relation between antenna bandwidth ,efficiency and diversity

I am not familiar with diversity. Maybe you can give its definition here.

I'd like to say sth about BW and Effi.

Known to all, the impedance BW may be the strictest (and hence the narrowest) bandwidth between several types of BWs. And the impedance BW is determined by the input impedance of antenna and the source impedance. Generally speaking, the wider the BW is, the smaller the S11 will be, which means the matching at particular frequencies will be degraded. As a result, we can infer that the matching Effi will be deteriorated.

Besides, the radiation resistance and loss resistance are all functions of the frequency. And the radiation resistance will decrease if the operating frequency differs from the resonance frequency. Therefore, generally speaking, the radiation Effi will be degraded with larger BW.

To sum up, a general inference will be that, the wider the BW is, the lower the Effi will be, no matter the Effi stands for the radiation Effi, or matching Effi, or the total Effi (=radiation Effi*matching Effi).