Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals

Abstract

In this paper, Jensen and Hardy inequalities, including Pólya–Knopp type inequalities for superquadratic functions, are extended using Riemann–Liouville delta fractional integrals. Furthermore, some inequalities are proved by using special kernels. Particular cases of obtained inequalities give us the results on time scales calculus, fractional calculus, discrete fractional calculus, and quantum fractional calculus.

Further Generalizations of Some Fractional Integral Inequalities

“…Adopting the same pattern of simplification as we did for ( 10) and ( 14), the following inequality can be observed for (11) and (14):…”

“…In recent past literature, several integral inequalities can be found for different kinds of fractional integral operators. For instance, in [6][7][8][9][10], Hadamard-like inequalities were studied, in [11], Ostrowski-like inequalities were studied, in [12,13], Chebyshev-like inequalities were studied, and Minkowksi-, Hardy-and Grüss-like inequalities were investigated in [14][15][16].…”

“…Under stated conditions, kernel (6) satisfies inequality (10). In addition, the (α, h − m)-p convex function satisfies inequality (14). Ultimately, one can have the following inequality:…”