On transforms of functions with bounded boundary rotation (Q1083576)

The authors consider the well-known class \(V_ k\) of normalized functions with bounded boundary rotation at most \(k\pi\) in the unit disc D. The major result of this paper is the following result. Let \(k\geq 2\), \(C_ j(k)\) defined by \[ \frac{1}{k}[(\frac{1+z}{1-z})^{k/2}- 1]=\sum^{\infty}_{j=1}C_ j(k)z^ j. \] Suppose \(f\in V_ k\), \(w\in C\setminus f(D)\) and \[ F(z)=wf(z)/(w- f(z))=\sum^{\infty}_{j=1}d_ jz^ j\quad (z\in D). \] Then there exists a constant \(g(k)<\infty\) such that \[ | d_ j| \leq g(k)C_ j(k),\quad j\geq 2. \] The authors remark that not much is known about the size of the constants g(k).