<s>[INST] Show $\frac{a^2}{b^2} + \frac{b^2}{a^2} +3 \ge 2\left(\frac{a}{b} + \frac{b}{a}\right)$ I want to verify the following inequality:
Let $a, b$ be non negative number
$$\frac{a^2}{b^2} + \frac{b^2}{a^2} +3 \ge 2\left(\frac{a}{b} + \frac{b}{a}\right)$$
I decided to analyse the sign of $$\frac{a^2}{b^2} + \frac{b^2}{a^2} +3  - 2\left(\frac{a}{b} + \frac{b}{a}\right)$$
But I'm not getting anywhere. 
I'm having trouble deciding whether $\dfrac{a^2(a-2)+b^2(b-2)}{ab} +3$ is positive or negative.
 [/INST] Let $t=a/b+b/a$. Then this becomes $t^2+1\geq 2t$, which is same as $(t-1)^2 \geq 0$.
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