Abstract
We establish certain new fractional integral inequalities involving the Raina function for monotonicity of functions that are used with some traditional and forthright inequalities. Taking into consideration the generalized fractional integral with respect to a monotone function, we derive the Grüss and certain other associated variants by using well-known integral inequalities such as Young, Lah–Ribarič, and Jensen integral inequalities. In the concluding section, we present several special cases of fractional integral inequalities involving generalized Riemann–Liouville, k-fractional, Hadamard fractional, Katugampola fractional, (k,s)-fractional, and Riemann–Liouville-type fractional integral operators. Moreover, we also propose their pertinence with other related known outcomes.
Highlights
Introduction and preliminariesThe fractional calculus has gained importance during recent years because of its applications in science and engineering
Fractional-order differential equations are widely used in the model problems of nanoscale flow and heat transfer, diffusion, polymer physics, chemical physics, biophysics, medical sciences, turbulence, electric networks, electrochemistry of corrosion, and fluid flow through porous media [1,2,3,4,5]
Fractional integral inequalities associating functions of two or more independent variables play a crucial role in the continuous growth of the theory, methods, and applications of differential and integral equations
Summary
It is well known that the Grüss-type inequalities in both continuous and discrete cases play a significant role in investigating the qualitative conduct of differential and difference equations, respectively, as well as several other fields of pure and applied analysis. It is necessary to propose the investigation of the generalized fractional integrals Throughout this investigation, we use the following suppositions: (i) θ : [0, ∞) → (0, ∞) is an increasing function with continuous derivative θ on the interval (0, ∞). × Fρσ,,λk ω θ (x) – θ (η) ρ Fρσ,,δk ω θ (x) – θ (η) ρ Q1(t)Q2(η) Integrating this inequality with respect to t and η over (0, x) gives α–1Aδ(x)Jρσ,λ,k,0,θ+;ωQα1 (x) + β–1Aλ(x)Jρσ,δ,k,0,θ+;ωQβ2 (x) ≥ Jρσ,λ,k,0,θ+;ωQ1(x)Jρσ,δ,k,0,θ+;ωQ2(x), which implies part (a). Theorem 3.3 Let ρ, λ > 0, ω ∈ R, Q1, Q2 ∈ L1,r[υ1, υ2], and let γ , Υ , θ , and be four integrable functions defined on [0, ∞) such that 0 < γ (x) ≤ Q1(x) ≤ Υ(x), 0 < θ (x) ≤ Q2(x) ≤ (x) Proof It follows from inequality (3.7) that γ ≤ Q1(t) ≤ Υ.
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