The order of starlikeness and close-to-convexity for certain analytic functions (Q2748616)

Given \(\alpha>0\), \(\beta\in[0,1)\) and \(n=1,2,\dots\). For the class \(M(n,\alpha,\beta)\) of holomorphic functions NEWLINE\[NEWLINEf(z)=z+a_{n+1}z^{n+1}+a_{n+2}z^{n+2}+\dots,\;|z|<1NEWLINE\]NEWLINE which satisfy the condition NEWLINE\[NEWLINE\text{Re}\left\{(1-\alpha)\frac{zf'(z)}{f(z)}+\alpha \left(1+\frac{zf''(z)}{f'(z)}\right)\right\}>\beta,\quad|z|<1NEWLINE\]NEWLINE the authors determine the sharp value NEWLINE\[NEWLINE\min_{f\in M(n,\alpha,\beta)}\text{Re}\left\{\frac{zf'(z)}{f(z)}\right\}.NEWLINE\]NEWLINE The obtained result generalize several previously known results for particular values of \(\alpha\), \(\beta\) and \(n\).