Problem 4

Let

c>0

be a given positive real and

\mathbb{R}_{>0}

be the set of all positive reals. Find all functions

f:\mathbb{R}_{>0}\to\mathbb{R}_{>0}

such that

f((c+1)x+f(y))=f(x+2y)+2cx\quad\text{for all }x,y\in\mathbb{R}_{>0}.

Step 3 of 3: Iterate to force g(y)=0

Previous step Next step

z>2y\ \Longrightarrow\ g(z)\ge 2m\,g(y)\quad\forall m\ge1\ \Longrightarrow\ g(y)=0\ \Longrightarrow\ f(x)=2x

Detailed analysis

Since $g\ge0$ , the identity gives $g(z)\ge2g(y)$ whenever $z=x+2y>2y$ . By induction on $m\ge1$ , $z>2y$ implies $g(z)\ge2m\,g(y)$ : indeed, for $z=x+2y>2y$ we have $(c+1)x+2y+g(y)>2y$ , so the induction hypothesis gives $g((c+1)x+2y+g(y))\ge2(m-1)g(y)$ , and adding $2g(y)$ gives $g(z)\ge2m\,g(y)$ . Fixing $z>2y$ and letting $m\to\infty$ forces $g(y)=0$ for every $y>0$ . Hence $f(x)=2x$ for all $x>0$ , which directly satisfies the equation.