Let \(A_n\) denote the set of functions \(f_{(z)}=z+a_{n+1}z^{n+1}+\dots\), \(n\geq 1\), that are analytic in the unit disc \(U=\{z:| z| <1\}, A=A_1\). Let \(S^*\) be the usual class of starlike (univalent) functions in \(U\), and let \(R=\{f\in A_1:\operatorname{Re} f^{\prime}(z)>0, z \in U\}\) be the class of functions with bounded turning. It is well known that \(S^*\) does not contain \(R\) and \(R\) does not contain \(S^*\).
Together with S. S. Miller, the author developed the well-known method of differential subordination [J. Differ. Equations 67, 199–211 (1987; Zbl 0633.34005); Differential subordinations: theory and applications. New York, NY: Marcel Dekker (2000; Zbl 0954.34003)]. This method is also used in the present paper to obtain the main result, which gives a subclass of \(S^*\) that is contained in \(R\). We cite here the main result:
Theorem: Let \(\alpha,M>0\) be such that \(| 1-\alpha| <M< \alpha\). If \(f\in A_{n}\) satisfies the condition \[ \left|\frac{zf^{\prime}(z)}{f(z)}-\alpha \right| <M ,\quad z \in U, \] where \(M=M(\alpha, n)\) is given by the equation \[ \frac{M^{2}-(1-\alpha)^{2}}{n| 1-\alpha|} \cdot \arctan \left(\frac{| 1-\alpha|}{\sqrt{M^{2}-(1-\alpha)^{2}}}\right)=\arctan\left(\frac{\sqrt{\alpha^{2}-M^{2}}}{M}\right) \] when \(\alpha \neq 1\) and by \(\cos \frac{M}{n}=M\) when \(\alpha=1\), then \(f\in R\), i.e., \(\operatorname{Re} f^{\prime}(z)>0\) for \(z \in U\).
Also, the author proposes an interesting conjecture:
If \(f\in A_{n}\) and \(zf^{\prime}(z)/f(z)\in E_{n}\) for all \(z\in U\), where \(E_{n}\) is the reunion of all discs with center \(\displaystyle{\alpha> \frac{1}{2}}\) and radius \(M=M(\alpha, n)\), then \(\operatorname{Re}f^{\prime}(z)>0\) for \(z\in U\).