On certain sufficient conditions for starlikeness and convexity (Q1390178)

Let \({\mathcal A}\) denote the class of functions \(f\) analytic in the unit \(E=\{z: | z|<1\}\). The class of functions \(f\in{\mathcal A}\), satisfying the condition \(\text{Re} \{\alpha {z^2f''(z) \over f(z)} +{zf'(z) \over f(z)}\}>0\), \(z\in E\) for some \(\alpha\geq 0\) and where \(f(z)\mid z\neq 0\) was investigated by the reviewer along with two others about three years back. Recently Nunokawa and others studied the same class and refined some of the earlier results. In this paper, the condition involving the real part is replaced by the assumption \[ \left\{\alpha {z^2f''(z) \over f(z)}+ {zf'(z)\over f(z)}\right\} \prec 1+\lambda z,\quad {f(z)\over z} \neq 0,\;z\in E,\;\alpha>0, \] \(\lambda>0\), where \(\prec\) denotes subordination and sufficient conditions for starlikeness and convexity are derived for \(f\) in \({\mathcal A}\) satisfying the above subordination.