Prove that for positive real numbers $p,x_1,\cdots, x_n, y_1,\cdots, y_n,$ we have $$\sum_{i=1}^n \frac{x_i^{p+1}}{y_i^p} \ge \frac{\left(\sum_{i=1}^n x_i\right)^{p+1}}{\left(\sum_{i=1}^n y_i\right)^p}$$
By the power-mean inequality, we have that for positive real numbers $x_1,\cdots, x_n$ and numbers $k\ge m$, $\left( \frac{\sum_{i=1}^n x_i^{k}}{n}\right)^{1/k} \ge \left(\frac{ \sum_{i=1}^nx_i^{m}}{n}\right)^{1/m}.$ Holder's inequality says that for real numbers $x_1,\cdots, x_n, y_1,\cdots, y_n,$ we have $\sum_{k=1}^n |x_k y_k| \leq (\sum_{k=1}^n |x_k|^p)^{1/p} (\sum_{k=1}^n |y_k|^q)^{1/q}$, where $p,q \in (1,\infty), 1/p + 1/q = 1.$ There's also the Cauchy-Schwarz inequality, which solves the problem in the special case where $p=1$. I'm not sure if Jensen's inequality might be useful for the general case.