Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings

: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings. some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. , Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Abstract: The principles of convexity and symmetry are inextricably linked. Because of the consid-erable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions ( IV-Fs ) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs ). Then, we obtain Hermite–Hadamard ( 𝓗 - 𝓗) and Hermite–Hadamard–Fejér ( 𝓗 - 𝓗 -Fejér ) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results.


Introduction
Hanson [1] defined the class of invex functions as one of the most significant extensions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex. Moreover, Işcan [4] also used fractional integrals for preinvex functions to obtain various -type inequalities. See [5][6][7][8] for other generalizations of the -inequality.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,

Introduction
Hanson [1] defined the class of invex functions as one of the most significant extensions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,

Introduction
Hanson [1] defined the class of invex functions as one of the sions of convex functions. Weir and Mond [2], in 1988, used the no tions to demonstrate adequate optimality criteria and duality in no For a differentiable mapping, the concept of fractional integral id mann-Liouville fractional and Hadamard fractional integrals integ Wang et al. [3], who identified some inequalities using standard convex, -convex, (s, m)-convex, and ( , )-convex.
Moreover, Iş tional integrals for preinvex functions to obtain various -type for other generalizations of the -inequality.
For accurate solutions to various problems in practical mathe interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitr lower limits. Moore [9] showed that, if a real-valued mapping Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , sion is a Lipschitz interval extension in . To combine the study o ous dynamical systems, Hilger [10] introduced a time scales theor of dynamic equations and integral inequalities on time scales, in electrical engineering, quantum physics, heat transfer, neural net and population dynamics [11], has highlighted the need for this th ity, Minkoswki's inequality, Jensen's inequality, Hölder's inequ

Introduction
Hanson [1] defined the class of invex functions as one of th sions of convex functions. Weir and Mond [2], in 1988, used the tions to demonstrate adequate optimality criteria and duality in For a differentiable mapping, the concept of fractional integral mann-Liouville fractional and Hadamard fractional integrals in Wang et al. [3], who identified some inequalities using standa convex, -convex, (s, m)-convex, and ( , )-convex.
Moreover tional integrals for preinvex functions to obtain various -ty for other generalizations of the -inequality.
For accurate solutions to various problems in practical mat interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arb lower limits. Moore [9] showed that, if a real-valued mapping Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ sion is a Lipschitz interval extension in . To combine the stud ous dynamical systems, Hilger [10] introduced a time scales the of dynamic equations and integral inequalities on time scales, electrical engineering, quantum physics, heat transfer, neural and population dynamics [11], has highlighted the need for this ity, Minkoswki's inequality, Jensen's inequality, Hölder's ine 1. Introduction Hanson [1] defined the class of invex functions as one of the most significant extensions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, r-convex, m-convex, S-convex, (s, m)-convex, and (β, m)-convex. Moreover, Işcan [4] [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Rie-

Introduction
Hanson [1] defined the class of invex functions as one of the most significant extensions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Rie-

Introduction
Hanson [1] defined the class of invex functions as one sions of convex functions. Weir and Mond [2], in 1988, used tions to demonstrate adequate optimality criteria and duali For a differentiable mapping, the concept of fractional inte mann-Liouville fractional and Hadamard fractional integra Wang et al. [3], who identified some inequalities using sta convex, -convex, (s, m)-convex, and ( , )-convex. More tional integrals for preinvex functions to obtain variousfor other generalizations of the -inequality.
For accurate solutions to various problems in practical interval arithmetic, IV-Fs, and integrals of IV-Fs to establis lower limits. Moore [9] showed that, if a real-valued mapp Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , sion is a Lipschitz interval extension in . To combine the ous dynamical systems, Hilger [10] introduced a time scale of dynamic equations and integral inequalities on time sca electrical engineering, quantum physics, heat transfer, neu and population dynamics [11], has highlighted the need for ity, Minkoswki's inequality, Jensen's inequality, Hölder's

Introduction
Hanson [1] defined the class of invex functions as o sions of convex functions. Weir and Mond [2], in 1988, u tions to demonstrate adequate optimality criteria and du For a differentiable mapping, the concept of fractional mann-Liouville fractional and Hadamard fractional integ Wang et al. [3], who identified some inequalities using convex, -convex, (s, m)-convex, and ( , )-convex. M tional integrals for preinvex functions to obtain various for other generalizations of the -inequality.
For accurate solutions to various problems in practi interval arithmetic, IV-Fs, and integrals of IV-Fs to estab lower limits. Moore [9] showed that, if a real-valued m Lipschitz condition in , | ( ) − ( )| ≤ | − |, for sion is a Lipschitz interval extension in . To combine t ous dynamical systems, Hilger [10] introduced a time sc of dynamic equations and integral inequalities on time electrical engineering, quantum physics, heat transfer, and population dynamics [11], has highlighted the need ity, Minkoswki's inequality, Jensen's inequality, Hölde inequality, Steffensen's inequality, Opial type inequality andČhebyšhev's inequality were all explored by Agarwal et al. [11]. Srivastava et al. [12] discovered some generic time scale weighted Opial type inequalities in 2010. Srivastava et al. [13] also proposed several time-based expansions and generalizations of Maroni's inequality. Under certain proper conditions, some new local fractional integral analogue of Anderson's inequality on fractal space was introduced by Wei et al. [14], demonstrating that for classical Anderson's inequality, it was a novel extension on fractal space. Tunç et al. [15] also constructed an identity for local fractional integrals and derived numerous modifications of the well-known Steffensen's inequality for fractional integrals. The papers [11,16] and the references therein might be consulted for further information. Bhurjee and Panda [17] identified the parametric form of an IV-F and devised a technique to investigate the existence of a generic interval optimization issue solution. Using the notion of the generalized Hukuhara difference, Lupulescu [18] developed differentiability and integrability for IV-Fs on time scales. Cano et al. [19] developed a novel form of the Ostrowski inequality for gH differentiable IV-Fs in 2015 and achieved an extension of the class of real functions that are not always differentiable. For gH-differentiable IV-Fs, Cano et al. [19] found error limitations to quadrature rules. In addition, Roy and Panda [20] developed the idea of the -monotonic property of IV-Fs in the higher dimension and used extended Hukuhara differentiability to obtain various conclusions. We refer to [21][22][23][24][25], and the references therein, for further information on IV-Fs. An et al. [26] and Zhao et al. [27] recently proposed an (h1, h2)-convex IV-F and harmonically h-convex IV-F, respectively. Moreover, they found certain interval

Introduction
Hanson [1] defined the class of invex sions of convex functions. Weir and Mond tions to demonstrate adequate optimality c For a differentiable mapping, the concept mann-Liouville fractional and Hadamard f Wang et al. [3], who identified some ineq convex, -convex, (s, m)-convex, and ( , tional integrals for preinvex functions to o for other generalizations of the -inequ For accurate solutions to various prob interval arithmetic, IV-Fs, and integrals of lower limits. Moore [9] showed that, if a r Lipschitz condition in , | ( ) − ( )| ≤ sion is a Lipschitz interval extension in . ous dynamical systems, Hilger [10] introdu of dynamic equations and integral inequa electrical engineering, quantum physics, h and population dynamics [11], has highligh ity, Minkoswki's inequality, Jensen's ineq

Introduction
Hanson [1] defined the class of inv sions of convex functions. Weir and Mo tions to demonstrate adequate optimalit For a differentiable mapping, the conce mann-Liouville fractional and Hadama Wang et al. [3], who identified some in convex, -convex, (s, m)-convex, and ( tional integrals for preinvex functions to for other generalizations of the -ine For accurate solutions to various p interval arithmetic, IV-Fs, and integrals lower limits. Moore [9] showed that, if Lipschitz condition in , | ( ) − ( ) sion is a Lipschitz interval extension in ous dynamical systems, Hilger [10] intr of dynamic equations and integral ineq electrical engineering, quantum physic and population dynamics [11], has high ity, Minkoswki's inequality, Jensen's i

Introduction
Hanson [1] defined the class of invex functions as one of the most sign sions of convex functions. Weir and Mond [2], in 1988, used the notion of p tions to demonstrate adequate optimality criteria and duality in nonlinear p For a differentiable mapping, the concept of fractional integral identities in mann-Liouville fractional and Hadamard fractional integrals integrals was c Wang et al. [3], who identified some inequalities using standard convex, convex, -convex, (s, m)-convex, and ( , )-convex.
Moreover, Işcan [4] a tional integrals for preinvex functions to obtain various -type inequali for other generalizations of the -inequality.
For accurate solutions to various problems in practical mathematics, M interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sha lower limits. Moore [9] showed that, if a real-valued mapping ( ) meet Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the sion is a Lipschitz interval extension in . To combine the study of discrete ous dynamical systems, Hilger [10] introduced a time scales theory. The wi of dynamic equations and integral inequalities on time scales, in domains electrical engineering, quantum physics, heat transfer, neural networks, c and population dynamics [11], has highlighted the need for this theory. You ity, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -    [2], in 1988, used the notion o tions to demonstrate adequate optimality criteria and duality in nonlinea For a differentiable mapping, the concept of fractional integral identitie mann-Liouville fractional and Hadamard fractional integrals integrals wa Wang et al. [3], who identified some inequalities using standard conve convex, -convex, (s, m)-convex, and ( , )-convex.
Moreover, Işcan [4 tional integrals for preinvex functions to obtain various -type inequ for other generalizations of the -inequality.
For accurate solutions to various problems in practical mathematics interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily s lower limits. Moore [9] showed that, if a real-valued mapping ( ) me Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, t sion is a Lipschitz interval extension in . To combine the study of discr ous dynamical systems, Hilger [10] introduced a time scales theory. The of dynamic equations and integral inequalities on time scales, in domai electrical engineering, quantum physics, heat transfer, neural networks and population dynamics [11], has highlighted the need for this theory. Y ity, Minkoswki's inequality, Jensen's inequality, Hölder's inequality,  inequality for a convex IV-F and its product. For more information related to generalized convex functions and fractional inequalities in interval-valued settings, see  and the references therein.
Inspired by the ongoing research, we introduce the concept of left and right preinvex IV-F and establish the

Introduction
Hanson [1] defined the class of invex functions as one of the most significant extensions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,   [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,   [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,

Introduction
Hanson [1] defined the class of invex functions as one of the most significant extensions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,  -Fejér inequality for left and right preinvex IV-Fs and the product of two left and right preinvex IV-Fs using Riemann integrals in interval-valued settings, which are motivated by the above studies and ideas. We also provide some examples to support our ideas.

Preliminaries
First, we offer some background information on interval-valued functions, the theory of convexity, interval-valued integration, and interval-valued fractional integration, which will be utilized throughout the article.
We offer some fundamental arithmetic regarding interval analysis in this paragraph, which will be quite useful throughout the article.
C be the set of all closed intervals of R, the set of all closed positive intervals of R and the set of all closed negative intervals of R. Then, K C , K + C , and K − C are defined as Remark 1. [36] The relation " ≤ p " defined on K C by for all [Q * , Q * ], [Z * , Z * ] ∈ K C , is a pseudo-order relation.
The collection of all Riemann integrable real valued functions and Riemann integrable IV-Fs is denoted by R [µ,υ] and IR [µ,υ] , respectively. Definition 1. A set K ⊂ R n is said to be a convex set, if, for all ω, κ ∈ K, t ∈ [0, 1], we have Definition 2. [36] Let K be a convex set. Then, IV-F Y : K → K + C is said to be left and right for all ω, κ ∈ K, t ∈ [0, 1]. Y is called left and right concave on K if Equation (3) is reversed.

Definition 3. [7]
A set A ⊂ R n is said to be an invex set, if, for all ω, κ ∈ A, t ∈ [0, 1], we have where ζ : R n × R n → R n .

Definition 4.
[6] Let A be an invex set. Then, IV-F Y : A → K + C is said to be left and right preinvex on A with respect to ζ if for all ω, κ ∈ A, t ∈ [0, 1], where ζ : R n × R n → R n . Y is called left and right preincave on A with respect to ζ if inequality (4) is reversed. Y is called affine if Y is both convex and concave. In the case of ζ(κ, ω) = −ω, we obtain (4) from (3).
The following outcome is very important in the field of interval-valued calculus because, by using this result, we can easily handle IV-Fs. Basically, Theorem 2 establishes the relation between IV-F Y(ω) and lower function Y * (ω) and upper function Y * (ω).
The following assumption will be required to prove the next result regarding the bifunction ζ : R n × R n → R n , which is known as:

Introduction
Hanson [1] defined the class of invex functions as one of the most significant exte sions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex fun tions to demonstrate adequate optimality criteria and duality in nonlinear programmin For a differentiable mapping, the concept of fractional integral identities involving Ri mann-Liouville fractional and Hadamard fractional integrals integrals was considered b Wang et al. [3], who identified some inequalities using standard convex, -convex, convex, -convex, (s, m)-convex, and ( , )-convex. Moreover, Işcan [4] also used fra tional integrals for preinvex functions to obtain various -type inequalities. See [5-for other generalizations of the -inequality.
For accurate solutions to various problems in practical mathematics, Moore [9] use interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper an lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinar Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united exte sion is a Lipschitz interval extension in . To combine the study of discrete and continu ous dynamical systems, Hilger [10] introduced a time scales theory. The widespread u of dynamic equations and integral inequalities on time scales, in domains as diverse electrical engineering, quantum physics, heat transfer, neural networks, combinatoric    [2], in 1988, used the notion of preinvex f tions to demonstrate adequate optimality criteria and duality in nonlinear programm For a differentiable mapping, the concept of fractional integral identities involving mann-Liouville fractional and Hadamard fractional integrals integrals was considere Wang et al. [3], who identified some inequalities using standard convex, -convex convex, -convex, (s, m)-convex, and ( , )-convex. Moreover, Işcan [4] also used tional integrals for preinvex functions to obtain various -type inequalities. See for other generalizations of the -inequality.
For accurate solutions to various problems in practical mathematics, Moore [9] interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordi Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united ex sion is a Lipschitz interval extension in . To combine the study of discrete and con ous dynamical systems, Hilger [10] introduced a time scales theory. The widespread of dynamic equations and integral inequalities on time scales, in domains as diver electrical engineering, quantum physics, heat transfer, neural networks, combinato If Y is left and right preincave, then, we achieve the following coming inequality: Proof. Let Y : [υ, υ + ζ(µ, υ)] → K + C be a left and right preinvex IV-F. Then, by hypothesis, we have Therefore, we have (µ, υ)).
That is Thus, In a similar way to the above, we have Combining (10) and (11), we have This completes the proof.
Proof. Since Y, D ∈ IR ([υ, υ+ζ(µ, υ)]) , then we have And From the definition of left and right preinvex IV-F, it follows that 0 ≤ p Y(ω) and 0 ≤ p D(ω), Integrating both sides of the above inequality over [0,1], we obtain It follows that, Thus, 1 and the theorem has been established.
Proof. Using condition C, we can write υ)).
Integrating over [0, 1], we have This completes the proof.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,

Introduction
Hanson [1] defined the class of invex functions as one of the most significant exten sions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex func tions to demonstrate adequate optimality criteria and duality in nonlinear programming For a differentiable mapping, the concept of fractional integral identities involving Rie mann-Liouville fractional and Hadamard fractional integrals integrals was considered b Wang et al. [3], who identified some inequalities using standard convex, -convex, convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] use interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper an lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinar Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united exten sion is a Lipschitz interval extension in . To combine the study of discrete and continu ous dynamical systems, Hilger [10] introduced a time scales theory. The widespread us of dynamic equations and integral inequalities on time scales, in domains as diverse a electrical engineering, quantum physics, heat transfer, neural networks, combinatoric and population dynamics [11], has highlighted the need for this theory. Young's inequa ity, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality

Introduction
Hanson [1] defined the class of invex functions as one of the most significant extensions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,   [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,  [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,

Introduction
Hanson [1] defined the class of invex functions as one of the most significant extensions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,

Introduction
Hanson [1] defined the class of invex functions as one of the most significant extensions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics,

Introduction
Hanson [1] defined the class of invex functions as one of the most significant extensions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics,

Introduction
Hanson [1] defined the class of invex functions as one of the most significant extensions of convex functions. Weir and Mond [2], in 1988, used the notion of preinvex functions to demonstrate adequate optimality criteria and duality in nonlinear programming. For a differentiable mapping, the concept of fractional integral identities involving Riemann-Liouville fractional and Hadamard fractional integrals integrals was considered by Wang et al. [3], who identified some inequalities using standard convex, -convex,convex, -convex, (s, m)-convex, and ( , )-convex. Moreover, Işcan [4] also used fractional integrals for preinvex functions to obtain various -type inequalities. See [5][6][7][8] for other generalizations of the -inequality.
For accurate solutions to various problems in practical mathematics, Moore [9] used interval arithmetic, IV-Fs, and integrals of IV-Fs to establish arbitrarily sharp upper and lower limits. Moore [9] showed that, if a real-valued mapping ( ) meets an ordinary Lipschitz condition in , | ( ) − ( )| ≤ | − |, for , ∈ , then, the united extension is a Lipschitz interval extension in . To combine the study of discrete and continuous dynamical systems, Hilger [10] introduced a time scales theory. The widespread use of dynamic equations and integral inequalities on time scales, in domains as diverse as electrical engineering, quantum physics, heat transfer, neural networks, combinatorics, and population dynamics [11], has highlighted the need for this theory. Young's inequality, Minkoswki's inequality, Jensen's inequality, Hölder's inequality, -inequality,