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<p>In this paper, we introduce the concept of strongly multiplicative convex functions and establish Hermite–Hadamard (<i>HH</i>)-type integral inequalities using the Atangana–Baleanu (AB) fractional integral operator within the framework of multiplicative calculus. We further derive HH-type integral inequalities for the product and quotient of strongly multiplicative convex and strongly multiplicative concave functions using the same operator. By setting <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2061_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>α</mi> <mo>=</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$\alpha =1$</EquationSource> </InlineEquation>, we obtain simplified versions of these inequalities for the product and quotient of such functions, which represent novel results not previously explored. The validity of the proposed theorems is demonstrated graphically through well-chosen examples. Additionally, we present a proof of Milne’s inequality utilizing the same operator within the context of multiplicative calculus. To further enhance the study, we explore applications of these functions in relation to special functions and derive new fractional recurrence relations, adding depth and significance to the findings.</p>

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Fractional integral inequalities for strongly convex functions via multiplicative calculus with applications

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月25日

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  • Saad Ihsan Butt,
  • Dawood Khan,
  • Valer-Daniel Breaz,
  • Luminita-Ioana Cotîrlǎ,
  • Bandar Bin Mohsin

摘要

In this paper, we introduce the concept of strongly multiplicative convex functions and establish Hermite–Hadamard (HH)-type integral inequalities using the Atangana–Baleanu (AB) fractional integral operator within the framework of multiplicative calculus. We further derive HH-type integral inequalities for the product and quotient of strongly multiplicative convex and strongly multiplicative concave functions using the same operator. By setting α = 1 $\alpha =1$ , we obtain simplified versions of these inequalities for the product and quotient of such functions, which represent novel results not previously explored. The validity of the proposed theorems is demonstrated graphically through well-chosen examples. Additionally, we present a proof of Milne’s inequality utilizing the same operator within the context of multiplicative calculus. To further enhance the study, we explore applications of these functions in relation to special functions and derive new fractional recurrence relations, adding depth and significance to the findings.