In this paper, we establish new refined versions of the Heinz inequality and sharper improvements of the classical Young inequality for positive real numbers. Our results provide tighter bounds that enhance several known inequalities in the literature. Furthermore, we extend these refinements to the matrix setting, deriving new matrix inequalities and improving a number of previously established results. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\left\Vert \cdot \right\Vert _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mfenced close="∥" open="∥"> <mo>·</mo> </mfenced> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> denote the Hilbert–Schmidt norm of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> complex matrices. It is proved that <Equation ID="Equ21"> <EquationSource Format="TEX">$\begin{aligned} \left\Vert A^v-B^v\right\Vert _2^2\le \left\Vert A^v\right\Vert _2^2-\left\Vert B^v\right\Vert _2^2, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msubsup> <mfenced close="∥" open="∥"> <msup> <mi>A</mi> <mi>v</mi> </msup> <mo>-</mo> <msup> <mi>B</mi> <mi>v</mi> </msup> </mfenced> <mn>2</mn> <mn>2</mn> </msubsup> <mo>≤</mo> <msubsup> <mfenced close="∥" open="∥"> <msup> <mi>A</mi> <mi>v</mi> </msup> </mfenced> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mfenced close="∥" open="∥"> <msup> <mi>B</mi> <mi>v</mi> </msup> </mfenced> <mn>2</mn> <mn>2</mn> </msubsup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where A, B are positive semidefinite matrices such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A-B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>-</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> is also positive semidefinite and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$0\le v\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>v</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The above inequality improve the following inequality <Equation ID="Equ22"> <EquationSource Format="TEX">$\begin{aligned} \left\Vert A-B\right\Vert _2^2\le \left\Vert A\right\Vert _2^2+\left\Vert B\right\Vert _2^2, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msubsup> <mfenced close="∥" open="∥"> <mi>A</mi> <mo>-</mo> <mi>B</mi> </mfenced> <mn>2</mn> <mn>2</mn> </msubsup> <mo>≤</mo> <msubsup> <mfenced close="∥" open="∥"> <mi>A</mi> </mfenced> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mfenced close="∥" open="∥"> <mi>B</mi> </mfenced> <mn>2</mn> <mn>2</mn> </msubsup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>which has been proved in [<CitationRef CitationID="CR20">20</CitationRef>].

Refinements of Young and Heinz inequalities with applications to matrix inequalities

期刊论文
发表日期：

2026年7月28日

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Bashar Mayyas,
Jorge Losada,
Aliaa Burqan,
Ahmad Al-Natoor

摘要

In this paper, we establish new refined versions of the Heinz inequality and sharper improvements of the classical Young inequality for positive real numbers. Our results provide tighter bounds that enhance several known inequalities in the literature. Furthermore, we extend these refinements to the matrix setting, deriving new matrix inequalities and improving a number of previously established results. Let $\left\Vert \cdot \right\Vert _2$ ${∥\cdot∥}_{2}$ denote the Hilbert–Schmidt norm of $n\times n$ $n \times n$ complex matrices. It is proved that $\begin{aligned} \left\Vert A^v-B^v\right\Vert _2^2\le \left\Vert A^v\right\Vert _2^2-\left\Vert B^v\right\Vert _2^2, \end{aligned}$ $\begin{matrix} {∥A^{v} - B^{v}∥}_{2}^{2} \leq {∥A^{v}∥}_{2}^{2} - {∥B^{v}∥}_{2}^{2}, \end{matrix}$ where A, B are positive semidefinite matrices such that $A-B$ $A - B$ is also positive semidefinite and $0\le v\le 1$ $0 \leq v \leq 1$ . The above inequality improve the following inequality $\begin{aligned} \left\Vert A-B\right\Vert _2^2\le \left\Vert A\right\Vert _2^2+\left\Vert B\right\Vert _2^2, \end{aligned}$ $\begin{matrix} {∥A - B∥}_{2}^{2} \leq {∥A∥}_{2}^{2} + {∥B∥}_{2}^{2}, \end{matrix}$ which has been proved in [20].