I am having trouble proving the following inequality:
Let $f,h: [0,1] \to \mathbb{R}$ continuous, $h >0$ and $\phi: \mathbb{R} \to \mathbb{R}$ continuous and convex. Show that: $$ \phi(\frac{\int^1 _0 f(x) h(x) dx}{\int _0 ^1 h(y) dy}) \leq \frac{\int^1 _0 \phi(f(x)) h(x) dx}{\int _0 ^1 h(y) dy} $$ We are supposed to use the general mean value theorem for integrals and the fact that: $ \phi(\int_0 ^1 f(x) dx) \leq \int_0 ^1 \phi(f(x)) dx$. Furthermore we can assume that for a segmentation $0=x_0, \ldots x_n=1$ it holds that for: $\alpha_k := \frac{\int_{x_k} ^{x_{k+1}} h(x) dx}{\int_0 ^1 h(y) dy}$ $\alpha_k > 0$ and $\sum_{k=0} ^{n-1} \alpha_k = 1$.
However the last thing seems quite trivial to me and not that useful.
So far I (think that I) was able to show that: $$ \phi(\frac{\int^1 _0 f(x) h(x) dx}{\int _0 ^1 h(y) dy}) = \phi(\frac{f(\xi) \int^1 _0 h(x) dx}{\int _0 ^1 h(y) dy}) = \phi(f(\xi)) $$ (For some $\xi \in [0,1]$ with the mean value theorem). However I don't know whether that's useful or not. Any tips/help is appreciated.