The univalence of a linear combination of convex mappings (Q1110673)

Let \(S^*(1/2)\) denote the set of functions that are analytic in \(\{z:| z| <1\}\) and satisfy \(f(0)=0\), \(f'(0)=1\) and \[ Re\{zf'(z)/f(z)\}>1/2\quad for\quad | z| <1. \] The author finds the sharp radius of univalence for each of the following classes: (1) the set \(\lambda f+(1-\lambda)g\) where \(f,g\in S^*(1/2)\) and \(\lambda\) is a fixed real number satisfying \(0\leq \lambda \leq 1;\) (2) the set \(\lambda_ 1f_ 1+\lambda_ 2f_ 2\) where \(f_ 1,f_ 2\in S^*(1/2)\), \(\lambda_ 1,\lambda_ 2\) are complex numbers for which \(| \arg \lambda_ k| \leq \alpha /2\) for \(k=1,2\) and \(| \lambda_ 1| +| \lambda_ 2| =1\) and \(\alpha\) is a fixed real number satisfying \(0\leq \alpha <\pi;\) (3) the set \(\sum^{n}_{k=1}\lambda_ kf_ k\) where \(f_ k\in S^*(1/2)\), \(\lambda_ k\geq 0\), \(\sum^{n}_{k=1}\lambda_ k=1\) and \(\max_{1\leq k\leq n}\lambda_ k\) is given and assumed to be at least 1/2. The result for (1) generalizes a theorem of the reviewer which asserts that if f and g are normalized convex mappings then \(\lambda f+(1- \lambda)g\) is univalent in \(\{z:| z| <\frac{1}{\sqrt{2}}\}\) for each \(\lambda\), \(0\leq \lambda \leq 1\) [J. Lond. Math. Soc. 44, 210-212 (1969; Zbl 0162.376)].