# Is this inequality correct?

Let $a_1,a_2,b_1,b_2\in R$ such that $a_1b_1=-a_2b_2$. Is it correct that $$|\alpha_1a_1b_1+\alpha_2a_2b_2|\le 2\frac{\alpha_1-\alpha_2}{\alpha_1+\alpha_2}\frac{(\alpha_1a_1^2+\alpha_2a_2^2)(\alpha_1b_1^2+\alpha_2b_2^2)}{\alpha_1(a_1^2+b_1^2)+\alpha_2(a_2^2+b_2^2)}$$ where $\alpha_1>\alpha_2>0$?

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Try it with $a_1 = 1$, $b_1 = 2$, $\alpha_1 = 1$, and $\alpha_2 \to 0$.