On Some New Fractional Type Heinz Inequalities

Yu Mei Liao (School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, China), Xuezhu Li (Department of Mathematics, Guizhou University, Guiyang, China) and Jin Rong Wang (Department of Mathematics, Guizhou University, Guiyang, China)
DOI: 10.4018/japuc.2013010102
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Abstract

The authors shall discuss Heinz inequalities involving Riemann-Liouville fractional integrals for certain unitarily invariant norms. By using the convexity of function and fractional Hermit-Hadamard integral inequality, some refinements of Heinz inequalities are derived.
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1. Introduction

It is well known that Heinz means is one of special means which can interpolate between the geometric and the arithmetic mean. The Heinz means are defined as:

(1) for positive numbers a and b. For , the inequality (1) is equal to the arithmetic mean, and for , that inequality (1) is equal to the geometric mean.

Moreover, one can show that:

The general matrix version of the above asserts were proved in Bhatia and Davis (1993). Let A, B and X are three operators on a complex separable Hilbert space such that A and B are positive. For every unitarily invariant norm , the function:

(2) is convex on the interval [0, 1], arrives its maximum at and , attains its minimum at and satisfies for . Thus, we have the following standard operator version Heinz inequalities:

(3)

It is easy to see that the old Heinz operator version inequality in Heinz (1951) is just the special case of the inequalities (3) whose norm is the operator bound norm.

For more refinements and applications on the inequalities (3), the reader can refer to Bhatia (2007), Hiai and Kosaki (1999), Hiai and Kosaki (2003), Kittaneh (2010), and Feng (2012) and references therein. Among these obtained results, the following known Hermit-Hadamard integral inequality for a convex function :

(4) which plays a very important role in the research. In fact, the inequality (4) was firstly discovered by Hermite in 1881 in the journal Mathesis (Mitrinovic & Lackovic, 1985). However, this beautiful result was nowhere mentioned in the mathematical literature and was not widely known as Hermite’s result (Pecarić, Proschan, & Tong, 1992). For more recent results which generalize, improve, and extend the inequality (4), one can see (Abramovich, Barić, & Pecarić, 2008; Cal, Carcamob, & Escauriaza, 2009; Demir, Avci, & Set, 2010; Dragomir, 2011; Dragomir, 2012; Sarikaya & Aktan, 2011; Xiao, Zhang, & Wu, 2011; Bessenyei, 2010; Tseng, Hwang, & Hsu, 2012; Niculescu, 2012) and references therein.

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