The Holder’s inequality states that: if \(1\leq p,q<\infty,\frac{1}{q}+\frac{1}{q}=1\) and if \(x_k,y_k,k=1,2,cdots,n\) are complex numnbers then
\(\left(\sum_{k=1}^{n}\left|x_k+y_k\right|^{p}\right)^{\frac{1}{p}\leq\left(\sum_{k=1}^{n}\left|x_k\right|^{p}\right)^{\frac{1}{p}+\left(\sum_{k=1}^{n}\left|y_k\right|^{p}\right)^{\frac{1}{p} \)
\(\left(\sum_{n=1}^{n}\left|x_k+y_k\right|^{p}\right)^{r}\leq\left(\sum_{k=1}^{n}\left|x_p\right|^{p}\right)^{\frac{1}{2}+\left(\sum_{k=1}^{n}\left|y_p\right|^{p}\right)^{\frac{1}{2} \)
—>> \(\left(\sum_{k=1}^{n}\left|x_k+y_k\right|^{p}\right)^{\frac{1}{p}\leq\left(\sum_{k=1}^{n}\left|x_k\right|^{p}\right)^{\frac{1}{p}+\left(\sum_{k=1}^{n}\left|y_k\right|^{q}\right)^{\frac{1}{q} \)
\(\left(\sum_{k=1}^{n}\left|x_k+y_k\right|^{p}\right)^{\frac{1}{p}\leq\left(\sum_{k=1}^{n}\left|x_p\right|\right)^{\frac{1}{p}+\left(\sum_{k=1}^{n}\left|y_p\right|\right)^{\frac{1}{q} \)
