Questions tagged [functional-equations]
The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).
4,191 questions
1 vote
2 answers
247 views
Find all functions $f : \mathbb R \to \mathbb R$ such that: $f (x f ( y ))+y f ( x) = x f ( y ) + f ( xy ),$ $\forall\ x , y \in \mathbb{R}$
Find all functions $f : \mathbb R \to \mathbb R$ such that: $f (x f ( y ))+y f ( x) = x f ( y ) + f ( x y )$, $\forall\ x , y \in \mathbb{R}$ and b) $\exists M \in \mathbb R$ such that $f(x)<M$ ...
- 81
1 vote
1 answer
42 views
Generalization of Cauchy's functional equation. What are the general solutions, $f$?
Consider a function $f : (0,\infty) \to (0,\infty)$ satisfying the identity $$ f(x^a y^b) \;=\; f(x)^{1/a}\, f(y)^{1/b} \qquad\text{for all } x,y>0 \text{ and all real } a,b\neq 0. $$ This can be ...
- 1,199
3 votes
2 answers
109 views
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(y^2+1)+f(xy)=f(x+y)f(y)+1$ holds for all $x,y \in \mathbb{R}$
Problem: Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(y^2+1)+f(xy)=f(x+y)f(y)+1$ holds for all $x,y \in \mathbb{R}$. My approach: I have got $f(0)=0,1$. Taking $f(0)=1$, and setting $...
- 469
2 votes
0 answers
67 views
Is there an easy way to know if a rational function is an n-th (compositional) iteration of a power series with indices in $\mathbb{Z}$?
I am asking this question mainly to probe the knowledge of people already familiar with this problem, otherwise I would advise caution to the unfamiliar trying to use computation, this can be really ...
- 2,911
0 votes
1 answer
65 views
Yet another nice functional inequality [duplicate]
Need to find out all the functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $$f(x+y) \leq f(xy).$$ Since $f(x) \leq f(0)$ for every $x \in \mathbb{R}$ and $f(0) \leq f(-x^2)$, it follows that $f(...
- 629
-9 votes
0 answers
73 views
Classification of monotone associative laws on a bounded interval with Möbius translations fixing the boundary (unique $\tanh$ linearizer)
Let $\kappa>0$ and $I_\kappa=\bigl(-\tfrac{1}{\kappa},\tfrac{1}{\kappa}\bigr)$. Consider a binary operation $\oplus:I_\kappa^2\to I_\kappa$ with: (A1) $\oplus$ is continuous, strictly increasing in ...
2 votes
1 answer
270 views
A strange functional equation
Suppose that an unbounded function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(a+b)=f(a)+f(b)$ whenever $f(a+b)$ is the maximum of $f(0), f(1), \dots, f(a+b)$, where $a, b\in \mathbb{N}$. Show ...
- 111
0 votes
0 answers
102 views
Interesting case of a system of functional equations
Building upon this question, I would like to extend the unanswered question in the comments. That is, do there exist $g,h: \mathbb{N} \rightarrow \mathbb{N}, y \in \mathbb{N}$, such that the following ...
