Development of Fractional Newton-Type Inequalities Through Extended Integral Operators

Hyder A., Almoneef A. A., Barakat M. A., BUDAK H., Aktaş Ö.

Fractal and Fractional, vol.9, no.7, 2025 (SCI-Expanded, Scopus) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 9 Issue: 7
  • Publication Date: 2025
  • Doi Number: 10.3390/fractalfract9070443
  • Journal Name: Fractal and Fractional
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, INSPEC, Directory of Open Access Journals
  • Keywords: generalized fractional operators, Newton-type inequalities, well-behaved functions
  • Kocaeli University Affiliated: Yes

Abstract

This paper introduces a new class of Newton-type inequalities (NTIs) within the framework of extended fractional integral operators. This study begins by establishing a fundamental identity for generalized fractional Riemann–Liouville (FR-L) operators, which forms the basis for deriving various inequalities under different assumptions on the integrand. In particular, fractional counterparts of the classical (Formula presented.) and (Formula presented.) Simpson rules are obtained when the modulus of the first derivative is convex. The analysis is further extended to include functions that satisfy a Lipschitz condition or have bounded first derivatives. Moreover, an additional NTI is presented for functions of bounded variation, expressed in terms of their total variation. In all scenarios, the proposed results reduce to classical inequalities when the fractional parameters are specified accordingly, thus offering a unified perspective on numerical integration through fractional operators.