In this paper, we present a novel estimate of Fejèr-type inequalities for a function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2130_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>ψ</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>ϱ</mi> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </math></EquationSource> <EquationSource Format="TEX">$\psi : [\varrho , \varepsilon ] \to \mathbb{R}$</EquationSource> </InlineEquation>, where ψ is continuous on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2130_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">[</mo> <mi>ϱ</mi> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">]</mo> </math></EquationSource> <EquationSource Format="TEX">$[\varrho , \varepsilon ]$</EquationSource> </InlineEquation> and twice differentiable on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2130_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>ϱ</mi> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\varrho , \varepsilon )$</EquationSource> </InlineEquation>, without requiring the convexity of ψ on the interval <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2130_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>ϱ</mi> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\varrho , \varepsilon )$</EquationSource> </InlineEquation>.

New bounds and generalizations of Hermite-Hadamard-Fejèr inequalities via second derivatives

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2025年9月30日

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Hasan Öğünmez

摘要

In this paper, we present a novel estimate of Fejèr-type inequalities for a function $ψ : [ϱ, ε] \to R$ $\psi : [\varrho , \varepsilon ] \to \mathbb{R}$ , where ψ is continuous on $[ϱ, ε]$ $[\varrho , \varepsilon ]$ and twice differentiable on $(ϱ, ε)$ $(\varrho , \varepsilon )$ , without requiring the convexity of ψ on the interval $(ϱ, ε)$ $(\varrho , \varepsilon )$ .