I'll repost the inequality using LaTeX code:
I hope it will be better this way.
Hi, there's a proof I need to complete which requires that the following inequality holds:
(1/4)*((b-a[1])^2+(b-a[2])^2+(1-a[3])^2+(1-a[4])^2) >= 1/6*((b-a[1])^2+(b-a[2])(2*b-a[1]-a[2])+(b-a[3])(2*b-a[4]-a[3])+(b-a[4])^2)
where:
0<=a[1]<=a[2]<=a[3]<=a[4]<=b<=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 a[1]=a[2] and a[3]=a[4], and also for the case where 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 0<=(2*b-a[4]-a[3])<=(2*b-a[1]-a[2])<=2. I also know that (b-a[3])(2*b-a[4]-a[3]) <= (b-a[2])(2*b-a[1]-a[2]). Finally, I also know that the case where both sides of the inequality are maximums occurs when a[1]=a[2]=a[3]=a[4]=0 and b=1.
I hope somebody here can help me with this. Best regards.