Disproof of a conjecture on univalent functions (Q2769204)

Gruenberg, Rønning and Ruscheweyh made in 1900 the following conjecture. Conjecture. Let \({\mathcal S}\) denote the class of normalized univalent functions in \(\Delta= \{z:|z|<1\}\) and let \({\mathcal D}\) be the class of functions \(d(z)\), \(d(0)=1\), \(|d'(z)|\leq\text{Re} d(z)\) for \(z\in \Delta\). If \(f_1,f_2 \in{\mathcal S}\) and \(d\in {\mathcal D}\) we have NEWLINE\[NEWLINE\text{Re} \left \{d(z)* {1 \over z}\int^z_0 \bigl(f_1(t) *f_2(t)\bigr) {dt\over t}\right\} >0, \quad z\in\Delta,NEWLINE\]NEWLINE where \(*\) stands for the Hadamard product. In this paper the authors disproof this conjecture. They construct the special functions which give the counterexample.