If $(\displaystyle\frac{2(a-1)}{a+1}\lt\ln a\lt\frac{2(a-1)}{2+\ln a})$ equals $(\displaystyle\frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1})$. Does that mean that $-1 + \sqrt{2a-1}$ = $\displaystyle\frac{2(a-1)}{2+\ln a}$?
The reason I ask is because if $a=4$ for example, then $-1 + \sqrt{2*4-1}$ $\ne$ $\displaystyle\frac{2(4-1)}{2+\ln 4}$
Thanks.
Generally if ($a<b<c$)=($a<b<d$), by no means you can conclude that c=d; In fact nothing specific can be said about the relationship between c and d.