We prove that a harmonic quasiconformal mapping defined on finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits extension to whole plane if its Schwarzian derivative is small. also make observation univalence criterion for mappings holds uniform domains.

For normalized analytic functions in the unit disk, we consider subclasses of bounded turning. The geometric representation is introduced, the radii of convexity (close to convex) are calculated and some subordination relations are suggested. Moreover, the upper bound of the pre-Schwarzian norm for these functions is computed. AMS subject classifications: 30C45

Let the function $$\varphi $$ be holomorphic in unit disk $${\mathbb {D}}$$ of complex plane {C}}$$ and let ({\mathbb {D}})\subset {\mathbb . We study level sets critical points hyperbolic derivative , $$\begin{aligned} |D_{\varphi }(z)|:=\frac{(1-|z|^2)|\varphi '(z)|}{1-|\varphi (z)|^2}. \end{aligned}$$ In particular, we show how Schwarzian reveals nature points.

These are well known and important theorems of Nehari, Ahlfors and Weill, and Krauss. We refer to Lehto's book [8] for a discussion of these results and for the properties of the Schwarzian that we shall need. The constants 2 in (1.1) and 6 in (1.3) are sharp. An example for the latter is the Koebe function k(z) = z(l — z)~ which has Schwarzian Sk(z) = — 6/(1 —z). We also remark that the class ...

The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is bounded and the map f restricted to its closure is expanding, the property refered to as subexpanding. We first show the existence, uniqueness, conservativity ...

The essential operation that characterizes most power electronics circuits is the cyclic switching of the circuit con guration from one linear system to another. Such periodic switching operation naturally permits discrete-time modeling of power electronics systems in the form of iterative functions. In this section we review iterative functions of rst order and in particular discuss a techniqu...

In every textbook on elementary function theory you can find a definition of Schwarzian derivative {z; x} = 1 2 z z − 1 4 z z 2 , where = d/dx, of a non-constant function z = z(x) with respect to x. You would also find an exercise to show (0.1) [PGL(2)-invariance] If a, b, c, and d are constants satisfying ad − bc = 0, then az + b cz + d ; x = {z; x}. (0.2) {z; x} = 0 if and only if z(x) = (ax ...