Proposition 2

Let $f_i$ ($1 \leq i \leq p$) be $p$ convex functions on an interval $I \subset \mathbf{R}$, and $c_i$ ($1 \leq i \leq p$) be $p$ arbitrary positive numbers; then the function $f = \sum_{i=1}^p c_i f_i$ is convex on $I$. Further, if for at least one index $j$ the function $f_j$ is strictly convex on $I$, and $c_j > 0$, then $f$ is strictly convex on $I$.

This follows immediately by applying the inequality (1) (resp. (2)) to each of the $f_i$, multiplying the inequality for $f_i$ by $c_i$, and then adding term-by-term.