Let H(U) denote the linear space of holomorphic functions on the unit disk U endowed with the topology of normal convergence. Furthermore let \[ L=\{f(z)=k_ 1(z)(1-h(z)): h\in H(\bar U)\} \] and \(L_ 0=\{f\in L:\) h(1)\(\neq 1\}\), where \(k_ 1(z)=z(1-z)^{-2}\) is the Koebe function. The authors first show that \(\overline{L_ 0\cap S}=S\) and \(\overline{L_ 0\cap S_{{\mathbb{R}}}}=S_{{\mathbb{R}}}\), where \(S_{{\mathbb{R}}}\) denote the normalized univalent functions of class S which have real coefficients.
In view of these two results it is evidently useful to study variations of \(k_ 1\) in \(L\cap S\) and \(L\cap S_{{\mathbb{R}}}\). The authors construct explicit variations of \(k_ 1\), and show an interesting connection their construction has with the class \[ T_{{\mathbb{R}}}=\{t(z)=z+\sum^{\infty}_{n=2}a_ n(t)z^ n\in H(U): \] \(a_ n(t)\in {\mathbb{R}}\) for all \(n\in {\mathbb{N}}\) and \(Re\{(1-z^ 2)t(z)/z\}>0\) in \(U\}\). A well known theorem of Rogosinski shows that \(S_{{\mathbb{R}}}\subset T_{{\mathbb{R}}}.\)
The concluding section demonstrates two applications of the variations. Letting \(S_ n(\theta)=\sin (n\theta)-n \sin (\theta)\) the first of these gives a proof of \[ \inf_{\theta \in [0,2\pi]}S_ n(\theta)/S_ m(\theta)\leq [n-a_ n(f)]/[m-a_ m(f)]\leq \sup_{\theta \in [0,2\pi]}S_ n(\theta)/S_ m(\theta) \] for all \(f\in T_{{\mathbb{R}}}\), \(f(z)\neq z(1\pm z)^{-2}\), with these bounds best possible in \(S_{{\mathbb{R}}}\). The second application verifies a conjecture of E. Bombieri [Invent. Math. 4, 26-67 (1967; Zbl 0174.123)] for the variations of \(k_ 1\).