbl 1151.30307) Full Text: EuDML