Some Inequalities for LR-$$\left({h}_{1}, {h}_{2}\right)$$-Convex Interval-Valued Functions by Means of Pseudo Order Relation

Abstract

Abstract In both theoretical and applied mathematics fields, integral inequalities play a critical role. Due to the behavior of the definition of convexity, both concepts convexity and integral inequality depend on each other. Therefore, the relationship between convexity and symmetry is strong. Whichever one we work on, we introduced the new class of generalized convex function is known as LR- $$\left({h}_{1}, {h}_{2}\right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfenced><mml:msub><mml:mi>h</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mfenced></mml:math> -convex interval-valued function (LR- $$\left({h}_{1}, {h}_{2}\right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfenced><mml:msub><mml:mi>h</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mfenced></mml:math> -IVF) by means of pseudo order relation. Then, we established its strong relationship between Hermite–Hadamard inequality ( HH -inequality)) and their variant forms. Besides, we derive the Hermite–Hadamard–Fejér inequality ( HH –Fejér inequality)) for LR- $$\left({h}_{1}, {h}_{2}\right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfenced><mml:msub><mml:mi>h</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mfenced></mml:math> -convex interval-valued functions. Several exceptional cases are also obtained which can be viewed as its applications of this new concept of convexity. Useful examples are given that verify the validity of the theory established in this research. This paper’s concepts and techniques may be the starting point for further research in this field.

Topics & Concepts