To mitigate the adverse effects of external multi-source time-varying disturbances and to address the trajectory tracking problem of quadrotor unmanned aerial vehicle (QUAV) under unknown initial conditions, a novel prescribed performance control (PPC) scheme based on sliding mode control is proposed. This method incorporates a saturated gain conversion function and a triple-inflection-point hyperbolic function. The conversion error is obtained via the saturated gain conversion function and is used for PPC. Subsequently, a composite dynamic sliding surface is constructed using the triple-inflection-point hyperbolic function. By appropriately designing the parameters, the tracking error can be guaranteed to rapidly converge within the predefined performance boundaries and eventually reach the equilibrium point on the sliding surface, thereby achieving the desired transient and steady-state performance. Numerical simulation results validate that the proposed performance function enables dynamic adjustment of the control gain in response to real-time system errors, effectively overcoming the dependency of conventional PPC methods on known initial conditions. The results demonstrate that the proposed PPC framework offers superior tracking performance and strong robustness against external disturbances, showing promising applicability in complex and uncertain flight environments.