Theorem 2.1. Let such that the partial delta derivative of order 2 exists and there exists a constant K with . Suppose that is a division of the interval , while J
: c = y0 < y1 < ⋯ < yk-1< y
= d is a division of the interval . Then we have
The inequality (1) is sharp in the sense that the right side of (1) can not be replaced by a smaller one.
Then we obtain
On the other hand, we have
Combining (3) and (4) we get the desired inequality (1).
In order to prove the sharpness of (1), we take k = 1, α1 = b, β1 = d, f(s, t) = st. Then the left side of (1) becomes
On the other hand, Using K = 1, the right side of (1) reduces to h2 (a, b)h2 (c, d), which implies (2) holds for equality form, and then the sharpness of (1) is proved.
Remark 2.2. Theorem 2.1 is the 2D extension of [, Theorem 3].
From Theorem 2.1 we can obtain some particular Ostrowski type inequalities on time scales. For example, if we take k = 1, α1 = b, β1 = d, then we have
If we take k = 1, α1 = a, β1 = c, then we have
If we take , then we have
If we take k = 2, α1 = a, α2 = b, β1 = c, β2 = d, x1 = x, y1 = y, then we have
If we take , then we have
If we furthermore take in the inequality above, then we obtain the time scale version of Simpson's inequality , which is omitted here.
In Theorem 2.1, if we take for some special time scales, then we immediately obtain the following three corollaries.
Corollary 2.3 (Continuous case). Let in Theorem 2.1, then , and we obtain
Corollary 2.4 (Discrete case). Let in Theorem 2.1. Suppose that x
∈ [m1, m2]ℤ, y
∈ [n1, n2]ℤ, i = 0,1, ...,k. I
: m1 = x0 < x1 < ⋯ < xk-1< x
= m2 is a division of [m1, m2]ℤ, while J
: n1 = y0 < y1 < ⋯ < yk-1< y
= n2 is a division of [n1, n2]ℤ. α
∈ [xi-1, x
∈ [yi-1, y
]ℤ, i = 1,2,...,k. Then we have
where K denotes the maximum value of the absolute value of the difference Δ1Δ2f over [m1, m2-1]ℤ × [n1, n2-1]ℤ.
As long as we notice for ∀t, s ∈ ℤ, we can easily get the desired result.
Corollary 2.5 (Quantum calculus case). Let in Theorem 2.1, where m1, m2, n1, n2 ∈ ℕ0 and q
> 1, i = 1, 2. Suppose that is a division of , while is a division of . Then we have
where K denotes the maximum value of the absolute value of the q1 q2-difference over .
Proof. Since for , we have
Substituting (8) into (1) we get the desired result.
Theorem 2.6. Under the conditions of Theorem 2.1, if there exist constants K1, K2 such that , then we have the following inequality
Proof. We notice that
We also have , and
Then combining (4), (10) and (11) we obtain the desired inequality (9).
Remark 2.7. If we take , where λ ∈ [0,1], then Theorem 2.6 reduces to [, Theorem 4].
In the following, we will establish a generalized Ostrowski-Grüss type integral inequality on time scales based on the result of Theorem 2.1.
Lemma 2.8 (2D Grüss' inequality on time scales). Let such that ϕ ≤ f(x, y) ≤ Φ and γ ≤ g(x, y) ≤ Γ for all , where ϕ, Φ, γ, Γ are constants. Then we have
The proof for Lemma 2.6 is similar to [, pp. 295-296], and we omit it here.
Theorem 2.9. Under the conditions of Theorem 2.1, if there exist constants K1, K2 such that , then we have the following inequality