Why $\frac{(1-x)\log(1-x)}{x\log x}$ is increasing on (0, 1)? Let $f(x)=\frac{(1-x)\log(1-x)}{x\log x}$. I find one claims this function is increasing on $(0,1)$. While I find the numerator of $f'(x)$ is
$$-x\log x -\log x\log(1-x)-(1-x)\log(1-x).$$
I am wondering why this is non-negative?
Using taylor series:
$$\frac {\ln(1-x)}{x}=-\frac {x+\frac {x^2}2+...}x=-
(1+\frac x2+...)$$
Which is decreasing, similarly:
$$\frac {\ln(x)}{1-x}=-\frac{\ln(1-(1-x))}{1-x}=-(1+\frac{1-x}2+...)$$
Which is increasing, then its reciprocal $\frac {1-x}{\ln(x)}$ is decreasing.
Since these two are negative, then they multiplied together is increasing.
(Not a complete answer but hopefully close enough)
You want to show that
$$ x \log(x)+ (1-x)\log(1-x) \le - \log(x) \log(1-x)$$
Divide on both sides by $\log(x) \log(1-x)$ and this is
$$ \frac{x}{\log(1-x)} + \frac{1-x}{\log(x)} \le -1 $$
This is of the form
$$ f(x) + f(1-x) \le c$$
where $f(x) = \frac{x}{\log(1-x)}$. I don't know if there is a nice way to show this.
But the not-so-nice way is to Taylor expand
$$ \frac{x}{\log(1-x)} = -1 + \frac{x}{2} + \frac{x^2}{12} + \frac{x^3}{24} + \dotsm $$
$$ \frac{1-x}{\log(x)} = -1 + \frac{1-x}{2} + \frac{(1-x)^2}{12} + \frac{(1-x)^3}{24} + \dotsm $$
Now you add them up, you see that the RHS becomes
$$ -2 + \frac{1}{2} + \frac{x^2 + (1-x)^2}{12} + \frac{x^3 + (1-x)^3}{24} + \dotsm $$
$$ \le -2 + 1 + \frac{1}{12} + \frac{1}{24} + \dotsm $$
But then showing that the last sum is $\le -1$, seems clear to me but maybe you might want more concrete justification...