As I understand it, the radiation belts are pretty much entirely dependent on the solar wind. If the jovian is closer to the star then there will be a lot more radiation because the wind will be stronger (due to the inverse square law). Cool stars will have habitable zones that are much closer than warmer stars, so if anything the problem could be even worse there.
If the jovian had a weaker/smaller magnetic field then that would offset the problem somewhat.
If the jovian had a weaker/smaller magnetic field then that would offset the problem somewhat.