Strongly starlike functions (Q2739113)

Let \({\mathcal A}\) denote the class of functions \(f(z)=z+ \sum^\infty_{n=2} a_nz^n\) analytic and univalent in the unit disk \(\Delta\) and for \(\alpha\geq 0\), \(0\leq \beta<1\) and \(0<\gamma\leq 1\) let \(\overline S^*_{ \alpha, \beta}(\gamma)\) be the subclass of \({\mathcal A}\) consisting of such \(f\) that \({zf'(z)\over f(z)} \neq\beta\) and NEWLINE\[NEWLINE\left|\arg\left((1-\alpha) {zf' (z)\over f(z)}+ \alpha{\bigl(zf'(z) \bigr)'\over f'(z)}-\beta\right) \right |<\gamma {\pi\over 2},(z\in\Delta).NEWLINE\]NEWLINE Results: 1. A function \(f\) from \({\mathcal A}\) belongs to \(\overline S^*_{\alpha,\beta} (\gamma)\) \((a>0)\), if and only if NEWLINE\[NEWLINEf(z)= \left[{1\over \alpha}\int^z_0 \bigl[\varphi (t)\bigr]^{1 \over \alpha}{dt \over t}\right]^\alphaNEWLINE\]NEWLINE for some \(\varphi\) from \(\overline S^*_{0, \beta} (\gamma)\), (which satisfy \(|\arg ({z\varphi'(z)\over \varphi(z)}-\beta) |<\gamma {\pi \over 2}\), \((z\in\Delta)\).) 2. Sharp estimates on \(|a_n|= |a_n(f)|\), \(f\in\overline S^*_{\alpha,\beta} (\gamma) \), \(n=2,3\) as well as on \(|a_3-\mu a^3_2 |\) \((u\in R)\) are obtained.