According to the Appendix of the book: Robust Nonlinear Control Design by Randy A.Freeman and Petar V. Kokotovic, we have the following result (in fact, it is much more general). Denote the real line by $\mathcal{R}$. We say a function $\gamma:\mathcal{R}\longrightarrow\mathcal{R}$ is of class $K$ if $\gamma(0)=0$, it is strictly increasing and continuous.
Suppose that the function $f:\mathcal{R}\longrightarrow\mathcal{R}$ is continuous. Then, there exist a continuous function $\rho:\mathcal{R}\longrightarrow\mathcal{R}$ and a class $K$ function $\gamma:\mathcal{R}\longrightarrow\mathcal{R}$ such that $$|f(x)-f(y)|\leq\rho(x)\gamma(|{x-y}|),$$ for all real $x,y$.
Is it possible to find $\rho$ and $\gamma$ explicitly for $f(x):=x^3+x$?
Write $y = x+h$, then $$|f(x) - f(y)| = |3 x h^2 + 3 x^2 h + h^3 + h| \le (2 + 3|x|+3x^2) \max(|h|, |h|^2, |h|^3)$$