Let $\varphi: \mathbb{R} \to \mathbb{R}$ be a bounded, Borel-measurable function with $$\varphi\biggl(\int_{0}^{1}f(x) d\lambda(x)\biggr) \leq \int_{0}^{1}\varphi(f(x))d\lambda(x)$$ for every bounded, Lebesgue-measurable function $f: \mathbb{R} \to \mathbb{R}$. Show that $\varphi$ is convex.
I am struggling with this one. I know I have to show that $\varphi(tx + (1-t)y) \leq t\varphi(x) + (1-t)\varphi(y)$ holds and I somehow wanted to express $tx + (1-t)y$ via a function such that I can use the given property of $\varphi$. I even thought about splitting up this $xt + (1-t)y$ into two seperate functions $f_1$ and $f_2$ and then using the property of $\varphi$ but since I have no linearity given for $\varphi$, that seems rather unfruitful to me.
So far, all of my attempts have been a complete mess. Could anybody help me?