Putnam 1980 B5

Problem B-5 For each $t \geqslant 0$, let $S$, be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$f(1)-2 f(2 / 3)+f(1 / 3) \geqslant t[f(2 / 3)-2 f(1 / 3)+f(0)] .$$

Develop necessary and sufficient conditions on $\boldsymbol{t}$ for $S$, to be closed under multiplication. (This closure means that, if the functions $f(x)$ and $g(x)$ are in $S_{t}$, so is their product $f(x) g(x)$. A function $f(x)$ is convex if and only if $f(s u+(1-s) v)<s f(u)+(1-s) f(v)$ whenever $0<s \leqslant 1$.)