A Note on Improved Young Type Inequalities with Kantorovich Constant

Leila Nasiri,Mahmood Shakoori

doi:10.3844/jmssp.2016.201.205

Abstract

In this article, we first present some improved Young type inequalities for scalars, then according to these inequalities we give the Hilbert-Schmidt norm and the trace norm versions.

Highlights

The scalar Young inequality says that if a and b be nonnegative real numbers and 0 ≤ ν ≤ 1, : aν b1−ν ≤ν a + (1 −ν )b (1.1)With equality if and only if a = b: Inequality (1.1) is called the ν-weighted arithmetic-geometric mean inequality
This paper obtained some refinements of Young type inequalities for scalars, as applications of them, we presented some norm and trace inequalities
Interested readers are encouraged to find the applications for the derived scalar inequalities

Summary

Introduction

The scalar Young inequality says that if a and b be nonnegative real numbers and 0 ≤ ν ≤ 1, : aν b1−ν ≤ν a + (1 −ν )b With equality if and only if a = b: Inequality (1.1) is called the ν-weighted arithmetic-geometric mean inequality. Zuo et al (2011) refined Young inequality (1.1) as follows: ν a + (1 −ν )b ≥ K (h, 2)Υ aν b1−ν We need to recall that Kantorovich constant satisfies the following properties: (i) K(1, 2) = 1, (ii)

Results

Conclusion