Properties of GA-h-Convex Functions in Connection to the Hermite–Hadamard–Fejér-Type Inequalities
Abstract
:1. Introduction
2. The Hermite–Hadamard–Fejér Inequalities for a --Convex Function
- (i)
- The function φ is convex, that is, if , then
- (ii)
- The function φ is s-convex, that is, if , , then
- (i)
- If h is multiplicative or
- (ii)
- If h is supermultiplicative and φ is non-negativeand if φ is a -h-convex function, then inequality (15) holds for
- (a)
- If φ is -convex, i.e., , then inequality (15) holds for .Furthermore, if , for and if h is multiplicative or if h is supermultiplicative and φ is non-negative, then inequality (15) holds for
- (b)
- If φ is -s-convex, i.e., , then inequality (15) holds for .Furthermore, if , for and if h is multiplicative or if h is supermultiplicative and φ is non-negative, then inequality (15) holds for
3. Mappings Connected with the Hermite–Hadamard-Type Inequalities for -Convex Functions
4. Applications to Special Means
- (i)
- The function is -convex if
- (a)
- and ;
- (b)
- and .
- (ii)
- The function is -concave if
- (a)
- and ;
- (b)
- and .
- (i)
- The function is --convex if
- (a)
- and ;
- (b)
- and .
- (ii)
- The function is --concave if
- (a)
- and ;
- (b)
- and .
5. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Amer Latif, M. Properties of GA-h-Convex Functions in Connection to the Hermite–Hadamard–Fejér-Type Inequalities. Mathematics 2023, 11, 3172. https://doi.org/10.3390/math11143172
Amer Latif M. Properties of GA-h-Convex Functions in Connection to the Hermite–Hadamard–Fejér-Type Inequalities. Mathematics. 2023; 11(14):3172. https://doi.org/10.3390/math11143172
Chicago/Turabian StyleAmer Latif, Muhammad. 2023. "Properties of GA-h-Convex Functions in Connection to the Hermite–Hadamard–Fejér-Type Inequalities" Mathematics 11, no. 14: 3172. https://doi.org/10.3390/math11143172
APA StyleAmer Latif, M. (2023). Properties of GA-h-Convex Functions in Connection to the Hermite–Hadamard–Fejér-Type Inequalities. Mathematics, 11(14), 3172. https://doi.org/10.3390/math11143172