Extending the minc-brègman upper bound for the permanent
George W. Solues · Linear and Multilinear Algebra · 2000
Let A=(aij ) be an n by n nonnegative matrix, where for each ros i the row sum is ri ≥0, the maximum element is (ti ), and the number of positive elements is mi . We prove the permanental inequality where for 1≤x≤m,μm(x) is the geometric mean of m equally spaced numbers from 1 to x;namely, μ1(1)=1 and for m>1, Our bound, which we denote by U(A), has the following properties. • When A is a (0,1)-matrixti =1 and mi =ri U(A)reduces to the Minc-Brègman upper bound . • The case of equality U(A)=per Aagrees, except for trivial row-scaling, with the case M(A)=perA. • U(A)is generally smaller than each of the best previous upper bounds{U l≥perA}for nonnegative matrices, except for very small n and for matrices close to the cases where U l= per A holds.