Newbetuts

About the polynomials satisfying $P\left((x-y)^2\right)+P\left((y-z)^2\right)+P\left((z-x)^2\right)=18P\left(\left(\frac{(x+y+z)^2}3\right)^2\right)$

Finding all $f:[0, \infty) \to [0, \infty)$ differentiable and convex with $f(0)=0$ and $f'(x)\cdot f\bigl(f(x)\bigr)=x$

functional equation $ 1= f(x)+\left(\frac x2\right)+\dots+f\left(\frac xN\right) $

Determining all $f : \mathbb R^+ \to \mathbb R^+$ that satisfy $f(x + y) = f\left(x^2 + y^2\right)$

Continuous $f : \mathbb R \to \mathbb R$ satisfying $f\left(\sqrt{\frac{x^2 + y^2}{2}}\right) = \frac{f(x) + f(y)}{2}$

Determining all $f : \mathbb R \to \mathbb R$ that satisfy $f\bigl(xf(y)\bigr) = x^{2002}f\bigl(f(y)\bigr)$

Solutions of the functional equation $f(x+1)= xf(x)$

$f:\mathbb N_0\to\mathbb N_0$ with $2f\left(m^2+n^2\right)=f(m)^2+f(n)^2$ and $f\left(m^2\right)\geqslant f\left(n^2\right)$ when $m\geqslant n$

Showing $f$ constant if it is continuous and $f(2x) = f(x)$

Determining all $f : \mathbb R \to \mathbb R$ that satisfy $xf(x) - yf(y) = (x-y)f(x+y)$

Does $f\big(x^2-y^2\big)=x\cdot f(x)-y\cdot f(y)$ imply $f\big(x^2\big)=x^2\cdot f(1)$?

I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive. [duplicate]

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Find all strictly monotone $f:(0,+\infty) \to (0, +\infty)$ such that $f(\frac{x^2}{f(x)})=x.$

Finding the family of functions satisfying $f(x + y)=f(x)f(y) - f(x-y)$

Find all functions such that $f(x^2+y^2f(x))=xf(y)^2-f(x)^2$

Solving functional equation $f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y))$

New posts in functional-equations

Find continuous functions such that $f(x+y)+f(x-y)=2[f(x)+f(y)]$

Solution of functional equation $f(x/f(x)) = 1/f(x)$?

Functional Equations, Linear Symmetry, Group Theory

About the polynomials satisfying $P\left((x-y)^2\right)+P\left((y-z)^2\right)+P\left((z-x)^2\right)=18P\left(\left(\frac{(x+y+z)^2}3\right)^2\right)$

Finding all $f:[0, \infty) \to [0, \infty)$ differentiable and convex with $f(0)=0$ and $f'(x)\cdot f\bigl(f(x)\bigr)=x$

functional equation $ 1= f(x)+\left(\frac x2\right)+\dots+f\left(\frac xN\right) $

Determining all $f : \mathbb R^+ \to \mathbb R^+$ that satisfy $f(x + y) = f\left(x^2 + y^2\right)$

Continuous $f : \mathbb R \to \mathbb R$ satisfying $f\left(\sqrt{\frac{x^2 + y^2}{2}}\right) = \frac{f(x) + f(y)}{2}$

Determining all $f : \mathbb R \to \mathbb R$ that satisfy $f\bigl(xf(y)\bigr) = x^{2002}f\bigl(f(y)\bigr)$

Solutions of the functional equation $f(x+1)= xf(x)$

$f:\mathbb N_0\to\mathbb N_0$ with $2f\left(m^2+n^2\right)=f(m)^2+f(n)^2$ and $f\left(m^2\right)\geqslant f\left(n^2\right)$ when $m\geqslant n$

Showing $f$ constant if it is continuous and $f(2x) = f(x)$

Determining all $f : \mathbb R \to \mathbb R$ that satisfy $xf(x) - yf(y) = (x-y)f(x+y)$

Does $f\big(x^2-y^2\big)=x\cdot f(x)-y\cdot f(y)$ imply $f\big(x^2\big)=x^2\cdot f(1)$?

I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive. [duplicate]

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Find all strictly monotone $f:(0,+\infty) \to (0, +\infty)$ such that $f(\frac{x^2}{f(x)})=x.$

Finding the family of functions satisfying $f(x + y)=f(x)f(y) - f(x-y)$

Find all functions such that $f(x^2+y^2f(x))=xf(y)^2-f(x)^2$

Solving functional equation $f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y))$