\left(\displaystyle\frac{|z_1+z_2|^n+|z_1-z_2|^n}{2}\right)^{1/n}\geq\left(\displaystyle\frac{|z_1+z_2|^2+|z_1-z_2|^2}{2}\right)^{1/2}=\\=\left(|z_1|^2+|z_2|^2\right)^{1/2}\geq\left(|z_1|^n+|z_2|^n\right)^{1/n}