How to implement this problem on CVX?

I am trying to solve the following optimisation problem in MATLAB CVX. I think it is Convex but I can’t think of a way to implement it.

min {x_i} -{\sum_{i=1}^{N} \log(1+h_i x_i)} - \frac{K}{\sum_{i=1}^{N}x_i^{-1}}
s.t. \sum_{i=1}^{N} x_i =< P
x_i>0, for all i in {1,…,N}

So far I had implemented the above expression in terms of diagonal matrices where the second term of the obj. function was represented with the harmonic mean function as follows:

K = constant;
A_diag is positive-real diagonal matrix


variable x_diag(N,1)

min {x_diag} -log_det (I_N + A_diag diag(x_diag)) - K*harm_mean(x_diag)
s.t. \sum x_diag =< P
x_diag > 0, for all i in {1,…,N}


The two terms in the above implementation seem disjoint. Any suggestions on implementing this in the first format?

Harmonic mean can be implemented as shown in section 3.2.7 “Harmonic mean” of Mosek Modeling Cookbook , and using rotated_loretnz, for which there are example on this forum, for the rotated 2nd order cone.