Converse of Jensen's inequality
Solution 1:
Given $x,y\in\mathbb R$ and $t\in(0,1)$, consider $$ f(s) = \begin{cases} x & s\leq t \\ y & s>t. \end{cases} $$ Then your inequality tells $$ \phi(tx+(1-t)y) \leq t\phi(x)+(1-t)\phi(y). $$
Converse of Jensen's inequality
Solution 1:
Given $x,y\in\mathbb R$ and $t\in(0,1)$, consider $$ f(s) = \begin{cases} x & s\leq t \\ y & s>t. \end{cases} $$ Then your inequality tells $$ \phi(tx+(1-t)y) \leq t\phi(x)+(1-t)\phi(y). $$