In this paper, we evaluate the Cramer-Rao bounds (CRB) for the estimation of the carrier frequency offset (CFO) for general QAM modulations with no knowledge of the transmitted sequence (blind operation) in frequency hopping (FH) based short burst mode mobile radios. We investigate the blind CFO estimation problem in FH based mobile system that uses multiple short narrowband bursts in different frequency channels for the transmission of one packet. Joint blind channel and CFO estimation over multiple frequency bursts is considered. In the first part of the paper, we determine the limiting performance by finding the CRB for blind estimation, and compare it to the CRB for known data (genie aided) case. The CRB relates the reliability of CFO estimation to the number of radio frequency bursts (RFB) used, constellation and SNR level, and is an important measure for system design without the need to simulate specific algorithms. It is shown that the linear combination (in MMSE sense) of multiple frequency estimates, each obtained from a single RFB independently, achieves the joint CRB over multiple bursts. In the second part of the paper, some practical blind estimation methods are proposed. They are shown to exhibit close performance to the true CRB.