The Poisson extension of \(K\)-quasihomography on the unit circle (Q2890393)

The present paper is devoted to the study of functions defined on the unit circle and their Poisson extension to the unit disk. Denote by \({\mathcal H}_T\) the family of all sense-preserving automorphisms on the unit circle \(T\). A mapping \(f\in {\mathcal H}_T\) is said to be a \(K\)-quasihomography, if and only if, for some \(K\), we have NEWLINE\[NEWLINE\Phi_{1/K}([z_1, z_2, z_3, z_4])\leq [f(z_1), f(z_2), f(z_3), f(z_4)]\leq \Phi_{K}([z_1, z_2, z_3, z_4])NEWLINE\]NEWLINE for any ordered four points \(z_1, z_2, z_3, z_4\in T\), where NEWLINE\[NEWLINE[z_1, z_2, z_3, z_4]:=\left(\frac{z_3-z_2}{z_3-z_1}: \frac{z_3-z_1}{z_4-z_1}\right)^{1/2}NEWLINE\]NEWLINE and \(\Phi_k\) is the Hersch-Pfluger function. Denote by \( A_T(K)\) the class of all such functions and let \(A_T^0(K)=\{f\in A_T(K): f(p_k)=p_k, k=0,1,2 \}\), where \(p_k=e^{k\cdot 2\pi i/3}.\) As already known, the Poisson extension of the function \(f\in {\mathcal H}_T\) to the unit disk can be expressed by the formula NEWLINE\[NEWLINEP[f](z)=a_0+\sum\limits_{n=1}^{\infty}a_nz^n+ \sum\limits_{n=1}^{\infty}a_{-n}\overline{z}^n.NEWLINE\]NEWLINE The main result of the paper is the following. Let \(f\in A^0_T(K)\), then \(-\frac{2}{3}\leq B(K)\leq \mathrm{Re} a_0\leq C(K)\leq \frac{1}{2}\), where \(C(K)\) and \(B(K)\) are some constants depending only on \(K\).