Mohammad Sababheh

Interpolated Young and Heinz inequalities

In this article, we interpolate the well-known Young’s inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. As an application of these interpolated versions, we study the monotonicity these interpolations obey.

REMOTALITY OF CLOSED BOUNDED CONVEX SETS IN REFLEXIVE SPACES

Let X be a Banach space and E be a closed bounded subset of X. For x ∈ X, we define D(x, E) = sup{‖ x − e‖:e ∈ E}. The set E is said to be remotal (in X) if, for every x ∈ X, there exists e ∈ E such that D(x, E) = ‖x − e‖. The object of this paper is to characterize those reflexive Banach spaces in which every closed bounded convex set is remotal. Such a result enabled us to produce a convex closed and bounded set in a uniformly convex Banach space that is not remotal. Further, we characterize Banach spaces in which every bounded closed set is remotal.

Means Refinements Via Convexity

The main goal of this article is to present multiple term refinements of the well-known matrix means’ inequalities. In particular, we present refinements of the matrix arithmetic-geometric, arithmetic-harmonic, geometric-harmonic, and Heinz means inequalities. These refinements will be presented in additive, squared, multiplicative, and reversed forms, generalizing and refining almost all known results in this direction. The main tool that we use to achieve our results is a general convexity argument, where refinements for convex functions happen to be the key behind all these refinements. Our first main result will be

Norm Inequalities Related to the Heron and Heinz Means

\begin{aligned} f(\nu )+\sum _{j=1}^{N}A_j(\nu )\Delta _jf(\nu ;0,1)\le (1-\nu )f(0)+\nu f(1), 0\le \nu \le 1 \end{aligned}

Heinz-type numerical radii inequalities

f(ν)+∑j=1NAj(ν)Δjf(ν;0,1)≤(1-ν)f(0)+νf(1),0≤ν≤1for the convex function f and certain positive summands. Then, variants of this inequality will be proved and means refinements will be obtained upon choosing certain convex functions.

New sharp inequalities for operator means

The main goal of this article is to present several quadratic refinements and reverses of the well known Heinz inequality, for numbers and matrices, where the refining term is a quadratic function in the mean parameters. The proposed idea introduces a new approach to these inequalities, where polynomial interpolation of the Heinz function plays a major role. As a consequence, we obtain a new proof of the celebrated Heron-Heinz inequality proved by Bhatia, then we study an optimization problem to find the best possible refinement. As applications, we present matrix versions including unitarily invariant norms, trace and determinant versions.

Exponential Inequalities for Positive Linear Mappings

In this article, we present several inequalities treating operator means and the Cauchy–Schwarz inequality. In particular, we present some new comparisons between operator Heron and Heinz means, several generalizations of the difference version of the Heinz means and further refinements of the Cauchy–Schwarz inequality. The techniques used to accomplish these results include convexity and Löwner matrices.

Graph indices via the AM–GM inequality

We prove that some inequalities, which are considered to be generalizations of Hardy’s inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line. Princess Sumaya University For Technology, Amman 11941-Jordan e-mail: sababheh@psut.edu.jo sababheh@yahoo.com Received by the editors April 16, 2008; revised December 29, 2008. Published electronically August 9, 2010. AMS subject classification: 42A05, 42A99.