[Paper Review] Some new inequalities involving Heinz operator means

A. G. Ghazanfari, Somayeh Malekinejad+1 more|arXiv (Cornell University)|Mar 7, 2017

Mathematical Inequalities and Applications8 references3 citations

TL;DR

This paper presents new refinements of Heinz and Young's inequalities for scalars, operators, and Hilbert-Schmidt norm of matrices, introducing improved reverse inequalities with explicit constants. It establishes operator and matrix versions of existing inequalities, unifying and simplifying results from Kittaneh-Mansarah and Kai, and derives sharp bounds using spectral theory and norm inequalities, particularly for positive definite matrices and the Hilbert-Schmidt norm.

ABSTRACT

We give some new refinements of Heinz inequality and an improvement of the reverse Young's inequality for scalars and we use them to establish new inequalities for operators and the Hilbert-Schmidt norm of matrices. We give a uniformly and abbreviated form of the inequalities presented by Kittaneh and Mansarah, and the inequalities presented by Kai and we obtain some of their operator and matrix versions.

Motivation & Objective

• To derive new scalar refinements of Heinz and reverse Young's inequalities with improved constants.
• To extend these scalar inequalities to operator inequalities on positive definite matrices.
• To establish matrix versions of inequalities from Kittaneh-Mansarah and Kai using a unified and abbreviated formulation.
• To analyze the Hilbert-Schmidt norm of matrix expressions involving Heinz means and operator convex combinations.
• To provide sharp bounds for the norm of $ A^\nu X B^{1-\nu} + A^{1-\nu} X B^\nu $ using spectral decomposition and norm inequalities.

Proposed method

• Derive a new reverse Young-type inequality: $ (1-\nu^2+\nu^3)a + (1-\nu^2)b \leq \nu^{\nu-2}a^\nu b^{1-\nu} + (\sqrt{a}-\sqrt{b})^2 $, proven via completing the square and non-negativity of squared terms.
• Apply the spectral theorem to diagonalize positive definite matrices $ A $ and $ B $, reducing operator inequalities to scalar inequalities on eigenvalues.
• Use the Hilbert-Schmidt norm $ \|\cdot\|_2 $ to translate scalar inequalities into matrix norm inequalities via summation over matrix entries.
• Introduce a parameter $ \alpha = \min\{\|A\|^{-1}, \|B\|^{-1}\} $ to refine the matrix Heinz inequality with a correction term involving $ A^2X + XB^2 - 2AXB $.
• Employ the triangle inequality and norm subadditivity to bound the norm of linear combinations of matrix terms, particularly $ \|A^\nu X B^{1-\nu} + A^{1-\nu} X B^\nu + \cdots\|_2 \leq \|AX + XB\|_2 $.
• Use the identity $ \|Y\|_2^2 = \sum_{i,j} |y_{ij}|^2 $ to express matrix norms in terms of eigenvalues and matrix entries, enabling term-by-term comparison.

Experimental results

Research questions

• RQ1 Can tighter reverse inequalities be established for Heinz and Young means using scalar refinements with explicit constants?
• RQ2 How can scalar refinements of Heinz and reverse Young’s inequalities be extended to operator and matrix settings?
• RQ3 Can existing inequalities by Kittaneh-Mansarah and Kai be unified and simplified into a single concise form?
• RQ4 What is the role of the Hilbert-Schmidt norm in deriving sharp bounds for matrix Heinz-type expressions?
• RQ5 How do spectral decomposition and norm inequalities interact to yield new matrix norm inequalities?

Key findings

• A new reverse Young-type inequality is proven: $ (1-\nu^2+\nu^3)a + (1-\nu^2)b \leq \nu^{\nu-2}a^\nu b^{1-\nu} + (\sqrt{a}-\sqrt{b})^2 $, with equality when $ a = b $.
• The matrix inequality $ \|\nu^{\nu-2}(A^\nu X B^{1-\nu} + A^{1-\nu} X B^\nu) + (A^{1/2} X B^{1/2})\|_2 \geq \|\nu^2(\nu-2)(AX + XB)\|_2 $ holds for positive definite matrices.
• A refined matrix Heinz inequality is established: $ \|A^\nu X B^{1-\nu} + A^{1-\nu} X B^\nu + \nu(1-\nu)\alpha(A^2X + XB^2 - 2AXB)\|_2 \leq \|AX + XB\|_2 $, where $ \alpha = \min\{\|A\|^{-1}, \|B\|^{-1}\} $.
• The inequality $ \|A^\nu X B^{1-\nu} + A^{1-\nu} X B^\nu\|_2 \leq \|A^\nu X B^{1-\nu} + A^{1-\nu} X B^\nu + \nu(1-\nu)\alpha(A^2X + XB^2 - 2AXB)\|_2 \leq \|AX + XB\|_2 $ is proven, showing the correction term improves the bound.
• A unified form of inequalities from Kittaneh-Mansarah and Kai is achieved, simplifying and generalizing their results into a single concise framework.
• For $ r = \min\{\nu, 1-\nu\} $, the inequality $ r^{2r}\|A^\nu X B^{1-\nu} + A^{1-\nu} X B^\nu + (2r-1)(AX + XB)\|_2 \leq 2r^2\|A^{1/2} X B^{1/2}\|_2 \leq R^{2R}\|A^\nu X B^{1-\nu} + A^{1-\nu} X B^\nu + (2R-1)(AX + XB)\|_2 $ is established, with $ R = \max\{\nu, 1-\nu\} $.