I was reading a paper and I found stated without proof or reference the following inequality:

$$\left(\sum_{i=1}^{m}\prod_{j=1}^{n}(\beta_{i}^j)^p\right)^{\frac{1}{p}}\leq \prod_{j=1}^{n}\left(\sum_{i=1}^{m}\left(\beta_{i}^{j}\right)^{p_j}\right)^{\frac{1}{p_j}}$$

where $\beta_i^j\geq 0$ and $\frac{1}{p_1}+\cdots\frac{1}{p_n}=\frac{1}{p}$

Anyone knows where I can find a proof for this?

Thanks in advance.


Use the generalized Hölder inequality with counting measure.

  • $\begingroup$ Thanks a lot for the answer, Milind! $\endgroup$ – Mark_Hoffman Dec 2 '15 at 9:12

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