  We study an optimum measurement for quantum state discrimination, which
maximizes the probability of correct results when the probability of
inconclusive results is fixed at a given value. The measurement describes
minimum-error discrimination if this value is zero, while under certain
conditions it corresponds to optimized maximum-confidence discrimination, or to
optimum unambiguous discrimination, respectively, when the fixed value reaches
a definite minimum. Using operator conditions that determine the optimum
measurement, we derive analytical solutions for the discrimination of two mixed
qubit states, including the case of two pure states occurring with arbitrary
prior probabilities, and for the discrimination of N symmetric states, both
pure and mixed. We also consider a case where the given density operators
resolve the identity operator, and we specify the optimality conditions for the
case of partially symmetric states. Moreover, we show that from the complete
solution for arbitrary values of the fixed rate of inconclusive results one can
always obtain the optimum measurement in another strategy where the error rate
is fixed, and vice versa.
