lean_workbook_plus_10224

By the Cauchy-Schwarz inequality $\sum u_{i}v_{i}\leqslant xy$ with the inequality if and only if the vectors $\vec{u}$ and $\vec{v}$ are parallel. Thus it remains to prove $1+x^{2}+y^{2}+2xy\leqslant \frac{4}{3}\left(1+x^{2}\right)\left(1+y^{2}\right).$