I have to prove that, given $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$, $\Omega$ open and bounded in $\mathbb R^n$ and $\lambda>0$ sufficiently small so that $2\lambda u<1$, the function $$ H(u)=\frac{1}{\lambda}\left(1-\sqrt{1-2\lambda u}\right) $$ is convex.
My attempt. Since $$ H'(u)=\frac{1}{\sqrt{1-2\lambda u}}, $$ then $$ H''(u)=\frac{\lambda}{(1-2\lambda u)^{\frac{3}{2}}}> 0. $$ Can we conclude that, since $H''(u)>0$, $H$ is convex?
Thank You