Additive growth amongst images of linearly independent analytic functions

Let $\mathcal{F}$ be a set of $n$ real analytic functions with linearly independent derivatives restricted to a compact interval $I$. We show that for any finite set $A \subset I$, there is a function $f \in \mathcal{F}$ that satisfies $$|2^{n-1}f(A)-(2^{n-1}-1)f(A)|\gg_{\mathcal{F},I} |A|^{\phi(n)},$$ where $\phi:\mathbb{N} \to \mathbb{R}$ satisfies the recursive formula $$\phi(1)=1, \quad \phi(n)=1+\frac{1}{1+\frac{1}{\phi(n-1)}} \quad \text{for } n\geq 2.$$ The above result allows us to prove the bound $$|2^nf(A-A)-(2^n-1)f(A-A)| \gg_{f,n,I} |A|^{1+\phi(n)}$$ where $f$ is an analytic function for which any $n$ distinct non-trivial discrete derivatives of $f'$ are linearly independent. This condition is satisfied, for instance, by any polynomial function of degree $m \geq n+1$. We also check this condition for the function $\arctan(e^x)$ with $n=3$, allowing us to improve upon a recent bound on the additive growth of the set of angles in a Cartesian product due to Roche-Newton.