The problem of prescribed performance finite-time bounded-$H_\infty$ tracking control is investigated for a class of nonlinear systems with unknown initial tracking conditions and external disturbances. A new error transformation idea is proposed aiming at the design of prescribed performance control. On this basis, the new prescribed performance functions are designed to solve the problem that the prescribed performance control depends on the initial condition of the constrained variable. By means of the prescribed performance functions, finite-time control theory and bounded-$H_\infty$ design method, a prescribed performance finite-time bounded-$H_\infty$ tracking controller is obtained, which is independent of the initial tracking condition. Meanwhile, the problem that it is difficult to design the $H_\infty$ controller for a nonlinear system with bounded stability is solved. The designed controller ensures that the tracking error converges to a small neighbourhood near the equilibrium point with prespecified dynamic performance within a finite time, and has stronger robustness to external disturbances.