The Inequality of Milne and Its Converse, III

The discrete version of Milne's inequality and its converse states that ( * ) ∑ j = 1 n w j 1 - p j 2 ≤ ∑ j = 1 n w j 1 - p j ∑ j = 1 n w j 1 - p j ≤ ( ∑ j = 1 n w j 1 - p j 2 ) 2 is valid for all wj > 0 (j = 1…, n) with w1+ … +wn = 1 and p j ∈ ( - 1 , 1 ) ( j = 1 , … , n ) . We present new upper and lower bounds for the product Σw/(1 – p) Σ w/(1 + p). In particular, we obtain an improvement of the right-hand side of (*). Moreover, we prove a matrix analogue of our double-inequality. Mathematical Reviews subject classification: Primary: 26D15; Secondary: 15A45 Key words: Milne's inequality, matrix inequalities