How can I express the following formula in CVX?

Where X is the variable,I tried many ways but it doesn’t work.

How to express log(1+ sum ( 1./x ) ) in CVX? and make the x_i be x_{i,m}/G_{i,m} and change the prepended 1 to 1/H_i

You can convert log2, which is not supported by CVX, to log by dividing by log(2).

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First of all, thank you very much for your reply.
I have read that topic,but the problem is Gi,m may be 0.Therefore conventional methods are not satisfactory,I have tried to discuss the number of parameters of log_sum_exp with a categorical discussion.But the effect is poor.

G_{i.m} is input data. So exclude terms having G_{i,m} = 0.

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