my understanding is that the author takes the positive square root i.e $E(+\sqrt(S^2))$ and not $E(-\sqrt(S^2))$.
If we were to take the negative square root value, the final expression would be negative, and 1- that value would be >1, in which case E(S) would over-estimate $\sigma$. Would this be possible?
However, from the plot, it seems that the bias seems to vary from 0 to 1, which suggests that the bias always an underestimate of $\sigma$
My query is whether there was any reason for taking the positive square root of $S^2$, and whether the bias of the sample standard deviation can theoretically represent either an underestimate or overestimate of $\sigma$