The derivative of a close-to-convex function
A function \(f(z)=a_ 1z+a_ 2z^ 2+...\), \(| a_ 1| =1\), analytic in the unit disc \(U=\{z: | z|<1\}\) is called close-to- convex if there is a function \(g(z)=b_ 1z+b_ 2z^ 2+...,\quad | b_ 1| =1,\) starlike in U such that \(Re\frac{zf'(z)}{g(z)}>0\) (\(z\in U)\). For a fixed function g, starlike in U, let K(g) denote the class of analytic functions f which are close-to-convex with respect to g. Finally, let \(M(r,F)=\sup \{| F(z)|:\;| z| =r\}\), \(0\leq r<1\), \(0 \leq t \leq 2\pi\). The author proves several interesting results concerning the growth of the derivative of a class of close-to-convex functions. The first two results are stated below: Theorem 1. Let \(f\in K(g)\). Then \[ | f'(re^{it})| \leq \frac{Af^*(r,t)\log [5r^{-1}g(re^{it})]} {(1-r)\log [2/(1-r)]}, \tag{\(*\)} \] for \(0<r<1\), \(0\leq t\leq 2\) and \(A\) is an absolute positive constant. For \(f\in K(g)\), therefore, we have \[ M(r,f')=O(\frac{m(r,f)\log [5r^{-1}M(r,g)]}{(1-r)\log [1/(1-r)]}) \tag{\(**\)} \] as \(r\to1\). The special case \(f=g\) is a starlike function, (*) part of Theorem 1 is known by the author [J. Lond. Math. Soc., II. Ser. 2, 99-110 (1970; Zbl 0194.386)], while (**) is also known by the author [op. cit.] when the starlike function f is bounded in U. In general the order O cannot be replaced by o for this special case as shown by the author [op. cit.]. However, a refinement of a special case of Theorem 1 is obtained when the domain U is replaced by a Stolz domain \(\Omega(\sigma,t)\), a domain defined to be the interior of the convex hull of the point \(e^{it}\) and the circle of radius \(\sigma\) with center at the origin. Theorem 2. Let \(f\in K(g)\) and suppose that \(| f(re^{it_ 0})| =O(1), r\to1\), for some fixed \(t_ 0\) in [0,2\(\pi]\). Then, with \(| z| =r,\) \[ | f'(z)| = o\left( \frac{\log[5r^{-1}| g(z)|]} {(1-r)\log [1/(1-r)]}\right) \] uniformly as \(z\to e^{it_0}\) inside any fixed Stolz domain \(\Omega(\sigma,t)\).
scientific article; zbMATH DE number 3845912
scientific article