I can proof it is convex, because this is a perspective function,

This is not a perspective function (denominator does not match multiplier of log).

Presuming A > 0., perhaps it is convex for B >= C, and concave for B <= C.

In my case, A> 0 and B>C, so it is a convex problem, how can i write it in cvx? Thanks !

@Mark_L_Stone please help me, I try to do it use (-rel_entr(ti, ti+C*a)/log(2))+rel_entr(ti, ti+B*a)/log(2)), but it appears to have reached the maximum recursion limit of 500.

You;'re adding a concave expression and a convex expression. That violates CVX’s DCP rules.

But how can i express t*log2((t+A*a)/(t+B*a)), where A>=B, 0<=t<=1, 0<=a<=1and a + t <=1? @Mark_L_Stone