In this article: http://rgmia.org/papers/v7e/RBKIIPS.pdf, the author claims that the inequality (after (2.4)) $$\frac{|\langle a,x\rangle \langle x,b\rangle|}{\|x\|^2} \leq \frac{1}{|\beta|}\left[|\langle a, b\rangle|+\frac{\|b\|}{\|x\|}\right]\left[|\beta-1|^2|\langle x,a\rangle|^2+\|x\|^2\|a\|^2-|\langle x,a\rangle|^2\right]^{\frac{1}{2}} $$ holds for arbitrary complex vectors $a,x,b$ and complex scalar $\beta \neq 0$ (The author uses $\alpha$ istead of $\beta$, which becomes a little difficult to read, so I wrote $\beta$ here.).
Well, let us set $a,x,b$ to be unit vectors that are all equal to each other, and suppose that $\beta$ is real and $\geq 1$. We get the inequality $$1 \leq \frac{2(\beta-1)}{\beta},$$ which fails for $1 \leq \beta < 2$. What am I missing here?
The inequality is indeed wrong. If you follow the derivation it should be
$$\frac{|\langle a,x\rangle \langle x,b\rangle|}{\|x\|^2} \leq \frac{|\langle a, b\rangle|}{|\beta|}+\frac{\|b\|}{|\beta|\|x\|}\left[|\beta-1|^2|\langle x,a\rangle|^2+\|x\|^2\|a\|^2-|\langle x,a\rangle|^2\right]^{\frac{1}{2}} $$
They misplaced one of the square brackets. Pay close attention to the square brackets in the first inequality in (2.5) - they are correct.