Convexity and the Schwarz-Christoffel mapping (Q1310133)

In 1952 [Mich. Math. J. 1, 169-185 (1952; Zbl 0048.311)] the authors introduced the following concept of close-to-convex functions. A function \(f(z)\) analytic in the unit disc \(\Delta\) is called close-to-convex if \(\text{Re} [f'(z)/\varphi'(z)]>0\) for some convex function \(\varphi(z)\) in \(\Lambda\). He gave the following criterion for close-to-convexity. Theorem. Let \(f(z)\) be locally univalent in \(\Delta\). Then \(f\) is close- to-convex in \(\Delta\) if and only if \[ \int_{\theta_ 1} ^{\theta_ 2} \text{Re}\left[ 1+{{zf''(z)} \over {f'(z)}}\right] d\theta>-\pi, \qquad z=re^{i\theta}, \] for each \(r\), \(0<r<1\) and each pair of real numbers \(\theta_ 1\), \(\theta_ 2\) with \(\theta_ 1<\theta_ 2\). In this paper the author shows that the above theorem is equivalent to the condition \(S_{f,\rho}(\theta_ 2)- S_{f,\rho}(\theta_ 1)> -\pi\), \(0<\rho<1\), \(\theta_ 1<\theta_ 2\), where \(S_{f,\rho}(\theta)= \arg f'(re^{i\theta})+\theta\), \(-\infty <\theta<\infty\). Another interesting result proved here is a necessary and sufficient condition for a Schwarz Christoffel mapping to be close-to-convex. Furthermore, it is shown that when \(f\) is close-to-convex, there is a convex Schwarz-Christoffel mapping \(\varphi\) on \(\Delta\) such that \(\text{Re} [f'/\varphi']>0\). The above result gives rise to a number of interesting questions which are discussed here. Let \(f\) be locally univalent in \(A\). Then \(f\) is said to be close-to- convex of order \(\beta\), \(\beta>0\) if \[ |\arg f'-\arg \varphi'|< \beta {\textstyle {\pi\over 2}} \text{ in } \Delta \] for some convex function \(\varphi\) in \(\Delta\). The author proves a theorem which gives several equivalent conditions for a function to be close-to-convex of order \(\beta\).