Document Zbl 1519.26010

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Univ. Mariae Curie-Skłodowska, Sect. A</a> <a href="/?q=in%3A496855" title="Articles in this Issue">76, No. 2, 15-32 (2022)</a>. </div> <div class="abstract">The authors give estimates for the modulus of the difference \[ D(f,a,b,\alpha):=\frac{1}{8}\big[3f((5a+b)/6)+2f((a+b)/2)+2f((a+3b)/6)\big]-\frac{\Gamma(\alpha+1)}{2(b-a)^\alpha}\big[J_{a+}^{\alpha} f(b)+J_{b-}^{\alpha} f(a)\big] \] where $f\in L_1[a,b]$ and \[ J_{a+}^{\alpha}f(x):=\frac{1}{\Gamma(\alpha)}\int_a^x (x-t)^{\alpha-1}f(t)dt,\,\,x>a \] \[ J_{b-}^{\alpha}f(x):=\frac{1}{\Gamma(\alpha)}\int_x^b (t-x)^{\alpha-1}f(t)dt,\,\,x<b \] are the Riemann-Liouville fractional integrals of order $\alpha>0$ with $a\ge 0$.<br class="zbmathjax-paragraph">Three theorems of similar nature are proved under various convexity conditions. A sample theorem is as follows.<br class="zbmathjax-paragraph">Suppose that $f:[a,b]\to\mathbb{R}$ is absolute continuous on $(a,b)$ such that $f'\in L_1[a,b]$ and $|f'|$ is convex on $[a,b].$ Then we have \[ |D(f,a,b,\alpha)|\le\frac{b-a}{2}\Omega(\alpha)[|f'(a)|+|f'(b)|] \] where the function $\Omega$ is given explicitly.<br class="zbmathjax-paragraph">Two examples show how the estimates work for the function $f(x)=x^2$.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10019">László Losonczi (Debrecen)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7690336">Cited in <strong>3</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A26D15" title="MSC2020">26D15</a> </td> <td class="space"> Inequalities for sums, series and integrals </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A26A33" title="MSC2020">26A33</a> </td> <td class="space"> Fractional derivatives and integrals </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65D32" title="MSC2020">65D32</a> </td> <td class="space"> Numerical quadrature and cubature formulas </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aconvex+function">convex function</a>; <a href="/?q=ut%3Afractional+calculus">fractional calculus</a>; <a href="/?q=ut%3Aquadrature+formulae">quadrature formulae</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1519.26010" data-ciurl="/ci/07690336" data-biburl="/bibtex/07690336.bib" data-amsurl="/amsrefs/07690336.bib" data-xmlurl="/xml/07690336.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07690336.pdf" title="Zbl 1519.26010 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.17951/a.2022.76.2.15-32" aria-label="DOI for “Maclaurin-type inequalities for Riemann-Liouville fractional integrals”" title="10.17951/a.2022.76.2.15-32">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Budak, H., Hezenci, F., Kara, H., On parametrized inequalities of Ostrowski and Simpson type for convex functions via generalized fractional integral, Math. 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