In a paper I find the authors say that function \log(1+\frac 1 x_1+\frac 1 x_2+\cdots+\frac1 x_N) is convex when x_1, x_2,\cdots,x_N>0, and it can be solved by CVX.

I know \log(1+\frac 1 x_1) is convex when x_1>0. But is this function \log(1+\frac 1 x_1+\frac 1 x_2+\cdots+\frac1 x_N) convex? How to write it in CVX ?

The left side is exp, the right side is the hypograph of the harmonic mean (rescaled). I don’t see that cvx has a harmonic mean atom but it definitely can be done with some quadratic cones: https://docs.mosek.com/modeling-cookbook/cqo.html#harmonic-mean