On the functional equation $F(A(z))=G(B(z)),$ where $A,B$ are polynomial$ and $F,G$ are continuous functions

We collect some classical results related to analysis on the Riemann surfaces. The notes may serve as an introduction to the field: we suppose that the reader is familiar only with the basic facts from topology and complex analysis. the treatment is organized to give a background for further applications to non-linear differential equations.

In this paper we give several conditions implying the irreducibility of the algebraic curve P(x)-Q(y)=0, where P,Q are rational functions. We also apply the results obtained to the functional equations P(f)=Q(g) and P(f)=cP(g), where c\in C. For example, we show that for a generic pair of rational functions P,Q the first equation has no non-constant solutions f,g meromorphic on C whenever (\deg P-1)(\deg Q-1) \geq 2.

Recently the author presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmuller spaces. It involves the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces. The aim of the present paper is to provide new applications of this approach and extend the indicated results to more general classes of functions

Thirty research questions on meromorphic functions and complex differential equations are listed and discussed. The main purpose of this paper is to make this collection of problems available to everyone.

We study some properties of the solutions of the functional equation $$f(x)+f(a_1x)+...+f(a_Nx)=0,$$ which was introduced in the literature by Mora, Cherruault and Ziadi in 1999, for the case $a_k=k+1$, $k=1,2,...,N$ and studied by Mora in 2008 and Mora and Sepulcre in 2009 and 2011.

We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.

This paper has been withdrawn by the author(s) and included into the new version of "An extension theorem for separately holomorphic functions with singularities", math.CV/0104089.

The purpose of this paper is to make an introduction to univalent function theory for readers of any level, assuming only foundational knowledge in real and complex analysis. In particular, we state and proof (with details) important theorems utilised in proving Bieberbach's conjecture, especially those that were missed or merely sketched by other texts on univalent functions. We will finally prove the conjecture following the proof given by Lenard Weinstein in a comprehensiv...

The Riemann Theorem states, that for any nontrivial connected and simply connected domain on the Riemann sphere there exists some its conformal bijection to the exterior of the unit disk. In this paper we find an explicit form of this map for a broad class of domains with analytic boundaries.

We prove a uniqueness theorem for a large class of functional equations in the plane, which resembles in form a classical result of Aczel. It is also shown that functional equations in this class are overdetermined in the sense of Paneah. This means that the solutions, if they exist, are determined by the corresponding relation being fulfilled not in the original domain of validity, but only at the points of a subset of the boundary of the domain of validity.