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

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.

Some Generalizations of Different Types of Quantum Integral Inequalities for Differentiable Convex Functions with Applications

“…Proof. The inequality (17) for function F(t) = t k , λ = 0 and r = y 1 +y 2 2 leads to the required result. Proposition 10.…”

“…Brahim et al established some new version of quantum Hermite-Hadamard inequality in [15]. On the other hand, some papers were devoted to fractional post-quantum inequalities [16,17]. Some authors generalized the quantum Hermite-Hadamard inequalities for coordinated convex functions in [18][19][20].…”

“…Proof. The inequality (17) for function F(t) = t k , λ = 1 and r = y ≤ M(y 2 − y 1 )∆ 7 (λ; q, r) 1 r…”

Coefficient Inequalities of a Comprehensive Subclass of Analytic Functions With Respect to Symmetric Points

“…It should be noted that everything in classical calculus cannot be generalized to quantum calculus, notably the chain rule needs adaptation. So recently there is renewed interest in all the fields of research to replace the classical derivative with a quantum derivative, refer to [3,7,21,31,37,46,47] for recent developments involving q-calculus. The duality theory of Quantum Calculus and Univalent Function Theory was introduced by Srivastava [42].…”

Uniqueness of continuous solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order

“…Moreover, the definition of q-derivative and q-integral is studied and gradually developed by many researchers (see [11,23,24,28]). The definitions of q-derivative and q-integral are developed, which are based on the q-Riemann-Liouville fractional integral.…”