Weak Schwarzians, bounded hyperbolic distortion, and smooth quasisymmetric functions
Let \(f\) be a real valued function which is continuous and increasing on a closed interval \(I\). For \(k\geq 1\), the function \(f\) is said to be a \(k\)-quasi symmetric if, for each \(x\in I\) and for each \(h>0\) such that both \(x+h\) and \(x-h\) are in the interval \(I\), we have \[ {1 \over k} \leq k_f(x,h) ={f(x+h) -f(x) \over f(z) -f(x-h)} \leq k. \] The authors give some sufficient conditions relating upper and lower bounds on the quantity \(k_f(x,h)\) to the Schwarzian derivative \(Sf= ({f''\over f'})' -{1\over 2} ({f'' \over f'})^2\). Typical of the sufficient conditions is the result that if there exist positive numbers \(r\) and \(s\), \(0<r\leq 1\), \(1\leq s<\infty\), such that \[ {-2(s^2-1) \over (1-x^2)^2} \leq Sf(x) \leq {2(1-r^2) \over(1-x^2)^2}, \quad x\in I, \] then \(f\) is \(k(r,s)\)-quasi symmetric where \[ k(r,s)= {s\over r} \max \left\{{2^{1-r} \over 2^r-1}, {2^s-1 \over 2^{1-s}} \right\}. \] Some examples are given to show that the sufficient conditions are not necessary. In addition, some results about the distortion of \(k\)-quasi symmetric functions in the hyperbolic metric are given in terms of the Schwarzian derivative. Further, it is shown that, under some reasonably general conditions, \(k\)-quasi symmetric mappings can be expressed as a composition of a finite number of \(p\)-quasi symmetric mappings, where \(p\) can be chosen as close to the number 1 as desired. The results express relationships suggested by the relationships between the complex valued quasi conformal functions, univalent functions, and the Schwarzian derivative.
scientific article; zbMATH DE number 927201
scientific article