Inequality $\left(1-\frac{x(1-x)+y(1-y)}{1-x+1-y}\right)^2+(1-\frac{x+y}{2})^2\geq (1-x)^2+(1-y)^2$

Question

it's a little problem found by myself

let $x,y\neq 1$ .be real numbers then we have : $$\left(1-\frac{x(1-x)+y(1-y)}{1-x+1-y}\right)^2+\left(1-\frac{x+y}{2}\right)^2\geq (1-x)^2+(1-y)^2$$

I tried Jensen's inequality and Am-Gm but without success . I also tried Karamata's inequality but it fails. I have a ugly proof using derivative and it's too long to be explain here .

I'm looking for a contest proof .

Thanks a lot to share your knowledge and your time.

Answer 1

For $x+y\neq2$ by AM-GM we obtain:$$\left(1-\frac{x(1-x)+y(1-y)}{1-x+1-y}\right)^2+\left(1-\frac{x+y}{2}\right)^2=$$ $$=\left(\frac{(1-x)^2+(1-y)^2}{2-x-y}\right)^2+\left(\frac{2-x-y}{2}\right)^2\geq$$ $$\geq2\sqrt{\left(\frac{(1-x)^2+(1-y)^2}{2-x-y}\right)^2\left(\frac{2-x-y}{2}\right)^2}= (1-x)^2+(1-y)^2$$

Answer 2

Hint: Your inequality is equivalent to $$\frac{(x-y)^4}{4 (x+y-2)^2}$$