Some New Fractional Estimates of Inequalities for LR-<inline-formula><math display="inline"><semantics><mrow><mi>p</mi></mrow></semantics></math></inline-formula>-Convex Interval-Valued Functions by Means of Pseudo Order Relation

• Main Authors: Muhammad Bilal Khan, Pshtiwan Othman Mohammed, Muhammad Aslam Noor, Dumitru Baleanu, Juan Luis García Guirao
• Format: Article
• Language: English
• Published: MDPI AG 2021-07-01
• Series: Axioms
• Subjects: LR-<i>p</i>-convex interval-valued function; Katugampola fractional integral operator; Hermite-Hadamard type inequality; Hermite-Hadamard-Fejér inequality
• Online Access: https://www.mdpi.com/2075-1680/10/3/175

It is a familiar fact that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis, both the inclusion relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mo>⊆</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and pseudo order relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><msub><mo>≤</mo><mi>p</mi></msub></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are two different concepts. In this article, by using pseudo order relation, we introduce the new class of nonconvex functions known as LR-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>p</mi></semantics></math></inline-formula>-convex interval-valued functions (LR-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>p</mi></semantics></math></inline-formula>-convex-IVFs). With the help of this relation, we establish a strong relationship between LR-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>p</mi></semantics></math></inline-formula>-convex-IVFs and Hermite-Hadamard type inequalities (<i>HH</i>-type inequalities) via Katugampola fractional integral operator. Moreover, we have shown that our results include a wide class of new and known inequalities for LR-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>p</mi></semantics></math></inline-formula>-convex-IVFs and their variant forms as special cases. Useful examples that demonstrate the applicability of the theory proposed in this study are given. The concepts and techniques of this paper may be a starting point for further research in this area.