Abstract
The purpose of this paper is to give a survey of the progress, advantages and limitations of various operator inequalities involving improved Young’s and its reverse inequalities related to the Kittaneh-Manasrah inequality. We also present our new progress to the related research topics. New scalar versions of Young’s inequalities are promoted, the operator version and the Hilbert-Schmidt form also get a promotion.
Highlights
As is well known, the famous Young’s inequality for real numbers is that ( – u)a + ub ≥ a –ubu, where a, b >, u ∈ [, ] ( . )which is called the u-weighted arithmetic-geometric mean inequality
2 New progress of Young’s and its reverse inequalities we mainly present the improved scalar Young and its reverse inequalities relating to the Kantorovich constant
4 New matrix versions of Young’s inequalities for the Hilbert-Schmidt norm In the last part, we focus on the matrix version of Young’s inequality for the HilbertSchmidt norm
Summary
The famous Young’s inequality for real numbers is that ( – u)a + ub ≥ a –ubu, where a, b > , u ∈ [ , ]. In [ , ], Kittaneh and Manasrah researched Young’s inequality and obtained the following results:. In [ ], Tominaga got the reverse Young inequality with the help of Specht’s ratio. Article [ ] pointed out that Specht’s ratio and the Kantorovich constant have the relationship as follows:. Based on this idea, in the article [ ], the authors got the refinement of Young’s inequality:. ) with the Kantorovich constant and gave the following results:. ). Let us take a closer look at [ ] where Liao et al made a reverse refinement for Young’s inequality as follows:. The operator versions A∇uB, A#uB, A!uB are called the arithmetic mean, geometric mean and harmonic mean, respectively
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