Generalized classes of starlike and convex functions of order \(\alpha\)
Let S denote the class of functions \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) which are analytic and univalent in the unit disk \(U=\{z:\) \(| z| <1\}\). The authors use the integrals of the fractional calculus to generalize \(S^*(\alpha)\) and K(\(\alpha)\) the starlike and convex functions of order \(\alpha\), respectively. The generalized classes \(S^*(\alpha,\lambda)\) and K(\(\alpha\),\(\lambda)\), originally introduced by Owa, are defined in the paper as follows: \(S^*(\alpha,\lambda)\) consists of functions f(z) in S which satisfy the inequality Re(\(\frac{\Lambda (\lambda,f)}{f(z)})>0\) for \(\lambda <1\), \(0\leq \alpha <1\) and all \(z\in U\), where \(\Lambda (\lambda,f)=\Gamma (1- \lambda)z^{1+\lambda} D_ z^{1+\lambda} f(z).\) Here \(D_ z^{\lambda} f(z)\) is the fractional derivative of order \(\lambda\) \((0\leq \lambda <1)\) given by \[ D_ z^{\lambda} f(z)=\frac{1}{\Gamma (1-\lambda)}\frac{d}{dz}\int^{z}_{0}\frac{f(\xi)d\xi}{(z- \xi)^{\lambda}}, \] and for \(n=0,1,2,..\). \[ D_ z^{n+\lambda} f(z)=\frac{d^ n}{dz^ n}D_ z^{\lambda} f(z) \] is the fractional derivative of order \(n+\lambda\). In addition, K(\(\alpha\),\(\lambda)\) is the class of all functions f(z) in S such that \(\Lambda (\lambda,f)\in S^*(\alpha,\lambda)\) for \(\lambda <1\) and \(0\leq \alpha <1\). For \(S^*(\alpha,\lambda)\) and K(\(\alpha\),\(\lambda)\) the authors establish subordination and argument theorems, together with some results for functions f(z) in these classes which admit the series expansion \(f(z)=z- \sum^{\infty}_{n=2}a_ nz^ n\) in U with \(a_ n\geq 0\) for all n. In particular, various results on the modified Hadamard product of such functions are included. Finally, some results on the generalized Libera operator \[ J_ c(f)=\frac{c+1}{z^ c}\int^{z}_{0}t^{c-1} f(t)dt\quad (c\geq 0) \] for f in the above classes are also given.
scientific article; zbMATH DE number 3944207