A new formulation of Hermite–Hadamard–Fejer inequalities connected with an extended fractional operator

• Published in: Journal of inequalities and applications Vol. 2025; no. 1; pp. 92 - 21
• Main Authors: Younis, Muhammad, Liu, Zhi Guo, Samraiz, Muhammad, Mehmood, Ahsan
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2025; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Calculus; Control and Optimization in Sustainability Applications; Convex Function; Fejer; Fractional calculus; Hadamard; Hermite; Inequalities; Integral equations; Integrals; Mathematics; Mathematics and Statistics; Modified ( k , s ) $(k,s)$ -fractional operator; Operators (mathematics); Hermite; Fejer; Hadamard; Modified; Convex Function; fractional operator
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-025-03342-2

In this work, we classify the extensions of Hermite–Hadamard(H–H)–Fejer-type inequalities for the fractional operators involving nonlinear kernel. By utilizing these inequalities, we develop many kinds of fractional integral (FI) inequalities. By considering the limiting cases of our main results, we attain the inequalities that already exist in the literature. In our work, we calculate the bounds of well-known fractional problems involving extended fractional operators. As implementations of the proved results, we calculate the midpoint-type inequalities. In the last section as the application of our defined operator, we present a generalized Abel integral equation and compute its solution. Also, we define the nonlinear form of a weakly singular Volterra-type integral equation and investigate its solution. These results might be useful in the investigation of the uniqueness of mathematical models and applied problems.