Let \(F_{\alpha}\) denote the class of normalized \(\alpha\)-spirallike functions in the unit disc D. Also, let C denote the class of normalized close-to-convex functions in D. M. R. Ziegler [Trans. Am. Math. Soc. 166, 59-70 (1972; Zbl 0245.30008)] introduced the class A which contains both \(F_{\alpha}\) and C as follows: \(f\in A\) if \(f(0)=0,\quad f'(0)=1\) and for some g(z) in \(\cup_{| \alpha |<\pi /2}F_{\alpha}\) and \(| \epsilon | =1\), \[ (*)\quad Re[\epsilon \frac{zf'(z)}{g(z)}]>0,\quad | z|<1. \] Also, \(A_{\alpha}\) denotes the subclass of A such that (*) is satisfied for some \(g(z)\in F_{\alpha}\), for some specified \(\alpha\), \(| \alpha |<\pi /2\). Similarly, let \(A_{\alpha \beta}\) denote the subclass of \(A_{\alpha}\) such that (*) is satisfied with \(\epsilon =e^{i\beta}\), for some fixed real \(\beta\), \(| \beta |<\pi /2\). Ziegler has shown that the class A, beside being an extension of the univalent classes \(F_{\alpha}\) and C, does contain some nonunivalent functions.
The authors extend the classes \(A,A_{\alpha}\) and \(A_{\alpha \beta}\) to the new classes \(T,T_{\alpha}\) and \(T_{\alpha \beta}\) by requiring the function g(z), appearing in (*), to be in \(A,A_{\alpha}\) and \(A_{\alpha \beta}\), respectively. The authors show that A is a proper subclass of T. As in the class A, there are univalent functions that are not in T. However, the main result, which is too long to state, is the finding of the sharp radius of close-to-convexity \(r_{\alpha}\) for the class \(T_{\alpha}\). Also approximate numerical values of \(r_{\alpha}\) for some particular values of \(\alpha\) are found.