Let λ i and μ i ( i = 1,2,…, N) be real constants and θ i ( i = 1,2,…, N) nonnegative variables satisfying Σθ i =1. It is shown that the ( N - 1)-dimensional extremum problem of finding the global maximum and minimum of the product (Σ θ i λ i )(Σ θ i μ i ) can be reduced to a one-dimensional problem defined on the boundary of the convex polygon spanned by the ordered pairs (λ i ,μ i ). For a given set of positive λ i ′s (λ 1⩾λ 2⩾ ⋯ ⩾λ N >0), the Kantorovich inequality states that max{(Σθ iλ i)(Σθ iμ i)}= (λ 1+λ N) 2 4λ 1λ N and min{( Σθ i λ i )( Σθ i μ i )}=1 if μ i= 1 λ i ,i=1,2,…,N . It is shown that the above new technique can be used to identify exactly how far the μ i ′s can vary from the Kantorovich values μ i = 1/λ i without invalidating the Kantorovich bounds. Let ν max=sup ( x, x) =1{( x, Ax)( x, Bx)} and ν min=inf ( x, x) =1{( x, Ax)( x, Bx)}, where x is an N-dimensional complex vector, and A and B are two N × N positive definite Hermitian matrices. For the special case in which B= A -1, the Kantorovich inequality implies that ν max ν min = [κ(A)+1] 2 4κ(A) , where κ( A) is the spectral condition number of A. It is shown that, in general, ν max ν min ⩾ max {[κ(A)+1] 2 4κ(A) , [κ(B)+1] 2 4κ(B)} . The conditions under which equality occurs are also established.