Some results for strongly starlike functions (Q1364831)

Let \(\alpha\), \(0\leq \alpha<1\), \(\beta,0 <\beta <1\) be given. A function \(f(z)= z+ \dots\), is said to belong to the class of functions strongly starlike of order \(\beta\) and type \(\alpha\) in the unit disk \(D\) provided \(|\arg({zf'(z) \over f(z)}- \alpha)|<{\pi \over 2} \beta\) holds in \(D\). In the case \(\alpha=0\) such functions have been studied by \textit{J. Stakiewicz} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A. 20 (1966) 59-75 (1971; Zbl 0247.30007), 20, 59-75 (1996; Zbl 0247.30007)]. The present authors derive here some sufficient conditions for \(f\) to belong to the class. Arguments are fairly straightforward.