The following result seems to be known in the classical or convex analysis but I cannot find the reference.
Let $f,g:\mathbb{R}\rightarrow\mathbb{R}$ and be twice differentiable and convex functions such that:
(1) $|f^\prime(x)|=|g^\prime(x)|\quad \forall x\in\mathbb{R}$,
(2) $\inf_{x\in\mathbb{R}}|f^\prime(x)|=0$.
Then, $f=g+c$ for some constant $c\in\mathbb{R}$.
Thank you for your solutions, references or geometrical explanation of the above result.
Can we generalize this result to a multivariable function?
Given your condition (2), there is some $x_0\in\mathbb{R}$ or $x_0=\pm\infty$ such that $$ \lim\limits_{x\to x_0}f'(x)=0 $$ where the limit is understood as a one-sided limit if $x_0=\pm\infty$. I will consider the case of a finite $x_0$ for simplicity. The analysis is similar in the other cases. By continuity of $f'$, in this case we have $f'(x_0)=0$ and, by convexity, $f'(x)\ge 0$ for $x>x_0$ and $f'(x)\le 0$ for $x<x_0$. Let $x>x_0$. Then, by condition (1) we have $f'(x)=\pm g'(x)$. Unless $f'(x)=0$, we necessarily have $g'(x)=f'(x)>0$, since otherwise we get a contradiction to the convexity of $g$ (a decreasing derivative $g'$). Analogously, for $x<x_0$ we necessarily have $g'(x)=f'(x)<0$ (unless $f'(x)=0$). Hence, for all $x$ we have $f'(x)=g'(x)$ which implies of course $f=g+c$ for some constant $c$.