New integral inequalities of Hermite–Hadamard’s and Simpson’s type for twice differentiable mappings

In this paper, some new interesting results based on quasi-geometrically convex mappings of both Hadamard’s and Simpson’s inequalities have been constructed defining a new identity for twice differentiable mappings.


Introduction and preliminary results
Any function f be continuous on an interval I such that its value at the midpoint of the interval does not exceed from the arithmetic mean of boundary values of the interval, termed as a convex function. The investigation of an important mathematical problem informs how function behaves using means. Jensen convex function is one of the eminent cases that deals with arithmetic mean [1, pp.2]. A function f is convex on an interval [p, q] if where , ∈ [p, q] and , ∈ ]0, 1] . Further, a real-valued function defined on a non-empty subinterval I of ℝ is called convex if we replace + = 1 for all points , ∈ I in the above inequality. It is called strictly convex if the above inequality holds strictly whenever and are distinct points.
In the field of applied sciences, the occurrence of new mathematical inequalities puts the foundation for the heuristic algorithms. Hermite-Hadamard's is one of the main inequalities that yields explicitly the error bounds of the trapezoidal and midpoint rules for a smooth convex function f ∶ [p, q] → ℝ , defined as For f to be concave, these inequalities also hold in reverse order. More precisely, Hermite-Hadamard's inequality (1.1) may depict the concept of convexity and follows from Jensen inequality. Inequality (1.1) has gained much attention among researchers due to its remarkable characteristics in refinements and generalizations, as well. Simpson's is another well-known type of inequality, defined as where f ∶ [p, q] → ℝ is a four times continuous differentiable mapping on (p, q) and ∥ f (iv) ∥ ∞ = sup ∈(p,q) | | f (iv) ( ) | | < ∞ . It is to be mentioned that the classical Simpson quadrature formula cannot be applied for f is neither differentiable four times nor f (iv) bounded on (p, q).
Both GA-convex and geometrically convex functions are quasi-geometrically convex functions, but there exist quasigeometrically convex functions which are neither GA-convex nor GG-convex discussed in [11,12].
Recently, İşcan et al. [13] developed some results based on single differentiability for quasi-geometrically convex functions using the identity 3) where This paper is in the continuation of [13]. The main purpose of the paper is to develop some new integral inequalities of both Hadamard and Simpson type for twice differentiable mappings using new integral identity.

Main results
The following identity is needed to prove main results.
where Proof Using integration rules and changing parameter, it leads to the result. ◻ for some fixed k ≥ 1 and 0 ≤ ≤ 1∕2 ≤ ≤ 1 , then the following inequality holds where and k, p, q) , for all ∈ [0, 1]

Using Lemma 2 and power mean inequality, it yields
Here we assume that Using substitution = p 2(1− ) q 2 in c 3 ( , k, p, q) , it leads to

3
Finally, we get