A new generalization of two refined Young inequalities and applications

Abstract

In this paper, we prove that if a, b > 0 and 0 ≤ α ≤ 1, then for m = 1, 2, 3, . . . ,\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 r0 = 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.

A new generalization of Young type inequality and applications

“…If α = 1 2 , the inequality (2.1) becomes an equality. Assume that α < 1 2 . Then, by the inequality (1.1) we have…”

“…Very recently, Ighachane and Akkouchi in [1] gave a new generalization of the refined Young inequalities (1.3) and (1.4) as follow: For m = 1, 2, 3, . .…”

“…where r 0 = min {α, 1 − α} . The authors in [1] gave the reverse of scalar Young type inequality (1.6) as follows…”

A new generalization of refined young inequalities for τ-measurable operators

“…For further reading related to generalized refinement of Young's inequality, the reader is referred to recent papers [4], [3], [9], [10] and [11]. One goal of this paper is to show the general refinements form governing Theorem 1.3.…”