with this inequality $\ln{x}\ln{(1-x)}<\sqrt{x(1-x)}$

Solution 1:

For $x\in (0,1)$ $$0<-\log(x)<\frac{1-x^2}{2x}.$$ This follows by estimating the integral $$-\log(x)=\int_x^1\frac{dt}{t}$$ with a trapezoid. Also $\log(x)=2\log\sqrt{x}$ so $$0<-\log(x)<\frac{1-x}{\sqrt{x}}$$ on this interval and your inequality follows.

Solution 2:

This is in fact a comment with a picture. The point is to obtain a formal symbolic proof of the inequality. Otherwise we could just offer

Solution 3:

Use well known inequality,we have $$\sqrt{ba}<\dfrac{b-a}{\ln{b}-\ln{a}},a>0,b>0$$ let $b=x,a=1$,then we have $$\Longrightarrow \ln{x}>\dfrac{x-1}{\sqrt{x}}$$

$$\Longrightarrow -\ln{x}<\dfrac{1-x}{\sqrt{x}}\tag{1}$$ simaler we have $$-\ln{(1-x)}<\dfrac{x}{\sqrt{1-x}}\tag{2}$$ $(1)\times(2)$ we have $$\ln{x}\ln{(1-x)}<\sqrt{x(1-x)}$$