See Lemma.
Let $k_i$, $\lambda_i$ be non-negative numbers such that that $\lambda1\ge \lambda _2\ge \cdots\ge \lambda _m$, and $\lambda _1\le k_1$, $\lambda _1\lambda _2 \le k_1 k_2$, $\cdots$, $\lambda _1 \cdots \lambda _m\le k_1\cdots k_m$, then for any function $\varphi$ such that $\varphi(e^t)$ is a convex function of $t$ and $\varphi(0)=\lim_{t\rightarrow 0}\varphi(t) =0$, we have $$ \sum_{i=1}^m \varphi(\lambda )\le \sum_{i=1}^m \varphi(k_i).$$
I would like to ask whether we have an integration version of this result. e.g., something similar to the following:
Let $k(t)$, $\lambda(t)$ be two non-negative decreasing functions on $(0,\infty)$ such that that $$\int_0^t\log \lambda (s)\,\mathrm ds \le \int_0^t \log k(s)\,\mathrm ds$$ for every $t>0$, then we have $$\int_0^t\log \varphi(\lambda (s))\,\mathrm ds \le \int_0^t \log \varphi(k(s))\,\mathrm ds$$