Regular functions in a semiplane which are topologically equivalent to quasi-isometric mappings (Q1920846)

A mapping \(f:H\to {\mathcal C}\), where \(H= \{x+iy \in{\mathcal C}: y>0\}\), is said to be uniformly locally univalent if there exists an \(r>0\) such that \(f\) is univalent in every hyperbolic disk \(G(z,r)= \{w\in H: l(z,w) <r\}\), where \(l(z_1,z_2) =ln[(|z_1- \overline z_2|+ |z_1-z_2 |)/(|z_1- \overline z_2|- |z_1- z_2|)]\), \(z\in H \cdot \{\text{not} l(z_1, z_2) =l(|z_1- \overline z_2 |)/(|z_1- \overline z_2|-|z_1- z_2|).\}\) A locally univalent mapping \(F:D \to {\mathcal C}\) is called \(K\)-quasi-isometric if \(\Lambda (F,z)\leq K\), \(\lambda(F,z) \geq 1/K\) for every \(z\) of a domain \(D\) of the complex plane \({\mathcal C}\), where \(K\geq 1\), \(\Lambda (f,z)= \varlimsup_{w\to z} (|f(z)- f(w)|/ |w-z|)\), \(\lambda (f,z)= \varliminf_{w\to z} (|f(z)- f(w)|/ |w- z|)\). A homeomorphism \(\varphi: H\rightleftharpoons H\) is called \(K\)-quasihyperbolic if \(l(z_1,z_2)/K \leq l[\varphi (z_1), \varphi(z_2)] \leq Kl(z_1,z_2) \forall z_1,z_2 \in H\). The author considers the following remark: ``A homeomorphism \(\varphi:H \rightleftharpoons H\) is quasihyperbolic iff the estimations \[ \Lambda (\varphi,z) \leq K\delta \bigl(\varphi (z)\bigr)/ \delta(z),\;\lambda (\varphi,z) \geq\delta \bigl(\varphi (z)\bigr)/ \bigl(K \delta(z) \bigr) \tag{1} \] hold \(\forall z\in H\)''. \{I think it would be more accurate to say ``A homeomorphism \(\varphi:H\rightleftharpoons H\) is quasihyperbolic iff there exists a constant \(K\geq 1\) such that (1) hold.''\} A mapping \(f_1:D\to{\mathcal C}\) is named topologically equivalent to a mapping \(f_2:D \to {\mathcal C}\), if there exists a homeomorphism \(\varphi :D \rightleftharpoons D\) such that \(f_2=f_1\circ\varphi\). The author obtains the following properties of a regular function \(F:H\to{\mathcal C}\) topologically equivalent to a \(K\)-quasi-isometric mapping \(F:H\to {\mathcal C}\) such that \(F=f \circ \varphi\): (i) \(f\) is uniformly locally univalent; (ii) \(\varphi\) is quasihyerbolic and (iii) if \(f\) is bounded over bounded sets of \(H\), then there exists a constant \(Q\geq 1\) such that estimates \[ 1/Q \leq\int^{x+y}_x \bigl|f'(s+it) \bigr|ds\left / \int^{x+y}_x \right.\bigl|f'(s+iy) \bigr|ds\leq Q \tag{2} \] hold \(\forall z\in H\) and \(\forall t\in (0,y]\). He establishes also that if \(f:H\to {\mathcal C}\) is a uniformly locally univalent regular function satisfying (2), then \(f\) is topologically equivalent to some quasi-isometric mapping \(F:H\to {\mathcal C}\).