A new criterion for starlike functions (Q1915533)

Let \(A\) be the class of functions \(f\), which are holomorphic in the unit disc \(D= \{z: |z|< 1\}\), with \(f(0)= f'(0)- 1= 0\). Let \(S^*\) be the set of starlike functions: \[ S^*= \{f\in A, \text{Re}(zf'(z)/f(z))> 0, z\in D\}. \] \textit{R. Singh} and \textit{S. Singh} [Proc. Am. Math. Soc. 106, No. 1, 145-152 (1989; Zbl 0672.30007)] proved that if \(f\in A\) and \(\text{Re}(f'(z)+ zf''(z))> - 1/4\), \(z\in D\), then \(f\in S^*\). In the present paper, it is proved that if \(f\in A\) and \(\text{Re}(f'(z)+ zf''(z))> 1- 3/[4(1- \ln 2)^2+ 2]\approx - 0,263\), \(z\in D\), then \(f\in S^*\).