Convex Function

Definition

Let $\varphi$ : Function defined on interval $(a,b)$, where $-\infty \le a<b\le \infty$

If $\varphi [\gamma x+(1-\gamma )y]\le \gamma \varphi (x)+(1-\gamma )\varphi (y),\forall x,y\in (a,b),0<\gamma <1$

Then we say $\varphi$ is a convex function

Note

If the inequality is strict (i.e., $<$ instead of $\le$), then we say $\varphi$ is strictly convex.