Remarks on a paper by Silverman
The author gives a very nice and short proof of the following result. Let \(f\) be analytic in the unit disc \(U\), \(f(0)=f'(0)- 1=0\) and \(b\in (0,1]\). Then \[ \Biggl|{1+ {zf''(z)\over f'(z)}\over {zf'(z)\over f(z)}}- 1\Biggr|< b,\qquad z\in U, \] implies \(\text{Re}({zf'(z)\over f(z)})> {1\over 1+b}\), \(z\in U\). The result is sharp and answers a question of \textit{H. Silverman} [Int. J. Math. Math. Sci. 22, No. 1, 75-79 (1999; Zbl 0921.30009)].
scientific article; zbMATH DE number 1751247