On a conjecture of Danikas and Ruschewyh (Q2712798)

Let \(f\) be in the Hardy class \(H^1\) and let NEWLINE\[NEWLINE\Phi_{a,b}(f)=\int\limits_a^bf\left( e^{it}\right)dt.NEWLINE\]NEWLINE Consider the following properties: NEWLINE\[NEWLINE\Phi_{a,b}(f')\neq 0,a<b\leq a+2\pi,\tag{i,}NEWLINE\]NEWLINE NEWLINE\[NEWLINEf \text{ is \;univalent in the unit disc} \Delta. \tag{ii}NEWLINE\]NEWLINE After the work of N. Danikas and V. Nestroridis (1985), where it was proved that for any \(z_0\in{\Delta}\) there exist a,b, \(a<b\leq a+2\pi\) such that \(\Phi_{a,b}(f)=(b-a)f(z_0)\), N. Danikas and S. Ruscheweyh in their work (1999) ask whether (i) implies (ii). The authors prove that the properties (i) and (ii) are independent giving negative answer to the question of Danikas and Ruscheweyh. To do this they study conformal maps of special think that such method can be applied for other problems .NEWLINENEWLINEFor the entire collection see [Zbl 0956.00046].