The closed convex perspective is subadditive

Put $A={\sum }_i{a}_i$ and $B={\sum }_i{b}_i$. If $B=0$, all $b_i=0$. If $A=0$, every pair is $(0,0)$ and both sides are zero. If $A>0$, both sides equal $A{f}_{\mathrm{\infty }}$ by the definition of ${\mathrm{\Pi }}_f$, including the convention that a positive coefficient times $+\mathrm{\infty }$ is $+\mathrm{\infty }$.

Suppose $B>0$ and no index has $b_i=0<a_i$. If any $b_i>0=a_i$ with $f(0)=+\mathrm{\infty }$, the right side is $+\mathrm{\infty }$ and the inequality holds. If $f(0)$ is finite, extend $f$ to the convex domain $[0,\mathrm{\infty })$ by that value. This extension is convex at the endpoint too: approximate a zero argument in any convexity inequality by positive arguments and pass to the limit using $f(0)={\lim }_{t\downarrow 0}f(t)$ from [F1] and continuity at positive arguments from [F3]. If $f(0)=+\mathrm{\infty }$, the excluded case shows that every index with $b_i>0$ has $a_i>0$, so all ratios used below lie in the original convex domain $(0,\mathrm{\infty })$. For indices with $b_i>0$, the weights $b_i\mathrm{/}B$ sum to one and $A\mathrm{/}B={\sum }_{b_i>0}(b_i\mathrm{/}B)(a_i\mathrm{/}{b}_i)$. Applying [F2] gives $Bf(A\mathrm{/}B)\le {\sum }_{b_i>0}{b}_{i}f(a_i\mathrm{/}{b}_i)$; terms $(a_i,b_i)=(0,0)$ contribute zero. This is the asserted inequality.

Suppose $B>0$ and $J:=\{i:b_i=0<a_i\}$ is nonempty. If $f_{\mathrm{\infty }}=+\mathrm{\infty }$, the right side is $+\mathrm{\infty }$, so the inequality holds. If any $b_i>0=a_i$ with $f(0)=+\mathrm{\infty }$, the right side is $+\mathrm{\infty }$ and the inequality holds. Otherwise, if $f_{\mathrm{\infty }}$ is finite, replace each $b_i=0$ with $i\in J$ by $b_i^{(\eta )}=\eta >0$, leaving all other second coordinates unchanged. the finite Jensen inequality [F2] then gives $(B+\mathrm{\mid }J\mathrm{\mid }\eta )f(A\mathrm{/}(B+\mathrm{\mid }J\mathrm{\mid }\eta ))\le {\sum }_{b_i>0}{b}_{i}f(a_i\mathrm{/}{b}_i)+{\sum }_{i\in J}\eta f(a_i\mathrm{/}\eta )$. As $\eta \downarrow 0$, the left side tends to $Bf(A\mathrm{/}B)$ by [F3] at the positive number $A\mathrm{/}B$, and each last summand tends to $a_i{f}_{\mathrm{\infty }}$ by the definition of $f_{\mathrm{\infty }}$. The resulting inequality is exactly ${\mathrm{\Pi }}_f(A,B)\le {\sum }_i{\mathrm{\Pi }}_f(a_i,b_i)$.