One more note on neighborhoods of univalent functions (Q2228044)

The paper under review is aimed to pay tribute to the late German mathematician Stephan Ruscheweyh by raising some open questions on the concept of neighbourhood of a univalent function. Let \(\mathcal{A}_0\) denote the class of analytic functions \(f\) in the unit disk \(\mathbb{D}=\{z\in\mathbb{C}:\, |z|<1\}\) that satisfy the conditions \[f(0)=f^\prime (0)-1=0.\] Let \(f(z)=z+\sum_{n=2}^\infty a_nz^n\) be an element of \(\mathcal{A}_0\). It is well known that the condition \[\sum_{n=2}^\infty |a_n|\le 1\] implies that \(f\) is one-to-one (univalent) in \(\mathbb{D}\), and \(f(\mathbb{D})\) is star-like with respect to the origin. Let \(\mathcal{S}\) denote the subclass of \(\mathcal{A}_0\) of star-like functions (functions \(f\) with the property that \(f(\mathbb{D})\) is star-like with respect to the origin. In [Proc. Am. Math. Soc. 81, 521-527 (1981; Zbl 0458.30008)], \textit{S. Ruschweyh} introduced the following notion of \(\delta\)-neighbourhood for a given function \(f(z)=z+\sum_{n=2}^\infty a_nz^n\): \[N_\delta(f)=\left \{g(z)=z+\sum_{n=2}^\infty b_nz^n\in \mathcal{A}_0: \sum_{n=2}^\infty n|a_n - b_n|\le \delta\right \}.\] It is clear that if \(I\) is the identity function, then \(N_1(I)\) coincides with the class of star-like functions. It is also clear that the class of convex functions \(K\) (those functions \(f\in\mathcal{A}_0\) with the property that \(f(\mathbb{D})\) is convex) is a subclass of \(\mathcal{A}_0\). A result of Ruscheweyh states that: If \(f\in\mathcal{A}_0,\, \delta>0\), and \[\frac{f(z)+\epsilon z}{1+\epsilon}\in\mathcal{S},\quad -\delta <\epsilon <\delta,\] then \(N_\delta(f)\subseteq \mathcal{S}\). Ruscheweyh asked in [loc. cit.] if this result is valid if \(\mathcal{S}\) is replaced by the class \(C\) of close-to-convex functions. The author discusses a partial answer to this question, and states that the question is still unsolved.