$\newcommand{\La}{\Lambda}$
The Legendre transform $\La^*_X$ of the log-moment generating function of a random variable $X$ is given by the formula 
$$\La^*_X(x):=\inf_{t\ge0}(-tx+\La_X(t)),
$$
where $\La_X(t):=\ln Ee^{tX}$. The function $\La^*_X$ is the pointwise infimum in $t\ge0$ of the concave functions $x\mapsto-tx+\La_X(t)$. So, $\La^*_X$ is concave. 

So, we may take $f_i=\La^*_i:=\La^*_{X_i}$ for all $i$. Then for $\La_i:=\La_{X_i}$ we have 
\begin{align}
\La^*_{\sum_{i=1}^n X_i}(x)&=\inf_{t\ge0}\sum_{i=1}^n(-t\tfrac xn+\La_i(t)) \\ 
&\ge\sum_{i=1}^n\inf_{t\ge0}(-t\tfrac xn+\La_i(t)) \\ 
&=\sum_{i=1}^n\La^*_i(\tfrac xn) \\ 
&=\sum_{i=1}^n f_i(\tfrac xn) \\ 
&\ge n\min_{i=1}^n f_i(\tfrac xn). 
\end{align}

So, we get the inequality opposite to what you suggested. Usually, this opposite inequality will be strict, if indeed the $X_i$'s are not identically distributed.