On the weighted fractional integral inequalities for Chebyshev functionals

Abstract

The goal of this present paper is to study some new inequalities for a class of differentiable functions connected with Chebyshev’s functionals by utilizing a fractional generalized weighted fractional integral involving another function $\mathcal{G}$ G in the kernel. Also, we present weighted fractional integral inequalities for the weighted and extended Chebyshev’s functionals. One can easily investigate some new inequalities involving all other type weighted fractional integrals associated with Chebyshev’s functionals with certain choices of $\omega (\theta )$ ω ( θ ) and $\mathcal{G}(\theta )$ G ( θ ) as discussed in the literature. Furthermore, the obtained weighted fractional integral inequalities will cover the inequalities for all other type fractional integrals such as Katugampola fractional integrals, generalized Riemann–Liouville fractional integrals, conformable fractional integrals and Hadamard fractional integrals associated with Chebyshev’s functionals with certain choices of $\omega (\theta )$ ω ( θ ) and $\mathcal{G}(\theta )$ G ( θ ) .

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“…Let us begin with the definition of fractional Lee-Wick Lagrangian density and then utilize the generalization formula of Euler -Lagrange equation (5) to derive the equations of motion from Lee-Wick Lagrangian density. Take the first field variable 𝜙, then: This represents the first homogeneous equation in fractional form.…”

“…Fractional derivatives have played significant roles in physics, engineering, and applied mathematics [4][5][6][7][8][9][10]. Hilfer fractional integro-differential equations were proposed due to the presence of nonlocal conditions [11].…”

Some New Beesack–Wirtinger-Type Inequalities Pertaining to Different Kinds of Convex Functions

“…Khan et al in [12] investigated the Hirota equation using the modified double Laplace decomposition method. Rahman et al in [13] obtained the weighted fractional integral inequalities for Chebyshev functionals. Khan et al in [14] established applications of the fixed-point theory to investigate a system of factional-order differential equations.…”

Certain Inequalities Pertaining to Some New Generalized Fractional Integral Operators

“…This has been intensively studied, with many book chapters and important research articles dedicated to the Chebyshev type inequalities, see [23][24][25][26][27][28]. We will develop in Section 4, some new results and basic examples as well using the same ideas as in recently published papers about certain generalized proportional fractional integrals from Rahman et al (see [29][30][31][32][33][34][35]) in the framework of the new class of generalized fractional integral operators which will be defined at the end of Section 2.…”