Moderator Edit: do not quote the whole post above you \[ \begin{array}{l} 1 + \sum\limits_{i = 1}^n ( u_i + v_i )^2 \le \frac{4}{3}\left( {1 + \sum\limits_{i = 1}^n {u_i^2 } } \right)\left( {1 + \sum\limits_{i = 1}^n {v_i^2 } } \right) \\ \Leftrightarrow 2\sum\limits_{i = 1}^n {u_i v_i } \le \frac{1}{3} + \frac{1}{3}\sum\limits_{i = 1}^n {\left( {u_i^2 + v_i^2 } \right)} + \frac{4}{3}\sum\limits_{i = 1}^n {u_i^2 } \sum\limits_{i = 1}^n {v_i^2 } \\ \Leftarrow 2\sum\limits_{i = 1}^n {u_i v_i } \le \frac{1}{3} + \frac{2}{3}\sum\limits_{i = 1}^n {u_i v_i } + \frac{4}{3}\left( {\sum\limits_{i = 1}^n {u_i v_i } } \right)^2 \\ \Leftrightarrow 0 \le \frac{1}{3}\left( {1 - 2\sum\limits_{i = 1}^n {u_i v_i } } \right)^2 \\ \end{array} \]

\begin{align*} 1+\lVert u+v\rVert^2&\le\frac43(1+\lVert u\rVert^2)(1+\lVert v\rVert^2)\\ \iff1+\lVert u\rVert^2+2\langle u,v\rangle+\lVert v\rVert^2&\le\frac43(1+\lVert u\rVert^2+\lVert v\rVert^2+\lVert u\rVert^2\lVert v\rVert)^2\\ \iff2\langle u,v\rangle&\le\frac13+\frac13\lVert u\rVert^2+\frac13\lVert v\rVert^2+\frac43\lVert u\rVert^2\lVert v\rVert^2\\ \iff6\langle u,v\rangle&\le1+\lVert u\rVert^2+\lVert v\rVert^2+4\lVert u\rVert^2\lVert v\rVert^2\\ \impliedby0&\le4\lVert u\rVert^2\lVert v\rVert^2-6\lVert u\rVert\lVert v\rVert+\lVert u\rVert^2+\lVert v\rVert^2+1\\ \impliedby0&\le4\lVert u\rVert^2\lVert v\rVert^2-4\lVert u\rVert\lVert v\rVert+1\\ \impliedby0&\le(2\lVert u\rVert\lVert v\rVert-1)^2. \end{align*}