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<p>This paper presents a unified framework for extending Jensen- and Mercer-type inequalities within the <i>h</i>-convex function space. By leveraging the supermultiplicative and superadditive properties of the weight function <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$h(t)$</EquationSource> </InlineEquation>, we refine classical inequalities like Jensen’s, Hölder’s, and Minkowski. The study introduces new characterizations, <i>h</i>-chord criteria, and partition smoothing techniques to provide sharper estimates for Jensen-Mercer gaps. These refinements have applications in optimization and information theory, offering improved bounds for classical mean inequalities and log-sum relations.</p>

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Jensen– and Mercer–type inequalities for h–convex functions: characterizations, refinements, and applications

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月16日

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  • Sajid Ali,
  • Rabia Bibi

摘要

This paper presents a unified framework for extending Jensen- and Mercer-type inequalities within the h-convex function space. By leveraging the supermultiplicative and superadditive properties of the weight function h ( t ) $h(t)$ , we refine classical inequalities like Jensen’s, Hölder’s, and Minkowski. The study introduces new characterizations, h-chord criteria, and partition smoothing techniques to provide sharper estimates for Jensen-Mercer gaps. These refinements have applications in optimization and information theory, offering improved bounds for classical mean inequalities and log-sum relations.