We study an important subclass of quasicircles, namely, symmetric quasicircles. Several characterizations for quasicircles, such as the reverse triangle inequality, the M -condition and the quasiconformal extension property, have been extended to symmetric quasicircles by Becker and Pommerenke and by Gardiner and Sullivan. In this paper we establish several relations among various domain constants such as quasiextremal distance constants, (local) reflection constants and (local) extension constants for this class. We also give several characterizations for symmetric quasicircles such as the strong quadrilateral inequality and the strong extremal distance property. They correspond to the quadrilateral inequality and the extremal distance property for quasicircles.

A sufficient and necessary condition is given for the continuity of Julia sets in the space of all rational maps with degree k>1.

By studying the mapping by heights for quadratic differentials introduced by Strebel, some relations have been established between the maximal norm sequence for quasisymmetric functions and the Hamilton sequence for extremal quasiconformal mappings in the unit disk. Consequently it is proved that a Hamilton sequence is only determined by e quasisymmetric function.

Let f1,f2 be two linearly independent solutions of the linear differential equation f"+A(z)f=0, where A(z) is transcendental entire, and assume that the exponents of convergence for the zero-sequences of f1,f2 satisfy max (λ(f1),λ(f2))=∞. Our main result proves that the zeros of E:=f1f2 are uniformly distributed in the sense that quite arbitrary large areas of the complex plane can be removed in such a way that if only zeros outside of these areas will be counted for the exponents of convergences, their maximum still remains infinite.

In this paper, some results on removable sets of entire and meromorphic functions in the sense of borel Theorem are obtained. Among them we prove that there exists a se-quence of discs (Dn) with radii tending to the infinity such that for any entire function f(z)with a positiv eand finite order, Borel Theorem always holds in C\∪∞n=1Dn.