Maximize $\lambda$ over $a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$ and $a,b,c,d\geq 0$.

Question

Find the largest real number $\lambda$ such that $$a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$$ for all nonnegative real numbers $a,$ $b,$ $c,$ $d.$

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I tried using AM-GM on like $a^2+\dfrac{b^2}{4}\geq ab$, and I tried combining the results, but it didn't get me anywhere. Could someone give me some guidance on how to proceed?

Thanks in advance!!!

Answer 1

You're very much on the right track. You can use \begin{align*} a^2+b^2+c^2+d^2 &=\left(a^2+\frac{b^2}4\right)+\left(\frac{c^2}4+d^2\right)+\frac{3(b^2+c^2)}4\\ &\geq ab+cd+\frac{3\cdot 2bc}4 \end{align*} to get that $\lambda=3/2$ works. Now, you just need to determine if any larger $\lambda$ will work. Can you look at the equality cases of the above process to show that any larger $\lambda$ will fail?