Assume you have $n$ positive interests $q_1\le q_2\le\ldots\le q_n$. And $N$ positive values $C_1,C_2,\ldots, C_N$. Do we then have
$$\frac{\sum\limits_{t=1}^N\frac{q_tC_t}{(1+q_t)^t}}{\sum\limits_{t=1}^N\frac{C_t}{(1+q_t)^t}}\ge\frac{\sum\limits_{t=2}^N\frac{q_{t-1}C_t}{(1+q_{t-1})^{t-1}}}{\sum\limits_{t=2}^N\frac{C_t}{(1+q_{t-1})^{t-1}}}?$$
I am able to show that it holds for $N=2$, like this:
$$\frac{\sum\limits_{t=1}^N\frac{q_tC_t}{(1+q_t)^t}}{\sum\limits_{t=1}^N\frac{C_t}{(1+q_t)^t}}=\frac{\frac{q_1C_1}{(1+q_1)}+\frac{q_2C_2}{(1+q_2)^2}}{\frac{C_1}{(1+q_1)}+\frac{C_2}{(1+q_2)^2}}\ge\frac{\frac{q_1C_1}{(1+q_1)}+\frac{q_1C_2}{(1+q_2)^2}}{\frac{C_1}{(1+q_1)}+\frac{C_2}{(1+q_2)^2}}=q_1.$$
And we also have
$$\frac{\sum\limits_{t=2}^N\frac{q_{t-1}C_t}{(1+q_{t-1})^{t-1}}}{\sum\limits_{t=2}^N\frac{C_t}{(1+q_{t-1})^{t-1}}}=\frac{\frac{g_1C_2}{(1+q_1)}}{\frac{C_2}{(1+q_1)}}=q_1.$$
But does it hold for $N>2$? I tried with induction, but I am not able to finish it.