$\frac{|ax_1+bx_2|}{(|x_1|^p+|x_2|^p)^\frac{1}{p}}\le(|a|^q+|b|^q)^\frac{1}{q}$

Question

I want to prove that:$$\frac{|ax_1+bx_2|}{(|x_1|^p+|x_2|^p)^\frac{1}{p}}\le(|a|^q+|b|^q)^\frac{1}{q}$$where a,b are arbitrary real numbers, $(x_1,x_2)\in R^2$ and $\frac{1}{p}+\frac{1}{q}=1$ $p,q\ge1$

I think I can use Holder or Minkowski inequalities somewhere in this question but I am not sure of where to use them in this question.

Answer 1

By Holder $$\left(|a|^q+|b|^q\right)^p\left(|x_1|^p+|x_2|^p\right)^q\geq\left(|ax_1|+|bx_2|\right)^{p+q}=\left(|ax_1|+|bx_2|\right)^{pq}$$ and we are done!