# Does the following inequality hold true?

I'm studying on a Jurimetrics problem where I want to design an optimal responsibility allocation mechanism between plaintiffs and defendants. But I'm right now facing the following problem, and I just have no idea how to prove it after several strives:

$$\frac{(1-x)^{2}-(1-y)^{2}}{1-x}>$$ $$\frac{(1-x-a)^{2}-(1-y-b)^{2}}{1-x-a}，$$ where $$x\in\left( 0,1 \right),y\in\left( 0,1 \right)$$, $$1>(x+a)>0, 1>(y+b)>0$$ and $$1>a>b>0$$ ?

• It would be helpful if you provided some context. How did this question arise? Jan 30, 2022 at 2:31

This inequality is false if e.g. $$x=\frac{5}{8},y=\frac{15}{16},a= \frac{7}{8},b=\frac{3}{4}.$$ (Then the left-hand side of your inequality is $$\dfrac{140}{384}$$ while its right-hand side is $$\dfrac{171}{384}$$.)

With the additional condition that $$1 > x + a > 0$$ and $$1 > y + b > 0$$, the inequality is true. According to Mathematica:

Here is a "human" proof of the inequality with the additional condition that $$1 > x + a > 0$$ and $$1 > y + b > 0$$:

Letting $$U:=1-x$$, $$V:=1-y$$, $$u:=U-a$$, $$v:=V-b$$, we reduce the inequality in question to the inequality $$U-\frac{V^2}U>u-\frac{v^2}u \tag{1}$$ if $$U>u>0$$, $$V>v>0$$, and $$U>V-v+u$$.

The left-hand side of (1) is increasing in $$U$$. So, (1) reduces to the inequality $$d:=V-v+u-\frac{V^2}{V-v+u}-\Big(u-\frac{v^2}u\Big)\ge0. \tag{2}$$ But $$d=\frac{(u - v)^2 (V-v)}{(V-v+u)u}\ge0,$$ and we are done.

Remark: Here we only used the conditions $$U>V-v+u$$, $$V>v$$, and $$u>0$$, which are equivalent to the conditions $$a>b$$, $$a>0$$, and $$1>x+a$$ (respectively). None of the conditions $$x>0$$, $$b>0$$, $$a<1$$, $$x+a>0$$, $$1 > y + b > 0$$ was used.

• What conditions do x and y need to meet to make the above inequality true?
– S.M.Hao
Jan 30, 2022 at 3:01
• @S.M.Hao : That would be an additional question, and it should be posted separately (preferably elsewhere). A simplest answer to this additional question is tautological: the inequality will be true if and only if it is true. Probably, you want a more informative answer. So, if you decide to post your additional question separately, you should specify the terms in which you want the additional conditions to be expressed. Jan 30, 2022 at 3:06
• thanks for your guidance, I will reorganize the problem. the current page should be deleted . thanks again!
– S.M.Hao
Jan 30, 2022 at 3:10
• @S.M.Hao : No, this page should not (and cannot) be deleted, as it has a valid answer to your question. Rather, let us have a closure here, and then you can work on an improved version of this question and perhaps post it separately. Jan 30, 2022 at 3:20
• @user7427029 : I used Mathematica's command FindInstance for that. Jun 8, 2022 at 19:31