We address monotone conditions for the convexity of trace function associated with circular cone and extend to parametrized problem in this talk. This structure was based on the Schur complement theorem and it played an important role in proving the positive-(semi)definite property on Hessian matrix. By smoothing the problem, the greatest benefit is spared the solving time and we believe it is useful on many fields. This is a joint work with T.-W. WU.