A new generalization of two refined Young inequalities and applications

Abstract

Abstract In this paper, we prove that if a, b > 0 and 0 ≤ α ≤ 1, then for m = 1, 2, 3, . . . , r 0 m ( a m 2 - b m 2 ) 2 ≤ r 0 m ( b m + 1 - a m + 1 b - a - ( m + 1 ) ( a b ) m 2 ) ≤ ( α a + ( 1 - α ) b ) m - ( a α b 1 - α ) m , \matrix{ {r_0^m{{\left( {{a^{{m \over 2}}} - {b^{{m \over 2}}}} \right)}^2}} & { \le r_0^m\left( {{{{b^{m + 1}} - {a^{m + 1}}} \over {b - a}} - \left( {m + 1} \right){{\left( {ab} \right)}^{{m \over 2}}}} \right)} \cr {} & { \le {{\left( {\alpha a + \left( {1 - \alpha } \right)b} \right)}^m} - {{\left( {{a^\alpha }{b^{1 - \alpha }}} \right)}^m},} \cr } where r 0 = min{α, 1 – α }. This is a considerable new generalization of two refinements of the Young inequality due to Kittaneh and Manasrah, and Hirzallah and Kittaneh, which correspond to the cases m = 1 and m = 2, respectively. As applications we give some refined Young type inequalities for generalized euclidean operator radius and the numerical radius of some well-know f -connection of operators and refined some Young type inequalities for the traces, determinants, and norms of positive definite matrices.