On functions starlike with respect to symmetric and conjugate points
For given fixed numbers \(\alpha,\beta, 0\leq\alpha\leq 1\), \(0\leq\beta\leq 1\), \(S^* (\alpha,\beta)\) denotes the class of functions of the form \(f(z)=z+ a_2z^2+ \dots\), that satisfy the condition \[ \Bigl| {zf'(z)\over f(z) -(-z)}-1 \Bigr|< \beta\Bigl| {\alpha zf' (z)\over f(z)-f(-z)} +1\Bigr| \] for \(| z| <1\). Using the well-known method of \textit{J. G. Clunie} and \textit{F. R. Keogh} [J. Lond. Math. Soc. 35, 229-233 (1960; Zbl 0092.07303)] the authors established several inequalities for coefficients and they gave a sufficient condition for \(f\) to belong to \(S^*(\alpha,\beta)\). Reviewer's remark. The authors claim the inequalities (i) and (ii) of Theorem 2.2 to be sharp for the function \(f(z)= {1\over 2} (1+\alpha \beta^2) {z\over 1-z}\). This function, however, is in \(S^*(\alpha, \beta)\) provided \(\alpha= \beta=1\).
scientific article; zbMATH DE number 1160427