General lower and upper bounds under minimum-error quantum state discrimination

For the optimal success probability under minimum-error discrimination between $r\geq2$ arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds specify relations these known in literature. We also present example where bounds, upper, on are tighter than most of The bound explicitly generalizes to $r>2$ form Helstrom bound. $r=2$, each our reduces