Presuming that the a_i are non-negative, then your objective function is indeed convex.
If t > 0, and if it were not for the linear constraints on t, then I think you could do this by defining your t as being the reciprocal of the way you defined it. and then using the exponential cone, as shown in Exponential perspective function on CVX . Then (I’ll let you put in the a_i and extend it to N dimensions)
minimize(z-t)
{1,t,z} == exponential
Hopefully I didn’t screw up that logic, including the subtracting t term from z. However, if you do the above, then the constraints which were linear in your (old) t will now be nonlinear in the new t, so you would have to be able to express them in a DCP-compliant manner, which perhaps you might not be able to do.