Help log,how to express it?

Hello everyone,I have a question to ask for help!

Here,g(m,k,n) and g(m,j,n) and y(m,k,n) are variables ,how can I express it correctly?
Can I introduce a new variable to replace ?

have you proven this constraint is convex?

Thanks for your reply! I see it from a paper and I have proven it .

Well, we haven’t seen it.

Is there a constructive “DCP-compliant” proof?