SOME WEIGHTED INEQUALITIES FOR HIGHER-ORDER PARTIAL DERIVATIVES IN TWO DIMENSIONS AND ITS APPLICATIONS

We establish some Ostrowski type inequalities involving higher-order partial derivatives for two dimensional integrals on Lebesgue spaces (L∞, Lp and L1). Some applications in Numerical Analysis in connection with cubature formula are given. Finally, with the help of obtained inequality, we establish applications for the kth moment of random variables.


Introduction
Let f : [a, b]→ R be a differentiable mapping on (a, b) whose derivative f : (a, b)→ R is bounded on (a, b), i.e., f ∞ = sup t∈(a,b) |f (t)| < ∞. Then, the inequality holds: for all x ∈ [a, b] [14]. The constant 1 4 is the best possible. This inequality is well known in the literature as Ostrowski inequality.
Recently in [2], Barnett and Dragomir proved the following Ostrowski type inequality for double integrals: In [2], the inequality (2) is established by the use of integral identity involving Peano kernels. In [17], Pecarić and Vukelić gave weighted Montgomery's identities for two variables functions. Recently, many authors have worked on the Ostrowski type inequalities for double integrals. For example, Pachpatte obtained a new inequality in the view (2) by using elementary analysis in [15] and [16]. In [7], [8] and [9], some Ostrowski type inequalities for double integrals and applications in numerical analysis in connection with cubature formula are given by researchers. Authors deduced weighted inequality of Ostrowski type for two dimensional integrals in [19] and [20]. Some researchers established some Ostrowski type inequalities for n-times differentiable mappings in [1], [6] and [11]. In [10], weighted integral inequalities for one variable mappings which are n−times differentiable are obtained by Erden and Sarıkaya. The researchers established some Ostrowski type inequalities involving higher-order partial derivatives for double integrals in [4], [12] and [21].
In this study, we first establish new integral equality involving higher-order partial derivatives. Then, some inequalities of Ostrowski type for two-dimensional integrals are attained by using this identity. Finally, some applications of the Ostrowski type inequality developed in this work for cubature formula and the kth moment of random variables are given.

Integral identity
In order to prove generalized weighted integral inequalities for double integrals, we need the following lemma: where M k (x) and M l (y) are defined by Applying integration by parts for partial derivatives ∂ n+m f (t,s) As we progress by this method, we get Similarly, applying integration by parts for partial derivatives ∂ n+l f (t,y) ∂t n ∂y l and ∂ n f (t,s) Substituting the identity (5) and (6) in (4), we deduce desired identity (3), and thus the theorem is proved.
3. Some inequalities for ∂ n+m f ∂t n ∂s m belongs to lebesgue space We give some results for functions whose n + m.th partial derivatives are bounded. We start with the following result.
if m and n are even numbers if m is odd number and n is even numbers if m is even number and n is odd number if m and n are odd numbers where Proof. If we take absolute value of both sides of the equality (3), because ∂ n+m f ∂t n ∂s m is a bounded mapping, we can write By using the change of order of integration, we obtain which completes the proof.

Remark 3.2.
Under the same assumptions of Theorem 3.1 with n = m = 1, then the following inequality holds: which is "weighted Ostrowski" type inequality for · ∞ −norm. This inequality was deduced by Sarikaya and Ogunmez in [19]. (8), then the inequality (8) reduce to the inequality (2).
which was given by Barnett and Dragomir in [2].
Remark 3.5. Under the same assumptions of Theorem 3.1 with g(u) = h(u) = 1, then we have the inequality and This inequality (10) was proved by Hanna et al. in [12].
Proof. Taking modulus of both sides of the equality (3), because ∂ n+m f ∂t n ∂s m is a bounded mapping, we have the inequality (7). Because of boundedness g and h, and by definitions of P n−1 (x, t) and If we calculate the above four integrals and also substitute the results in (14), we obtain desired inequality (13) which completes the proof.
Corollary 3.7. Under the same assumptions of Theorem 3.6 with n = m = 1, then the following inequality holds: which is "weighted Ostrowski" type inequality for · ∞ −norm.
which is Ostrowski type inequality for double integrals. Thus, (16) is a higher degree "weighted mid-point" inequality for · ∞ −norm.
Now, we deduce some inequalities for mappings whose higher-order partial derivatives belongs to either L p (∆) or L 1 (∆) .
Proof. Using the properties of modulus and from Hölder's inequality, from (3), we find that Owing to boundedness of g and h, and by definitions of P n−1 (x, t) and Q m−1 (y, s), we can write By simple calculations, we easily deduced required inequality, and thus the theorem is proved.
Corollary 3.13. Under the same assumptions of Theorem 3.12 with n = m = 1, then the following inequality holds: which is "weighted Ostrowski" type inequality for · p −norm.
Corollary 3.14. If we choose x = a+b 2 and y = c+d 2 in (17), then we have the inequality which is "weighted mid-point" inequality for two dimensional integrals. This inequality is a weighted Ostrowski type inequality for · p −norm.
where X k (x) and Y l (y) are defined as in (11) and (12), respectively. The inequality (18) was deduced by Hanna in [12].
Corollary 3.17. Under the same assumptions of Theorem 3.12 with x = a+b 2 and y = c+d 2 , then we have the inequality which is "weighted mid-point" inequality for double integrals. This inequality is a higher degree weighted Ostrowski type for · p −norm.
Theorem 3.18. Let f : ∆ ⊂ R 2 → R be a continuous on ∆ such that ∂ n+m f ∂t n ∂s m exist on (a, b) × (c, d) and assume that the functions g : Proof. By taking absolute value of (3), we find that By boundedness g and h, and because of definitions of P n−1 (x, t) and Q m−1 (y, s), we have We obtain desired inequality (19) using the identity The proof is thus completed.
Corollary 3.19. Under the same assumptions of Theorem 3.18 with n = m = 1, then the following inequality holds: which is "weighted Ostrowski" inequality for double integrals of the Ostrowski type inequality for · 1 −norm.
Remark 3.23. Under the same assumptions of Theorem 3.18 with g(u) = h(u) = 1, then we have the inequality where X k (x) and Y l (y) are defined as in (11) and (12), respectively. The inequality (22) was proved by Hanna et al. in [12].
Corollary 3.24. Under the same assumptions of Theorem 3.18 with x = a+b 2 and y = c+d 2 , then we have the inequality which is "weighted mid-point" inequality for double integrals. Thus, (23) is a heigher degree weighted Ostrowski type inequality for · 1 −norm.

Applications to Cubature Formulae
We now deal with applications of the integral inequalities developed in the previous section, to obtain estimates of cubature formula, which it turns out to have a markedly smaller error than that which may be obtained by the classical results. Thus the following applications in numerical integration are natural to be considered.

Some applications for the moments
Distribution functions and density functions provide complete descriptions of the distribution of probability for a given random variable. However, they do not allow us to easily make comparisons between two different distributions. The set of moments that uniquely characterizes the distribution under reasonable conditions is useful in making comparisons. Knowing the probability function, we can determine moments if they exist. Applying the mathematical inequalities, some estimations for the moments of random variables were recently studied (see, [3], [5], [13], [18]).
Set X to denote a random variable whose probability density function is g : [a, b] → [0, ∞) on the interval of real numbers I (a, b ∈ I, a < b) and Y to denote a random variable whose probability density function is h : [c, d] → R on the interval of real numbers I (c, d ∈ I, c < d). Denoted by M r (x) and M r (y) the r.th central moment of the random variable X and Y, respectively, defined as where E(x) and E(y) are the mean of the random variables X and Y, respectively. It may be noted that M 0 (x) = M 0 (y) = 1, M 1 (x) = M 1 (y) = 0, M 2 (x) = σ 2 (X) and M 2 (y) = σ 2 (Y ) where σ 2 (X) and σ 2 (Y ) are the variance of the random variables X and Y, respectively. Now, we reconsider the identity (3) by changing conditions given in Lemma 2.1. Herewith, we deduce an identity involving r.th moment.
Similarly, if we examine the other integral in (28), we obtain desired inequality (27). Thus, the proof is completed.
Similarly, using the other inequalities in section 3, we obtain similar results involving r.th central moment of the random variable X and Y.