On the reduction numbers of monomial ideals

Let I be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume I is contained in the integral closure of some ideal that is generated by two elements from the generating set of I. We produce sharp upper bounds for each of the reduction number and the Ratliff-Rush reduction number of the ideal I. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.