Inequality $(w_1^{1/p}|x|)w_1^{1/q}+(w_2^{1/p}|y|)w_2^{1/q} \leq (w_1|x|^p+w_2|y|^p)^{1/p}(w_1+w_2)^{1/q}$?

Question

How to prove that

$$(w_1^{1/p}|x|)w_1^{1/q}+(w_2^{1/p}|y|)w_2^{1/q} \leq (w_1|x|^p+w_2|y|^p)^{1/p}(w_1+w_2)^{1/q}$$

Where $x,y\in\mathbb{R},$ $\dfrac1p+\dfrac1q=1$ and $w_1,w_2>0$?

Thank you

Answer 1

For positives $p$ and $q$ it's just Holder inequality: $$\left(w_1|x|^p+w_2|y|^p\right)^{\frac{1}{p}}(w_1+w_2)^{\frac{1}{q}}\geq\left(\left(\left(w_1|x|^p\right)^{\frac{1}{p}}w_1^{\frac{1}{q}}\right)^{\frac{1}{\frac{1}{p}+\frac{1}{q}}}+\left(\left(w_2|y|^p\right)^{\frac{1}{p}}w_2^{\frac{1}{q}}\right)^{\frac{1}{\frac{1}{p}+\frac{1}{q}}}\right)^{\frac{1}{p}+\frac{1}{q}}=$$ $$=(w_1)^{\frac{1}{p}}|x|(w_1)^{\frac{1}{q}}+(w_2)^{\frac{1}{p}}|y|(w_2)^{\frac{1}{q}}=w_1|x|+w_2|y|.$$ About Holder see here: https://math.stackexchange.com/tags/holder-inequality/info