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

The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials and f,g are trancendental meromorphic functions in C. We find all the pairs p,q for which there exist nonconstant f,g satisfying our equation and there exist no rational f,g with this property. In fact, a more general problem is solved. In addition to algebraic methods, some results from local analytic dynamics are used.

We study the continuous solutions of several classical functional equations by using the properties of the spaces of continuous functions which are invariant under some elementary linear trans-formations. Concretely, we use that the sets of continuous solutions of certain equations are closed vector subspaces of $C(\mathbb{C}^d,\mathbb{C})$ which are invariant under affine transformations $T_{a,b}(f)(z)=f(az+b)$, or closed vector subspaces of $C(\mathbb{R}^d,\mathbb{R})$ whic...

In 1922 Ritt described polynomial solutions of the functional equation P(f)=Q(g). In this paper we describe solutions of the equation above in the case when P,Q are polynomials while f,g are allowed to be arbitrary entire functions. In fact, we describe solutions of the more general functional equation s=P(f)=Q(g), where s,f,g are entire functions and P,Q are arbitrary rational functions. Besides, we solve the problem of description of "strong uniqueness polynomials" for enti...

In this article we consider functions $f$ meromorphic in the unit disk. We give an elementary proof for a condition that is sufficient for the univalence of such functions. This condition simplifies and generalizes known conditions. We present some typical problems of geometrical function theory and give elementary solutions in the case of the above functions.

Recently, the functional equation \[ \sum_{i=0}^mf_i(b_ix+c_iy)= \sum_{i=1}^na_i(y)v_i(x) \] with $x,y\in\mathbb{R}^d$ and $b_i,c_i\in\mathbf{GL}_d(\mathbb{C})$, was studied by Almira and Shulman, both in the classical context of continuous complex valued functions and in the framework of complex valued Schwartz distributions, where these equations were properly introduced in two different ways. The solution sets of these equations are, typically, exponential polynomials and,...

We describe meromorphic solutions to the equations $f^n(z)+\left(f'\right)^n(z)=e^{\alpha z+\beta}$ and $f^n(z)+f^n(z+c)=e^{\alpha z+\beta}$ ($c\neq0$) over the complex plane $\mathbf{C}$ for integers $n\geq1$.

This is an appendix to the English translation of the book by A. A. Goldberg and I. V. Ostrovskii, Distribution of values of meromorphic functions, Moscow, Nauka, 1970. An English translation of this book is to be published soon by the AMS. In this appendix we survey the results obtained on the topics of the book after 1970.

We offer a solution to a functional equation using properties of the Mellin transform. A new criteria for the Riemann Hypothesis is offered as an application of our main result, through a functional relationship with the Riemann xi function.

In this sequence of work we investigate polynomial equations of additive functions. We consider the solutions of equation \[ \sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x)^{q_{i}}= 0 \qquad \left(x\in \mathbb{F}\right), \] where $n$ is a positive integer, $\mathbb{F}\subset \mathbb{C}$ is a field, $f_{i}, g_{i}\colon \mathbb{F}\to \mathbb{C}$ are additive functions and $p_i, q_i$ are positive integers for all $i=1, \ldots, n$. Using the theory of decomposable functions we descr...

In 1922 Ritt constructed the theory of functional decompositions of polynomials with complex coefficients. In particular, he described explicitly indecomposable polynomial solutions of the functional equation f(p(z))=g(q(z)). In this paper we study the equation above in the case when f,g,p,q are holomorphic functions on compact Riemann surfaces. We also construct a self-contained theory of functional decompositions of rational functions with at most two poles generalizing the...