Extended Jensen’s functional for diamond integral via Green’s function and Hermite polynomial

In this paper, with the help of Green’s function and Hermite interpolating polynomial, an extension of Jensen’s functional for n -convex functions is deduced from Jensen’s inequality involving diamond integrals. Special Hermite conditions, including Taylor two point formula and Lagrange’s interpolation, are also deployed to ﬁnd the further extensions of Jensen’s functional. This paper also includes discussion on bounds for Grüss inequality, Ostrowski inequality, and ˇCebyšev functional associated to the newly deﬁned Jensen’s functional.


Introduction
Johan Jensen proved Jensen's inequality in [9]. It serves as a tool in discrete and continuous analysis for generating classical and new inequalities. The discrete version of Jensen's inequality is given below as where (z 1 , . . . , z n ) ∈ S, S is an interval in R, (g 1 , . . . , g n ) ∈ R n + (i.e., nonnegative weights are taken into account) and the function ζ : S → R is convex on S. Steffensen in [16] extended it by using negative weights.
The researchers have devised several new functions for the refinements of Jensen's discrete (or integral) inequality. For instance, in [7,8,11,13] improvements of the operated version of Jensen's inequality are given. In [5] Čuljak et al. generalized Jensen's inequality via Hermite polynomial. Several researchers discussed and applied these inequalities on time scales. In [1], Anwar et al. proved Jensen's inequality for delta integrals: Suppose a 1 , a 2 ∈ T are such that a 1 < a 2 and F ∈ C rd ([a 1 , a 2 ] T , R) assures a 2 a 1 |F(s)| s > 0. If ζ ∈ C(S, R) is convex on an interval S ⊂ R and τ ∈ C rd ([a 1 , a 2 ] T , S), then Under a similar hypothesis, in [12], by replacing the delta integral with the nabla integral, same results are obtained.
Sheng et al. in [14] presented a convex combination of the delta and nabla integrals named as diamond-α integral, where α ∈ [0, 1]. In [15] the following Jensen's inequality for diamond-α integral is given: Suppose a time scale T, and a 1 , In [6] the authors introduced a more generalized version of the diamond-α integral, termed as diamond integral having a special interest even for T = R. These integrals get us nearer to building a true symmetric integral on time scales.
In [3] Jensen's inequality is proved for diamond integrals: Let a 1 , a 2 ∈ T with a 1 < a 2 , F ∈ C([a 1 , a 2 ] T , R + ), and τ ∈ C([a 1 , a 2 ] T , S), assuring Considering the conditions of (2), Jensen's-type linear functional defined on T is given below as Remark 1.1 Inequality (2) implies that J(ζ ) ≥ 0 for the family of convex mappings and J(ζ ) = 0 for identity or constant functions.
The aim of the present study is the extension of (3) for n-convex functions with Green's function and some types of interpolations introduced by Hermite. In the next section, after defining the diamond derivative and integral, we recall Hermite interpolating polynomial along with some of its special forms. Section 3 consists of the main results of the paper, and finally concluding remarks are given in the last section. • if ρ(s) < s and σ (s) > s, then s is isolated. A mapping : T → R is said to be rd-continuous if it is continuous ∀s ∈ T such that σ (s) = s and left-sided limit is finite ∀s ∈ T such that ρ(s) = s. The set of such functions is denoted by C rd . Definition 2.1 Let : T → R be a mapping and s ∈ T k k . Define ♦ (assuming it is a finite positive number) having characteristic that, for a given ε > 0, there exists a neighborhood W of s such that

Some essentials from diamond calculus
holds for all u ∈ W for which 2su ∈ W . Then ♦ (s) is known as the diamond derivative of at s. Definition 2.2 Let a 1 , a 2 ∈ T and τ : T → R be a function. The diamond integral of τ from a 1 to a 2 is given by for all s ∈ T, where γ τ and (1γ )τ are delta and nabla integrable on [a 1 , a 2 ] T , respectively. It is to be noted that for u ∈ T k k , ( s b τ (s)♦) ♦ = τ (s), in general. The fundamental theorem of calculus also does not hold for diamond integrals.
The properties of the diamond integrals are analogous to the properties of the delta, nabla, and diamond-α integrals, see [6].

Results on Hermite interpolating polynomial
there exists an (n -1)th degree polynomial P H (s) defined by It satisfies the following Hermite condition: Factors H uv represent essential polynomials of the Hermite basis which satisfy the relations: Also H uv (s) is given by with Hermite conditions encompass the following specific cases: Conditions for Type (z, nz): (Conditions for Taylor's Two-point Formula) For n = 2z, r = 2, k 1 = k 2 = z -1, we have a Taylor two-point interpolating polynomial P 2T (s), satisfying The next theorem is useful for our results and is given in [10].
where P H is the Hermite interpolating polynomial as defined in (4) and R H (ζ , s) denotes the remainder given by for all a b ≤ t ≤ a b+1 , b = 0, . . . , r with a 0 = μ and a r+1 = ν. Here P L (s) represents a Lagrange polynomial, which is and R L (ζ , s) is the remainder, defined by . . , n -1 along a 1 = μ and a n = ν.
Remark 2.6 Similarly, by imposing (z, nz) conditions on Theorem 2.4, one gets and The remainder R (z,n) (ζ , s) is given by

Remark 2.7 Theorem 2.4 in the form of Taylor two-point formula becomes
where Taylor two-point interpolating polynomial, P 2T (s), is defined by

Extension of Jensen's functional via Green's function and Hermite polynomial
This section begins with the proof of our key identity regarding Jensen's inequality extension. Green's function G : Because of symmetry, G satisfies the conditions of convexity and continuity with respect to both s and t. For where G(s, t) is defined in (9).

Theorem 3.3 Under the assumptions of Theorem
Proof Substituting (15) into (11), we have as required. where is nonnegative, then Proof Substituting (17) into (16), we get The use of type (z, nz) condition yields the result given below.

Corollary 3.5
Suppose that ξ u , η u are defined as in (7) and (8), respectively. If nz is even, then for every n-convex function ζ : [μ, ν] → R and where is nonnegative, we have Application of two-point Taylor conditions gives the following result. where is nonnegative, then Proof Substituting (21) into (20), we get As the right-hand side is nonnegative, we have Remark 3.7 As mentioned in Remark 3.2, we can deduce special cases for the results of this section for different time scales.

Bounds for identities associated to the extension of Jensen's functional
Here we use Čebyšev functional and Grüss-type inequalities to present a few important results. The Čebyšev functional is given by The next two theorems are given in [4]. Theorem 4.1 If g 1 , g 2 : [μ, ν] → R are functions such that g 1 is Lebesgue integrable and g 2 is absolutely continuous, along with (·μ)(ν -·)[g 2 ] 2 ∈ L[μ, ν], then we have where 1 2 is the best possible constant.

Conclusion
Jensen's functional for the diamond integral (3) is generalized for n-convex functions using Green's function and Hermite polynomial in the present article. Different conditions of Hermite polynomial are utilized to describe respective refinements of the functional. As applications, bounds for the quantities associated to the constructed functional are also discussed. Moreover, by defining the functional as the difference of the right-and lefthand sides of the extended inequality (14), it is possible to study n-exponential convexity, exponential convexity, and applications to Stolarsky-type means as discussed by Aras-Gazič et al. in [2,Sects. 5,6]. This article extends the results of [5] on time scales.