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<p>In this paper, we develop a new class of refined inequalities for log convex functions, motivated by and extending the classical Young-type inequality. Our method generalizes the recent refinements established by Hu, as well as their subsequent extensions by Ighachane <i>et al.</i>, through a piecewise refinement technique. The resulting inequalities provide sharp multiple-term improvements for log-convex functions on <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </math></EquationSource> <EquationSource Format="TEX">$[0,1]$</EquationSource> </InlineEquation> and further yield two-weight versions that explicitly capture the dependence on pairs of points <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mn>0</mn> <mo>&lt;</mo> <mi>x</mi> <mo>≤</mo> <mi>y</mi> <mo>&lt;</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$0 &lt; x \le y &lt; 1$</EquationSource> </InlineEquation>.</p><p>We further extend our results to general intervals and to weighted power means, obtaining new refined and reverse estimates for the <i>p</i>-power mean interpolation for <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>p</mi> <mo>≤</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$p\le 0$</EquationSource> </InlineEquation>. As applications, we derive strengthened Young-type inequalities for unitarily invariant norms, and related results for positive definite matrices. Finally, by exploiting the log-convexity of numerical radius mappings under unitarily invariant norms, we obtain refined Young-type bounds for the numerical radius and its generalized forms. Our results unify and significantly sharpen several known inequalities in convexity theory, operator means, and matrix analysis.</p>

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A new class of log-convexity inequalities and applications

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月2日

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  • Yonghui Ren,
  • Mohamed Amine Ighachane,
  • Duong Quoc Huy,
  • Otmane Benchiheb

摘要

In this paper, we develop a new class of refined inequalities for log convex functions, motivated by and extending the classical Young-type inequality. Our method generalizes the recent refinements established by Hu, as well as their subsequent extensions by Ighachane et al., through a piecewise refinement technique. The resulting inequalities provide sharp multiple-term improvements for log-convex functions on [ 0 , 1 ] $[0,1]$ and further yield two-weight versions that explicitly capture the dependence on pairs of points 0 < x ≤ y < 1 $0 < x \le y < 1$ .

We further extend our results to general intervals and to weighted power means, obtaining new refined and reverse estimates for the p-power mean interpolation for p ≤ 0 $p\le 0$ . As applications, we derive strengthened Young-type inequalities for unitarily invariant norms, and related results for positive definite matrices. Finally, by exploiting the log-convexity of numerical radius mappings under unitarily invariant norms, we obtain refined Young-type bounds for the numerical radius and its generalized forms. Our results unify and significantly sharpen several known inequalities in convexity theory, operator means, and matrix analysis.