AHZOLFAGHARI wrote: Prove that there are not function $f,g$ such that $ f(x^2+g(y)) -f(x^2)+g(y)-g(x)$ equal or less than $2y$ . ($f,g$ are two function from the real number to it ) . Wrong. Choose as counter-example $f(x)=-2x$ and $g(x)=x^2+1$

pco wrote: AHZOLFAGHARI wrote: Prove that there are not function $f,g$ such that $ f(x^2+g(y)) -f(x^2)+g(y)-g(x)$ equal or less than $2y$ . ($f,g$ are two function from the real number to it ) . Wrong. Choose as counter-example $f(x)=-2x$ and $g(x)=x^2+1$ Dear pco the main problem has also the condition $f(x)\ge x^2$.

AHZOLFAGHARI wrote: Prove that there are not function $f,g$ such that $ f(x^2+g(y)) -f(x^2)+g(y)-g(x)$ equal or less than $2y$ . ($f,g$ are two function from the real number to it ) . We get: $2y\geq f(x^2+g(y))-f(x^2)+g(y)-g(x)\geq (x^2+g(y))^2-f(x^2)+g(y)-g(x)$ $\Rightarrow 0\geq g(y)^2+(2x^2+1)g(y)+x^4-g(x)-f(x^2)-2y\Rightarrow 0 \leq \Delta=4g(x)+4f(x^2)+4x^2+8y+1$ If $y \rightarrow -\infty$ we have a contradiction.

prove $g(x)<0$ then see $g(0)$

AHZOLFAGHARI wrote: Prove that there are not function $f,g$ such that $$i)f(x)\geq x^2$$ $$ f(x^2+g(y)) -f(x^2)+g(y)-g(x)\leq 2y$$ ($f,g$ are two function from the real number to it ) .