Some Integral Inequalities for Generalized Convex Fuzzy-Interval-Valued Functions via Fuzzy Riemann Integrals
Muhammad Bilal Khan, Muhammad Aslam Noor, Pshtiwan Othman Mohammed, Juan L. G. Guirao, Khalida Inayat Noor
Abstract
Abstract In this study, we introduce the new concept of $$h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -convex fuzzy-interval-valued functions. Under the new concept, we present new versions of Hermite–Hadamard inequalities (H–H inequalities) are called fuzzy-interval Hermite–Hadamard type inequalities for $$h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -convex fuzzy-interval-valued functions ( $$h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -convex FIVF) by means of fuzzy order relation. This fuzzy order relation is defined level wise through Kulisch–Miranker order relation defined on fuzzy-interval space. Fuzzy order relation and inclusion relation are two different concepts. With the help of fuzzy order relation, we also present some H–H type inequalities for the product of $$h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -convex FIVFs. Moreover, we have also established strong relationship between Hermite–Hadamard–Fej´er (H–H–Fej´er) type inequality and $$h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -convex FIVF. There are also some special cases presented that can be considered applications. There are useful examples provided to demonstrate the applicability of the concepts proposed in this study. This paper's thoughts and methodologies could serve as a springboard for more research in this field.