On the sharp constant for starlikeness (Q5931268)

The main result of the paper is the following. Let NEWLINE\[NEWLINEf(z)= z^p+ \sum^\infty_{n=p+1} a_n z^n,\quad p\in\mathbb{N},NEWLINE\]NEWLINE be an analytic function in the unit disk and NEWLINE\[NEWLINE\Biggl|1+ {zf^{(p+ 1)}(z)\over f^{(p)}(z)}\Biggr|< A\Biggl|{zf^{(p)}(z)\over f^{(p- 1)}(z)}\Biggr|,NEWLINE\]NEWLINE where \(A\) is the root of the equation \(xe^{1/(x^2- 1)}= x+ 1\). Then \(\text{Re}\{{zf'(z)\over f(z)}\}> 0\) in the unit disk. The constant \(A\) is sharp.