Some trapezoid and midpoint type inequalities via fractional ( p , q ) $(p,q)$ -calculus

Abstract

Abstract Fractional calculus is the field of mathematical analysis that investigates and applies integrals and derivatives of arbitrary order. Fractional q-calculus has been investigated and applied in a variety of research subjects including the fractional q-trapezoid and q-midpoint type inequalities. Fractional ( p , q ) $(p,q)$ -calculus on finite intervals, particularly the fractional ( p , q ) $(p,q)$ -integral inequalities, has been studied. In this paper, we study two identities for continuous functions in the form of fractional ( p , q ) $(p,q)$ -integral on finite intervals. Then, the obtained results are used to derive some fractional ( p , q ) $(p,q)$ -trapezoid and ( p , q ) $(p,q)$ -midpoint type inequalities.

Keywords

• Trapezoid type inequalities
• Midpoint type inequalities
• Quantum calculus
• q-shifting operator
• ( p , q ) $(p,q)$ -calculus
• Fractional ( p , q ) $(p,q)$ -integral