Extremals for families of plane quasiconformal mappings (Q1433474)

Let \(K\geq 1\). The author considers the set \({\mathcal Q}={\mathcal Q}(K)\) of \(K-\)quasiconformal mappings from the Riemann sphere onto itself and the three following subsets of \({\mathcal Q}\) : \[ {\mathcal H}={\mathcal H}(K)=\{f\in {\mathcal Q}; f(0)=0, f(\infty)=\infty\}, \] \[ {\mathcal G}={\mathcal G}(K)=\{f\in {\mathcal H}; f(-1)=-1\} \] and \[ {\mathcal F}={\mathcal F}(K)=\{f\in {\mathcal G}; f({\mathbb R})= {\mathbb R}\}. \] For \(t\in {\mathbb R}\), the author sets \(\lambda(K,t)=\sup_{f\in {\mathcal F}(K)} f(t)\) and \(\nu(K,t)=\inf_{f\in {\mathcal F}(K)} f(t)\). Finally, for any \(t>0\) and for any complex-valued continuous function \(f\) on the circle \(\{| z|=t\}\), the author defines \(p(f,t)=\max_{| z|=t}| f(z)|\) and \(q(f,t)=\min_{| z|=t}| f(z)|\). The main results of this paper are the following Theorem 1: For \(t>0\), we have \[ \lambda(K,t)=\max_{f\in{\mathcal G}(K)} p(f,t) \] and \[ \nu(K,t)=\min_{f\in{\mathcal G}(K)} q(f,t). \] Theorem 2: For \(r>0\) and \(t>0\), we have \[ \lambda(K,t)=\max_{f\in{\mathcal H}(K)}{p(f,tr)\over q(f,r)} \] and \[ \nu(K,t)=\min_{f\in{\mathcal H}(K)}{q(f,tr)\over p(f,r)}. \] Theorem 1 improves a result of S. Agard and Theorem 2 improves a result of Lehto, Virtanen and Väisälä. Theorem 3: Given a domain \(D\) in the Riemann sphere and given \(t>0\), we have, for all \(z\) in \(D\), \[ \lambda(K,t)=\sup_{f\in {\mathcal Q}(K,D)}\Delta_t^+(f,z) \] and \[ \nu(K,t)=\inf_{f\in {\mathcal Q}(K,D)}\Delta_t^-(f,z) \] where \[ \Delta_t^+(f,z)=\limsup_{r\to 0}{\max_{|\zeta|=tr}| f(\zeta+z)-f(z)|\over \min_{|\zeta|=r}| f(\zeta+z)-f(z)|)}, \] and \[ \Delta_t^-(f,z)=\liminf_{r\to 0}{\min_{|\zeta|=tr}| f(\zeta+z)-f(z)|\over \max_{|\zeta|=r}| f(\zeta+z)-f(z)|)}. \]