Refined Error Estimates for Milne–Mercer-Type Inequalities for Three-Times-Differentiable Functions with Error Analysis and Their Applications

Munir A., Li S., BUDAK H., Kashuri A., Ciurdariu L.

Fractal and Fractional, cilt.9, sa.9, 2025 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 9 Sayı: 9
  • Basım Tarihi: 2025
  • Doi Numarası: 10.3390/fractalfract9090606
  • Dergi Adı: Fractal and Fractional
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, INSPEC, Directory of Open Access Journals
  • Anahtar Kelimeler: (α,m)-convex function, bounded function, Caputo–Fabrizio fractional operators, L-Lipschitzian function, Milne–Mercer-type inequalities, modified Bessel function, power mean inequality, q-digamma function, quadrature formula, s-convex function, special means
  • Kocaeli Üniversitesi Adresli: Evet

Özet

In this study, we examine the error bounds related to Milne-type inequalities and a widely recognized Newton–Cotes method, originally developed for three-times-differentiable convex functions within the context of Jensen–Mercer inequalities. Expanding on this foundation, we explore Milne–Mercer-type inequalities and their application to a more refined class of three-times-differentiable s-convex functions. This work introduces a new identity involving such functions and Jensen–Mercer inequalities, which is then used to improve the error bounds for Milne-type inequalities in both Jensen–Mercer and classical calculus frameworks. Our research highlights the importance of convexity principles and incorporates the power mean inequality to derive novel inequalities. Furthermore, we provide a new lemma using Caputo–Fabrizio fractional integral operators and apply it to derive several results of Milne–Mercer-type inequalities pertaining to (Formula presented.) -convex functions. Additionally, we extend our findings to various classes of functions, including bounded and Lipschitzian functions, and explore their applications to special means, the q-digamma function, the modified Bessel function, and quadrature formulas. We also provide clear mathematical examples to demonstrate the effectiveness of the newly derived bounds for Milne–Mercer-type inequalities.