Jensen–Mercer and Hermite–Hadamard–Mercer Type Inequalities for GA-h-Convex Functions and Its Subclasses with Applications

• DOI: 10.3390/math11020278
• Publication Date: 2023-01-05T09:12:56Z
• Authors: Asfand Fahad; Ayesha; Yuanheng Wang; Saad Ihsaan Butt
• Keywords: convex functions; <i>h</i>-convex functions; GA-<i>h</i>-convex functions; Jensen–Mercer inequality; Hadamard fractional integral; QA1-939; convex functions; <i>h</i>-convex functions; GA-<i>h</i>-convex functions; Jensen–Mercer inequality; Hermite–Hadamard–Mercer type inequalities; Hadamard fractional integral; 0101 mathematics; Hermite–Hadamard–Mercer type inequalities; 01 natural sciences; Mathematics

Many researchers have been attracted to the study of convex analysis theory due to both facts, theoretical significance, and the applications in optimization, economics, and other fields, which has led to numerous improvements and extensions of the subject over the years. An essential part of the theory of mathematical inequalities is the convex function and its extensions. In the recent past, the study of Jensen–Mercer inequality and Hermite–Hadamard–Mercer type inequalities has remained a topic of interest in mathematical inequalities. In this paper, we study several inequalities for GA-h-convex functions and its subclasses, including GA-convex functions, GA-s-convex functions, GA-Q-convex functions, and GA-P-convex functions. We prove the Jensen–Mercer inequality for GA-h-convex functions and give weighted Hermite–Hadamard inequalities by applying the newly established Jensen–Mercer inequality. We also establish inequalities of Hermite–Hadamard–Mercer type. Thus, we give new insights and variants of Jensen–Mercer and related inequalities for GA-h-convex functions. Furthermore, we apply our main results along with Hadamard fractional integrals to prove weighted Hermite–Hadamard–Mercer inequalities for GA-h-convex functions and its subclasses. As special cases of the proven results, we capture several well-known results from the relevant literature.