Sum of strictly convex and convex functions

Question

My prof mentioned that the sum of strictly convex and convex functions is strictly convex, Im having trouble swallowing that, is it accurate?

Answer 1

Assume $f$ is convex and $g$ is stricly convex. Let $0<\theta < 1$. Then calculate that $$ \begin{align} (f+g)(\theta x + (1-\theta) y) &= f(\theta x + (1-\theta) y) + g(\theta x + (1-\theta) y) \\ &< \theta f(x) + (1-\theta) f(y) + \theta g(x) + (1-\theta) g(y) \\ &= \theta (f+g)(x) + (1-\theta) (f+g)(y) \end{align} $$ where there is strict inequality because the inequality is strict in one case (and not necessarily strict in the other case).