This paper is concerned with the H2 state-feedback synthesis problem for discrete-time systems under positivity constraint on the closed-loop systems. This problem is believed to be a non-convex problem and hence exact treatment is not known to this date. With this difficulty in mind, in this paper, we first derive semidefinite programs (SDPs) for the computation of the upper bounds of the achievable H2 performance as well as suboptimal gains. However, if we merely rely on the upper bound computation, we cannot draw any definite conclusion on the quality of the computed suboptimal gains. Thus the main issue of the present paper is the lower bound computation, and to this end we derive an SDP for that computation via specific treatment of the finite impulse resonse (FIR) with the idea of time-varying gain synthesis. We show that the lower bounds become tighter as we increase the length of the FIR. By numerical examples we show the soundness of the proposed strategy with upper and lower bounds computation.