Result: On an asymptotically sharp variant of Heinz's inequality

The authors study quasiconformal automorphisms of the unit disk \(D\) and prove several results about this topic refining some of their earlier results e.g. in [\textit{D. Partyka, K. Sakan}, J. Comput. Appl. Math. 105, No. 1--2, 425--436 (1999; Zbl 0951.30017)]. In particular, they prove the following theorem. Given \(K \geq 1\,,\) let \(F\) be a \(K\)-quasiconformal and harmonic self-mapping of \(D\) satisfying \(F(0)=0 \,.\) Then the inequality \[ | \partial_x F(z) | ^2 + | \partial_y F(z)| ^2 \geq \frac{1}{2}\left(1+ \frac{1}{K}\right)^2 \max \{ \frac{4}{\pi^2}, L_K^2 \} \] holds for every \(z \in D \,.\) Some properties of the function \(L(K)\) are studied and the following explicit formula is given \[ L_K= \frac{2}{\pi} \int_0^u \frac{dt}{ \varphi_K(\sqrt{t}) \varphi_{1/K}(\sqrt{1-t}) } \,, u= \varphi_{1/K}(1/\sqrt{2})^2 \, , \] where \(\varphi_K:(0,1) \to (0,1)\) defined for \(K>1\), \(r \in (0,1), \;\;r' = \sqrt{1 - r^2},\) by \[ \varphi_K(r)=\mu^{-1}(\mu(r)/K);\;\mu(r)={\pi \over 2}{K(r') \over K(r)};\;K(r)=\int\limits_0^1 {dx \over \sqrt{(1-x^2)(1-r^2x^2)}} . \]