The notions of convex mappings and inequalities, which form a strong link and are key parts of classical analysis, have gotten a lot of attention recently. As a familiar extension of the classical one, interval-valued analysis is frequently used in the research of control theory, mathematical economy and so on. Motivated by the importance of convexity and inequality, our aim is to consider a new class of convex interval-valued mappings (I-V⋅Ms) known as left and right (L-R) $ \mathfrak{J} $-convex interval-valued mappings through pseudo-order relation ($ {\le }_{p} $) or partial order relation, because in interval space, both concepts coincide, so this order relation is defined in interval space. By using this concept, first we obtain Hermite-Hadamard (HH-) and Hermite-Hadamard-Fejér (HH-Fejér) type inequalities through pseudo-order relations via the Riemann-Liouville fractional integral operator. Moreover, we have shown that our results include a wide class of new and known inequalities for L-R $ \mathfrak{J} $-convex- I-V⋅Ms and their variant forms as special cases. Under some mild restrictions, we have proved that the inclusion relation "$ \subseteq $" is coincident to pseudo-order relation "$ {\le }_{p} $" when the I-V⋅M is L-R $ \mathfrak{J} $-convex or L-R $ \mathfrak{J} $-concave. Results obtained in this paper can be viewed as an improvement and refinement of classical known results.
Citation: Muhammad Bilal Khan, Savin Treanțǎ, Hleil Alrweili, Tareq Saeed, Mohamed S. Soliman. Some new Riemann-Liouville fractional integral inequalities for interval-valued mappings[J]. AIMS Mathematics, 2022, 7(8): 15659-15679. doi: 10.3934/math.2022857
The notions of convex mappings and inequalities, which form a strong link and are key parts of classical analysis, have gotten a lot of attention recently. As a familiar extension of the classical one, interval-valued analysis is frequently used in the research of control theory, mathematical economy and so on. Motivated by the importance of convexity and inequality, our aim is to consider a new class of convex interval-valued mappings (I-V⋅Ms) known as left and right (L-R) $ \mathfrak{J} $-convex interval-valued mappings through pseudo-order relation ($ {\le }_{p} $) or partial order relation, because in interval space, both concepts coincide, so this order relation is defined in interval space. By using this concept, first we obtain Hermite-Hadamard (HH-) and Hermite-Hadamard-Fejér (HH-Fejér) type inequalities through pseudo-order relations via the Riemann-Liouville fractional integral operator. Moreover, we have shown that our results include a wide class of new and known inequalities for L-R $ \mathfrak{J} $-convex- I-V⋅Ms and their variant forms as special cases. Under some mild restrictions, we have proved that the inclusion relation "$ \subseteq $" is coincident to pseudo-order relation "$ {\le }_{p} $" when the I-V⋅M is L-R $ \mathfrak{J} $-convex or L-R $ \mathfrak{J} $-concave. Results obtained in this paper can be viewed as an improvement and refinement of classical known results.
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Muhammad Bilal Khan, Savin Treanțǎ, Hleil Alrweili, Tareq Saeed, Mohamed S. Soliman. Some new Riemann-Liouville fractional integral inequalities for interval-valued mappings[J]. AIMS Mathematics, 2022, 7(8): 15659-15679. doi: 10.3934/math.2022857
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