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f(x^{n+1}+(y^{n+1})=x^n f(x)+y^n f(y) when x,y>0

Source: Ukraine TST 2009 p4

May 3, 2020
functional equationfunctionalalgebra

Problem Statement

Let nn be some positive integer. Find all functions f:R+ā†’Rf:{{R}^{+}}\to R (i.e., functions defined by the set of all positive real numbers with real values) for which equality holds f(xn+1+yn+1)=xnf(x)+ynf(y)f\left( {{x}^{n+1}}+ {{y}^{n+1}} \right)={{x}^{n}}f\left( x \right)+{{y}^{n}}f\left( y \right) for any positive real numbers x,yx, y
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