prove that an equation is bounded by 1

Question

I have an equation that is bounded by 1 and where $s<m$, how can I prove it that it actually bounded by 1. The equation is:

$\frac{-m}{s}(1-\frac{s}{m})\ln(1-\frac{s}{m})$

Any help will be useful to me.

thanks

Answer 1

Let $x=\frac{s}{m}$. So $0<x<1$, we want to prove that $-\frac{1}{x}(1-x)\ln(1-x)\leq 1$.

$$-\frac{1}{x}(1-x)\ln(1-x)-1 = (1-\frac{1}{x})\ln(1-x)-1=\frac{(x-1)\ln(1-x)-1}{x}$$

$x>0$ so it is enough to prove that $(x-1)\ln(1-x)\leq 1$ or $\ln(1-x)\geq \frac{1}{x-1}$ we changed sign because $x-1<0$.

Let $f(t)=\ln(1-t)-\frac{1}{t-1}$ than $f'(t)= \frac{-1}{1-t}+\frac{1}{(t-1)^2}=\frac{t-1+1}{(t-1)^2}\geq0$ for $t\geq 0$. $f(0)=0$, so $f(t)\geq0$ and we are done.