Generalized Jensen’s functional on time scales via extended Montgomery identity

In the paper, we use Jensen’s inequality for diamond integrals and generalize it for n-convex functions with the help of an extended Montgomery identity. Moreover, the bounds have been suggested for identities associated with the generalized Jensen-type functional.


Introduction and preliminaries
Montgomery identity is well known in the literature. It is utilized to obtain a number of revolutionary inequalities such as trapezoid inequality, Ostrowski-type inequality, Čebyšev inequality, Grüss inequality, and Mohajani inequality. Mitrinović et al. in [17] proved that if χ : [b 1 , b 2 ] → R is differentiable on [b 1 , b 2 ] and χ : [b 1 , b 2 ] → R is integrable on [b 1 , b 2 ], then the Montgomery identity is where Bohner et al. in [7] proved the Montgomery identity on time scales and discussed it for discrete, continuous, and quantum cases. In case of T = R, it becomes (1). Suppose that a probability density function ψ : [b 1 , b 2 ] → [0, ∞), i.e., an integrable function which satisfies b 2 b 1 ψ(p) dp = 1. Also ξ (p) = p b 1 ψ(t) dt when p ∈ [b 1 , b 2 ], ξ (p) = 0 when p < b 1 and ξ (p) = 1 when p > b 2 . Then the weighted generalization [19] of the Montgomery identity is where the weighted Peano kernel is Sarikaya et al. [20] proved the weighted Montgomery identity on time scales. To obtain our main results, we use the extended Montgomery identities by means of Taylor's formula I and II given in [1,2].

Theorem 1 Let E be an open interval in
Assume that a function χ : E → R, s.t., χ (n-1) is absolutely continuous for n ∈ N, then where
Divided differences are truly assigned to Newton, and Augustus de Morgan in 1842 used the term divided difference. The divided differences are beneficial if the functions have different degrees of smoothness. Divided differences are discussed in [13].
Definition 1 Suppose μ: [p, q] → R, n ∈ N and mutually distinct points η 0 , η 1 , . . . , η n of [p, q]. Then the nth order divided difference of the function μ is n-convex function is defined on the basis of nth-order divided difference [19]. Suppose n ≥ 0, then a function μ : [p, q] → R is termed n-convex iff ∀ (n + 1) distinct points y 0 , y 1 , . . . , y n ∈ [p, q], [y 0 , y 1 , . . . , y n ; μ] ≥ 0 holds. The reverse effect of the above inequality implies that μ is n-concave. The strict effect of the above inequality implies that μ is a strictly n-convex (n-concave) function. The n convexity of a function μ is examined by the following theorem [19].

Theorem 3
If μ n exists, then μ is n-convex if and only if μ n ≥ 0.

On time scales
In 1988, Hilger presented the notion of time scales calculus and proposed the unification of discrete and continuous time dynamical systems [14]. Gradually, this perspective of unification has been affixed by the extension and generalization characteristics. and ∇ calculus is the initial approach to study time scales calculus. For detailed study of calculus on time scales, readers are reffered to [6,8,9].
An arbitrary and nonempty closed subset of real numbers is called time scales T. The real numbers R and the integers Z are most familiar examples of time scales.
Suppose p ∈ T, then the mappings σ (t), ρ(t): T → T indicate the forward and backward jump operators respectively on time scale T and are defined as follows: Take a point t ∈ T, then t is stated as follows: • The graininess functions σ , ν : T → [0, +∞) are stated as Assume that a function μ: T → R, then it is termed: • rd-continuous if it is continuous at all right-dense points in T and its left-sided limits are finite at all left-dense points in T; • ld-continuous if it is continuous at all left-dense points in T and its right-sided limits are finite at all right-dense points in T.
We present the sets T k , T k , and T * that originated from the time scale T.
Suppose that a function μ: In the same way, for t ∈ T k , we say μ Suppose that μ(t) is differentiable in the and ∇ sense on the time scales T. Let t ∈ T, Hence, μ is α -differentiable iff and ∇ derivatives of μ exist.
For a complete advancement of the calculus on the α -derivative and α -integrals, refer to [21]. The refinement of α -derivative on time scales T is known as symmetric derivative given in [11], which is as follows: Let t ∈ T k k and μ: holds ∀y ∈ V for which 2ty ∈ V .

Remark 2 The symmetric derivative is
• classic symmetric derivative, when T = R.
A detailed discussion of symmetric derivatives on time scales is given in [11].
In the special case, for each t ∈ T k k , if a function is and ∇ differentiable at the same time, then μ is symmetric differentiable and is a real-valued function.

Jensen's inequality
Inequality proved by Jensen [15] in 1906 is popular in mathematical analysis. For continuous and discrete analysis, it is used to formulate many classical inequalities. Therefore the developments in many other inequalities are based on the developments in Jensen's inequality. Jensen's inequality in a discrete version [15] is as follows: Suppose that an interval E in R and a convex function χ : E → R, v = (v 1 , . . . , v n ) is a real n-tuple and y = (y 1 , . . . , y n ) is a positive n-tuple (n ∈ N). Then The strict convexity of χ implies that (12) is strict unless y 1 = · · · = y n and χ is concave if (12) holds in reverse direction. Jensen's inequality in an integral form [16] is as follows: , E) and a convex function χ ∈ C(E, R), then Anwar et al. in [4] extended the Jensen's inequality for -integrals as follows: Özkan et al. in [18] showed that if we apply nabla integrals instead of delta integrals, then (14) also holds. In [21], Sheng et al. presented α -dynamic derivative and α -dynamic integral for providing more balanced approximations with respect to computations. Jensen's inequality for α -integral is given in [3].
and a convex function χ ∈ C(E, R) with an interval E ⊂ R, then we have Da Cruz et al. in [11] described diamond integral (a generalization of diamond-α integral) in terms of "approximate symmetric integral" on time scales T. Bibi et al. proved Jensen's inequality related to diamond integrals in [5].
Under the suppositions of Theorem 4, inequality (16) produces the linear functional Remark 4 By Theorem 4, we conclude that J(χ) = 0 if χ is a constant function or an identity function and J(χ) ≥ 0 for the group of convex functions.
In the same manner, the linear functionals for (14) and (15) can be obtained.

Generalized Jensen-type functional by extended Montgomery identity via Taylor's formula I
where R n (t, s) is defined in (6) and Proof Putting (5) in (17), we obtain The linearity of the functional J(·) gives us

Theorem 6 Let all the suppositions of Theorem 5 be satisfied and
If χ is n-convex such that χ (n-1) is absolutely continuous, then we have that holds, where The reverse inequality in (21) gives rise to the reverse inequality in (22).
Proof Since χ (n-1) is absolutely continuous on [b 1 , b 2 ], therefore χ n exists almost everywhere. Now the n-convexity of χ implies that χ n (t) ≥ 0, ∀t ∈ [b 1 , b 2 ]; this fact together with (21) implies that Using (25) in (18), we get The linearity of J(·) yields which is the required result. The reverse inequality in (21) gives rise to the reverse inequality in (24); therefore, the reverse inequality in (22) is obtained.
Remark 5 The generalized form of Theorem 4 is Theorem 7.

Generalized Jensen-type functional by extended Montgomery identity via Taylor's formula II
Theorem 8 Let n ∈ N such that n ≥ 2 and all the suppositions of Theorem 4 be satisfied.
If χ is a convex function defined on [b 1 , b 2 ] such that χ (n-1) is absolutely continuous, then whereR n is defined in (8) and Proof Putting (7) in (17), we obtain By using the linearity of the functional J(·), we get Proof Since χ (n-1) is absolutely continuous on [b 1 , b 2 ], therefore χ n exists almost everywhere. Now the n-convexity of χ implies that χ n (t) ≥ 0, ∀t ∈ [b 1 , b 2 ]; this fact together with (21) implies that Using (36) in (29), we get By the linearity of J(·), we get which is the required result. The reverse inequality in (32) gives rise to the reverse inequality in (35); therefore, the reverse inequality in (33) is obtained. Proof Since U(t) is convex ∀t ∈ [b 1 , b 2 ], thus by means of Remark 4, we get J(U(t)) ≥ 0. As a result, (33) implies J(χ(t)) ≥ 0.
Remark 6 The generalized form of Theorem 4 is Theorem 10.

Bounds for identities associated with generalization of Jensen-type functional
Assume f, ν : [b 1 , b 2 ] → R as Lebesgue integrable functions, then the Čebyšev functional is The following theorems were proved by Cerone and Dragomir in [10].

Theorem 11
Consider a Lebesgue integrable function f : [b 1 , b 2 ] → R and an absolutely continuous function ν : . Then The constant 1 √ 2 in (41) is the best possible.
Then we have The constant 1 2 in (42) is the best possible.
For r 1 = 1, we prove that Then, for small enough, the statement follows.