Let me give you a counterexample showing that the given limits for $R_X$ in terms of $R_X$ and $R_X$ are not sufficient for guaranteeing the non-negativity of the corresponding power spectrum.
Let's fix $R_X$ at its given lower limit:
Furthermore, let's assume that $R_X[k]=0$ for $|k|>2$. The DC value of the corresponding power spectrum $S_X(\omega)$ is just the sum over all values $R_X[k]$, and is given by
It's straightforward to show that for
the value of $S_X(0)$ becomes negative, which shouldn't be the case for a power spectrum. Hence, it appears that the given lower bound for $R_X$ is not correct.
EDIT: As shown in GrapefruitIsAwesome's answer, it appears that the given limits on $R_X$ have been derived from the requirement that the autocorrelation matrix be positive semi-definite. However, for the sequence $R_X[k]$ to be a valid autocorrelation sequence, it is necessary that all Toeplitz matrices with first row $R_X,R_X,\ldots,R_X[n]$ are positive semi-definite for $n=1,2,\ldots,\infty$.
In the example above I've assumed $R_X[k]=0$ for $|k|>2$, in which case the given limits on $R_X$ are not sufficient, but only necessary. The given limits just guarantee that there exist valid autocorrelation sequences with the first three elements as given. However, not all sequences with those first three elements are valid autocorrelation sequences.