I'm sorry to repost this thread, but I put it in the pre-univ algebra section and I had no suggestions.

There's a proof I need to complete which requires that the following inequality holds:

$\displaystyle \frac{1}{4}((b-a_1)^2+(b-a_2)^2+(1-a_3)^2+(1-a_4)^2) \geq$ $\displaystyle \frac{1}{6}((b-a_1)^2+(b-a_2)(2\cdot b-a_1-a_2)+(b-a_3)(2\cdot b-a_4-a_3)+(b-a_4)^2)$

where:

$\displaystyle 0\leq a_1\leq a_2\leq a_3\leq a_4\leq b\leq 1$

I have made several plots and in all of them the inequality holds. Nonetheless, I need to finish this problem theoretically. I have already made the proof for the case of $\displaystyle a_1=a_2\neq a_3=a_4$, and also for the case where $\displaystyle a_1=a_2=a_3=a_4$, but these are straightforward, and the real deal comes when I want to solve for the general case.

I know $\displaystyle 0\leq (2\cdot b-a_4-a_3)\leq (2\cdot b-a_1-a_2)\leq 2$. I also know that $\displaystyle (b-a_3)(2\cdot b-a_4-a_3)\leq (b-a_2)(2\cdot b-a_1-a_2)$. Finally, I also know that the case where both sides of the inequality are maximums occurs when $\displaystyle a_1=a_2=a_3=a_4=0$ and $\displaystyle b=1$.

I hope somebody here can help me with this problem. Best regards.