QUASICONFORMAL HARMONIC MAPPINGS BETWEEN DOMAINS CONTAINING INFINITY

Assume that $\unicode[STIX]{x1D6FA}$ and $D$ are two domains with compact smooth boundaries in the extended complex plane $\overline{\mathbf{C}}$. We prove that every quasiconformal mapping between $\unicode[STIX]{x1D6FA}$ and $D$ mapping $\infty$ onto itself is bi-Lipschitz continuous with respect to both the Euclidean and Riemannian metrics.

Uniqueness of meromorphic functions with their reduced linear c-shift operators sharing two or more values or sets

In the paper, we introduce a new notion of reduced linear c-shift operator $L _{c}^{r}\,f$Lcrf, and with the aid of this new operator, we study the uniqueness of meromorphic functions $f(z)$f(z) and $L_{c}^{r}\,f$Lcrf sharing two or more values in the extended complex plane. The results obtained in the paper significantly improve a number of existing results. Further, using the notion of weighted sharing of sets, we deal the same problem. We exhibit a handful number of examples to justify certain statements relevant to the content of the paper. We are also able to determine the form of the function that coincides with its reduced linear c-shift operator. At the end of the paper, we pose an open question for future research.

Types of complete Infinitely sheeted Planes

We will answer negatively to the question whether the completeness of infinitely sheeted covering surfaces of the extended complex plane have anything to do with their types being parabolic or hyperbolic. This will be accomplished by giving a one parameter family {W[α]: α ∈ A} of complete infinitely sheeted planes W[α] depending on the parameter set A of sequences α = (an)n>1 of real numbers 0 < an ≤ 1/2 (n ≥ 1) such that W[α] is parabolic for ‘small’ α’s and hyperbolic for ‘large’ α’s.

Ergodic properties of random iterations of analytic functions

It is a known fact that an iterated function system (IFS) of entire functions is not necessarily ergodic. In this paper we show that if an IFS of analytic functions is defined in a domain whose boundary contains more than two points (in the extended complex plane) then the system possesses an ergodic property.

Simple proofs of some fundamental properties of the Julia set

Let $f$ be a holomorphic self-map of $\mathbb{C} \backslash \{ 0 \}, \mathbb{C}$, or the extended complex plane $\overline{\mathbb{C}}$ that is neither injective nor constant. This paper gives new and elementary proofs of the well-known fact that the Julia set of $f$ is a non-empty perfect set and coincides with the closure of the set of repelling cycles of $f$. The proofs use Montel–Caratheodory's theorem but do not use results from Nevanlinna theory.

Quasiconformal extensions for some geometric subclasses of univalent functions

LetSdenote the set of all functionsfwhich are analytic and univalent in the unit diskDnormalized so thatf(z)=z+a2z2+…. LetS∗andCbe those functionsfinSfor whichf(D)is starlike and convex, respectively. For0≤k<1, letSkdenote the subclass of functions inSwhich admit(1+k)/(1−k)-quasiconformal extensions to the extended complex plane. Sufficient conditions are given so that a functionfbelongs toSk⋂S∗orSk⋂C. Functions whose derivatives lie in a half-plane are also considered and a Noshiro-Warschawski-Wolff type sufficiency condition is given to determine which of these functions belong toSk. From the main results several other sufficient conditions are deduced which include a generalization of a recent result of Fait, Krzyz and Zygmunt.