Sampling of bandpass signal with bandpass filter
Assuming that we're really talking about an analytic complex-valued signal with no negative frequency components, then a sampling frequency $\omega_s&gt;\omega_h-\omega_l$ will guarantee that the shifted spectra don't overlap, i.e., there will be no aliasing. However, it's not guaranteed that you'll have an image of the spectrum centered at DC. This is only the case if the sampling frequency satisfies
$$k\omega_s=\frac{\omega_l+\omega_h}{2},\qquad k\in\mathbb{Z}^+\tag{1}$$
Eq. $(1)$ means that the sampling frequency must be an integer multiple of the signal's center frequency.
So in the case of an analytic signal, reconstruction with a bandpass filter is possible if there is no aliasing, i.e., if $\omega_s&gt;\omega_h-\omega_l$ holds.
But since you're given the extra condition $\omega_l&gt;\omega_h-\omega_l$ I strongly suspect that the signal is actually real-valued with a (conjugate) symmetric spectrum. In this case, this answer contains the relevant information. Since $\omega_l&gt;\omega_h-\omega_l$, we have
$$n_{max}=\left\lfloor{\frac{\omega_l}{\omega_h-\omega_l}}\right\rfloor\ge 1\tag{2}$$
And, from Eq. $(1)$ in the quoted answer, the lowest possible sampling frequency must satisfy
$$\frac{2\omega_h}{n_{max}+1}&lt;\omega_s&lt;\frac{2\omega_l}{n_{max}}\tag{3}$$