Univalent functions and Dirichlet integral (Q1262972)

For a real-valued integrable function \(\phi\) on (-\(\pi\),\(\pi)\), the \(\phi\)-outer function is defined in \(U=\{| z| <1\}\) by \[ \exp [(2\pi)^{-1}\int_{(-\pi,\pi)}\{(e^{it}+z)/(e^{it}-z)\}\phi (t)dt). \] For f holomorphic in U we denote \[ A(f)=\iint_{U}| f'(z)|^ 2/(1+| f(z)|^ 2)^ 2 dx dy. \] Furthermore, we call f Dirichlet-finite if \(\iint_{U}| f'(z)|^ 2 dx dy<\infty\). Let S be the family of functions f holomorphic and univalent in U with \(f(0)=f'(0)-1=0\). For \(f\in S\) we have A(f)\(\leq \pi\). Each \(f\in S\) has finite radial limit (\(\rho\) f)(t) at a.e. point \(e^{it}\) of the unit circle. It is known that each \(f\in S\) can be expressed as \(f(z)=zF(z)\) in U, where F is the log\(| \rho f|\)-outer function. Let \(\phi_ 1=\min (\log | \rho f|,0)\) and \(\phi_ 2=-\max (\log | \rho f|,0)\). Theorem 1. The \(\phi_ k\)-outer functions \(F_ k\) \((k=1,2)\) are bounded and Dirichlet-finite with \(f(z)/z=F_ 1(z)/F_ 2(z)\) for \(f\in S\). For the proof of Theorem 1 use is made of the general Theorem 2. If F is \(\phi\)-outer, then \[ A(F)=(16\pi)^{- 1}\iint_{(-\pi,\pi)\times (-\pi,\pi)}\Phi (s,t)ds dt, \] where \[ \Phi (s,t)=\{\sin (t/2)\}^{-2}(\phi (s+t)-\phi (s)\}\{\Theta (s+t)-\Theta (s)\}, \] \[ \Theta =\{\exp (2\phi)\}/\{1+\exp (2\phi)\}. \] Theorem 3. If \(f\in S\), then \[ A(f)=(1/2)\int_{(-\pi,\pi)}| \rho f(t)|^ 2/(1+| \rho f(t)|^ 2)dt+A(F). \] The first integral in the right-hand side is the integral mean of the spherical area of the disks \(\{| z| <| \rho f(t)| \}\) on the interval \(-\pi <t<\pi\). Set \(C_ s=\sup A(f(z)/z)\) (f\(\in S)\). We have the estimates: \(1.922<C_ s<2.057\). It would be interesting to find the exact value of this absolute constant.