Prove $ \frac{1}{a^{2}-ab+b^2}+\frac{1}{b^{2}-bc+c^2} +\frac{1}{c^{2}-ca + a^2} \leq \frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} $

Question

I'm not able to prove this inequality for all positive real numbers $a, b, c$.

Also I need to know when does the equality hold.

$ \dfrac{1}{a^{2}-ab+b^2}+\dfrac{1}{b^{2}-bc+c^2} +\dfrac{1}{c^{2}-ca + a^2} \leq \dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2} $

The teacher hinted us that in the end it could be either AM-GM inequality or Cauchy–Schwarz inequality, but I'm not sure.

Could anyone help me, please.

Thanks

Answer 1

Hint: $$a^2-ab+b^2\ge ab \text{ and }\frac{2}{ab}\le \frac1{a^2}+\frac{1}{b^2}.$$