$(\sum_{i=1}^{\infty} |x_i|^\beta)^{\frac{\alpha}{\beta}} \leq \sum_{i=1}^{\infty} |x_i|^\alpha$ where $x_i\in \mathbb{R}$ and $0<\alpha<\beta$.

Question

Show that $(\sum_{i=1}^{\infty} |x_i|^\beta)^{\frac{\alpha}{\beta}} \leq \sum_{i=1}^{\infty} |x_i|^\alpha$ where $x_i\in \mathbb{R}$ and $0<\alpha<\beta$.

This is a homework question. I'd like to know what material should I look at so that I can solve it by myself. So far, I guess understanding Minkowski inequality might be useful. Any other reading suggestions?

Take the $\alpha$-th root on both side, and this is exactly the monotonicity of $\ell_p$ norms. — Clement C., Feb 12 '18 at 23:25

score 1 · Accepted Answer · answered Feb 12 '18 at 23:28

1

I cannot really recommend "material". It is about $p$-norms, but the exercise is rather trivial. You should convince yourself that you can reduce it to the case where $0\leq x_j\leq 1$ for all $j$, and from there it is fairly straighforward.

answered Feb 12 '18 at 23:28

Martin Argerami

217,281

$(\sum_{i=1}^{\infty} |x_i|^\beta)^{\frac{\alpha}{\beta}} \leq \sum_{i=1}^{\infty} |x_i|^\alpha$ where $x_i\in \mathbb{R}$ and $0<\alpha<\beta$.

1 Answers1