ON WEIGHTED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS OF TWO VARIABLES WITH BOUNDED VARIATION

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HUSEYIN BUDAK∗, MEHMET ZEKI SARIKAYA

Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨uzce-Turkey

Abstract. In this paper, we obtain new weighted Ostrowski type inequalities for functions of two independent variables with bounded variation. Applications for qubature formulae are also given.

Keywords. Bounded variation; Ostrowski type inequality; Riemann-Stieltjes integral. 2010 Mathematics Subject Classification. 26D15, 26B30, 26D10, 41A55.

1. Introduction

Let f : [a, b] → R be a differentiable mapping on (a, b) whose derivative f0: (a, b) → R is bounded on (a, b), i.e., k f0k_∞:= sup

t∈(a,b)

| f0(t)| < ∞. From [19], we have the inequality

f(x) − 1 b− a b Z a f(t)dt ≤ " 1 4+ x−a+b₂ 2 (b − a)2 # (b − a) f0 ∞, ∀x ∈ [a, b]. (1.1)

The constant 1₄ is the best possible. This inequality is well known in the literature as the Os-trowski inequality.

In [15], Dragomir proved following Ostrowski type inequalities related functions of bounded variation.

∗_{Corresponding author.}

E-mail addresses: hsyn.budak@gmail.com (H. Budak), sarikayamz@gmail.com (M.Z. Sarikaya). Received December 2, 2016; Accepted March 10, 2017.

c

2017 Mathematical Research Press

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Theorem 1.1. Let f : [a, b] → R be a mapping of bounded variation on [a, b] . Then b Z a f(t)dt − (b − a) f (x) ≤ 1 2(b − a) + x−a+ b 2 b _ a ( f )

holds for all x∈ [a, b] . The constant 1₂ is the best possible.

In [22], Tseng et al. gave the following weighted Ostrowski type inequalities for functions of bounded variation.

Theorem 1.2. Let us have 0 ≤ α ≤ 1, let w : [a, b] → [0, ∞) continuous and positive on (a, b) and h_{: [a, b] → R be differentiable such that h}0(t) = w(t) on [a, b] . Let a₁= h−1 1 −α

2 h(a) + α

2h(b) ,

b₁= h−1 α₂h(a) + 1 −α

2 h(b) . If f : [a, b] → R be mapping of bounded variation on [a,b],

then for all x∈ [a1, b1] , we have the inequality

b Z a w(t) f (t)dt − (1 − α) f (x) + α f(a) + f (b) 2 Zb a w(t)dt ≤ K b _ a ( f ) , (1.2) where K:=                1−α 2 b R a w(t)dt + h(x) − h(a)+h(b) 2 , if0 ≤ α ≤ 1 2, max 1−α 2 b R a w(t)dt + h(x) − h(a)+h(b) 2 , α 2 b R a w(t)dt , if 1₂ < α <2₃, α 2 b R a w(t)dt, if 2₃ ≤ α ≤ 1 and b W a

( f ) denotes the total variation of f on interval [a, b] . In (1.2), the constant 1−α₂ for 0 ≤ α ≤1₂ and the constant α

2 for 2

3 ≤ α ≤ 1 are the best possible.

2. Preliminaries

In 1910, Fr´echet [17] has given the following characterization for the double Riemann-Stieltjes integral. Assume that f (x, y) and g(x, y) are defined over the rectangle Q = [a, b] × [c, d]; let R be the divided into rectangular subdivisions, or cells, by the net of straight lines x= xi, y = yi,

a= x0< x1< ... < xn= b, and c = y0< y1< ... < ym= d;

let ξi, ηjbe any numbers satisfying ξi∈ [xi−1, xi] , ηj∈yj−1, yj , (i = 1, 2, ..., n; j = 1, 2, ..., m);

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sum S = ∑n i=1 m ∑ j=1 f ξi, ηj

∆11g(xi, yj) tends to a finite limit as the norm of the subdivisions

approaches zero, the integral of f with respect to g is said to exist. We call this limit the restricted integral, and designate it by the symbol

b Z a d Z c f(x, y)dydxg(x, y). (2.1)

If S is replaced by the sum S∗= ∑n

i=1 m ∑ j=1 f ξi j, ηi j

∆11g(xi, yj), where ξi j, ηi j are numbers

satis-fying ξi j∈ [xi−1, xi] , ηi j ∈yj−1, yj , we call the limit, when it exist, the unrestricted integral,

and designate it by the symbol

(∗) b Z a d Z c f(x, y)dydxg(x, y). (2.2)

Clearly, the existence of (2.2) implies both the existence of (2.1) and its equality (2.2). On the other hand, Clarkson [13] has shown that the existence of (2.1) does not imply the existence of (2.2).

In [12], Clarkson and Adams gave the following definitions of bounded variation for func-tions of two variables.

The function f (x, y) is assumed to be defined in rectangle R(a ≤ x ≤ b, c ≤ y ≤ d). By the term net we shall, unless otherwise specified mean a set of parallels to the axes:

x = xi(i = 0, 1, 2, ..., m), a = x0< x1< ... < xm= b;

y = yj( j = 0, 1, 2, ..., n), c = y0< y1< ... < yn= d.

Each of the smaller rectangles into which R is devided by a net will be called a cell. We employ the notation

∆11f(xi, yj) = f (xi+1, yj+1) − f (xi+1, yj) − f (xi, yj+1) + f (xi, yj)

∆ f (xi, yj) = f (xi+1, yj+1) − f (xi, yj)

The total variation function, φ (x) [ψ(y)] , is defined as the total variation of f (x, y) [ f (x, y)] considered as a function of y [x] alone in interval (c, d) [(a, b)],or as +∞ if f (x, y) [ f (x, y)] is of unbounded variation.

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Definition 2.1. (Vitali-Lebesque-Fr´echet-de la Vall´ee Poussin). The function f (x, y) is said tobe of bounded variation if the sum

m−1 , n−1

∑

i=0 , j=0 ∆11f(xi, yj)

is bounded for all nets.

Definition 2.2. (Fr´echet). The function f (x, y) is said tobe of bounded variation if the sum

m−1 , n−1

∑

i=0 , j=0 εiεj ∆11f(xi, yj)

is bounded for all nets and all possible choices of εi= ±1 and εj= ±1.

Definition 2.3. (Hardy-Krause). The function f (x, y) is said tobe of bounded variation if it satisfies the condition of Definition 2.1 and if in addition f (x, y) is of bounded variation in y (i.e. φ (x) is finite) for at least one x and f (x, y) is of bounded variation in y (i.e. ψ(y) is finite) for at least one y.

Definition 2.4. (Arzel`a). Let (xi, yi) (i = 0, 1, 2, ..., m) be any set of points satisfiying the

condi-tions

a = x0< x1< ... < xm= b;

c = y0< y1< ... < ym= d.

Then f (x, y) is said tobe of bounded variation if the sum∑m

i=1

|∆ f (xi, yi)| is bounded for all such

sets of points.

Therefore, one can define the consept of total variation of a function of variables, as follows: Let f be of bounded variation on Q = [a, b] × [c, d], and let ∑ (P) denote the sum

n

∑

i=1 m

∑

j=1 ∆11f(xi, yj)

corresponding to the partition P of Q. The number

_ Q ( f ) := d _ c b _ a ( f ) := sup

_∑

(P) : P ∈ P(Q) , is called the total variation of f on Q.

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Lemma 2.5. (Integrating by parts) If f ∈ RS(g) on Q, then g ∈ RS( f ) on Q, and we have d Z c b Z a f(t, s)dtdsg(t, s) + d Z c b Z a g(t, s)dtdsf(t, s)

= f (b, d)g(b, d) − f (b, c)g(b, c) − f (a, d)g(a, d) + f (a, c)g(a, c).

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Lemma 2.6. Assume that Ω ∈ RS(g) on Q and g is of bounded variation on Q. Then d Z c b Z a Ω(x, y)dxdyg(x, y) ≤ sup (x,y)∈Q |Ω(x, y)|_ Q (g) . (2.4)

In [18], Jawarneh and Noorani obtained the following Ostrowski type inequality for functions of two variables with bounded variation.

Theorem 2.7. Let f : Q →→ R be mapping of bounded variation on Q. Then for all (x, y) ∈ Q, we have inequality (b − a) (d − c) f (x, y) − d Z c b Z a f(t, s)dtds ≤ 1 2(b − a) + x−a+ b 2 1 2(d − c) + y−c+ d 2 _ Q ( f ) , (2.5) whereW Q

( f ) denotes the total (double) variation of f on Q.

In [6], Budak and Sarikaya proved the following generalization of the inequality (2.5).

Theorem 2.8. Let f : Q → R be mapping of bounded variation on Q. Then for all (x, y) ∈ Q, we have inequality |(b − a) (d − c) [(1 − λ ) (1 − η) f (x, y) +(1 − λ ) η 2 [ f (a, y) + f (b, y)] + λ (1 − η ) 2 [ f (x, c) + f (x, d)] +λ η 4 [ f (a, c) + f (a, d) + f (b, c) + f (b, d)] − b Z a d Z c f(t, s)dsdt ≤ max λ b− a 2 , x−(2 − λ ) a + λ b 2 , (2 − λ ) b + λ a 2 − x × max ηd− c 2 , y−(2 − η) c + ηd 2 , (2 − η) d + ηc 2 − y b _ a d _ c ( f ) (2.6)

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for any λ , η ∈ [0, 1] and a + λb−a₂ ≤ x ≤ b − λb−a₂ , c+ ηd−c₂ ≤ y ≤ d − ηd−c₂ , where b W a d W c ( f ) denotes he total variation of f on Q.

For more information and recent developments on inequalities for mappings of bounded vari-ation, we refer authors to [1]-[11], [14]-[16], [18], [20]-[25] and the references therein. The aim of this paper is to establish new weighted Ostrowski type inequalities for functions of two inde-pendent variables with bounded variation.

3. Main results

Let us have 0 ≤ α, β ≤ 1 and let w1: [a, b] → [0, ∞) continuous and positive on (a, b) and

h₁_{: [a, b] → R be differentiable such that h}0₁(t) = w1(t) on [a, b] . Similarly, let w2: [c, d] → [0, ∞)

continuous and positive on (c, d) and h2: [c, d] → R be differentiable such that h0₂(t) = w2(t)

on [c, d] . Let a1 = h−1₁ 1 −α₂ h1(a) +α₂h1(b) , b1 = h−1₁ α₂h1(a) + 1 −α₂ h1(b) , c1=

h−1₂ 1 −β₂h₂(c) +β₂h₂(d)and d1= h−1₂ β 2h2(c) + 1 −β₂h₂(d).

Theorem 3.1. If f : [a, b] × [c, d] → R is a mapping of bounded variation on [a, b] × [c, d] , then we have the following inequality for all(x, y) ∈ [a1, b1] × [c1, d1] ,

  b Z a w₁(t)dt     d Z c w₂(t)dt  [(1 − α) (1 − β ) f (x, y) + (1 − α) β f(x, c) + f (x, d) 2 + α (1 − β ) f(a, y) + f (b, y) 2 +αβ f(a, c) + f (a, d) + f (b, c) + f (b, d) 4 − b Z a d Z c w1(t)w2(s) f (t, s)dsdt ≤ KL b _ a d _ c ( f ) , (3.1) where K=                1−α 2 b R a w₁(t)dt + h1(x) − h1(a)+h1(b) 2 , if0 ≤ α ≤ 1 2, max 1−α 2 b R a w₁(t)dt + h1(x) − h1(a)+h1(b) 2 , α 2 b R a w₁(t)dt , if 1₂ < α <2₃, α 2 b R a w1(t)dt, if 2₃ ≤ α ≤ 1,

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and L=                1−β 2 d R c w₂(t)dt + h2(y) − h2(c)+h2(d) 2 , if0 ≤ β ≤ 1 2, max 1−β 2 d R c w2(t)dt + h2(y) − h2(c)+h2(d) 2 , β 2 d R c w2(t)dt , if 1₂ < β < 2₃, β 2 d R c w₂(t)dt, if 2₃ ≤ β ≤ 1, and b W a d W c

( f ) denotes the total variation of f on interval [a, b] × [c, d] .

In (3.1), the constant (1−α)(1−β )₄ for α, β ∈0,1₂ and the constant α β₄ for α, β ∈2₃, 1 are the best possible.

Proof. For (x, y) ∈ [a1, b1] × [c1, d1] , we define the following mappings q, p by

q(t) =    h₁(t) − 1 −α 2 h1(a) + α 2h1(b) , t ∈ [a, x), h₁(t) −α 2h1(a) + 1 − α 2 h1(b) , t ∈ [x, b] , p(s) =    h₂(s) − h 1 −β₂ h₂(c) +β₂h₂(d) i , s ∈ [c, y), h₂(s) −hβ 2h2(c) + 1 −β₂ h₂(d)i, s ∈ [y, d] .

Using the q(t) and p(s) kernels, we have

b Z a d Z c q(t)p(s)dsdtf(t, s) = x Z a y Z c h1(t) − h 1 −α 2 h1(a) + α 2h1(b) i h2(s) − 1 −β 2 h2(c) + β 2h2(d) dsdtf(t, s) + x Z a d Z y h₁(t) − h 1 −α 2 h₁(a) +α 2h1(b) i h₂(s) − β 2h2(c) + 1 −β 2 h₂(d) d_sd_tf(t, s) + b Z x y Z c h₁(t) −hα 2h1(a) + 1 −α 2 h₁(b)i h₂(s) − 1 −β 2 h₂(c) +β 2h2(d) d_sd_tf(t, s) + b Z x d Z y h₁(t) −hα 2h1(a) + 1 −α 2 h₁(b)i h₂(s) − β 2h2(c) + 1 −β 2 h₂(d) d_sd_tf(t, s) = I1+ I2+ I3+ I4.

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By integrating by parts, we get I1= x Z a y Z c h1(t) − h 1 −α 2 h1(a) + α 2h1(b) i × h2(s) − 1 −β 2 h2(c) + β 2h2(d) dsdtf(t, s) =hh₁(x) −1 −α 2 h₁(a) −α 2h1(b) i × h₂(y) − 1 −β 2 h₂(c) −β 2h2(d) f(x, y) +hh₁(x) −1 −α 2 h₁(a) −α 2h1(b) i βh2(d) − h2(c) 2 f(x, c) + αh1(b) − h1(a) 2 h₂(y) − 1 −β 2 h₂(c) −β 2h2(d) f(a, y) + αh1(b) − h1(a) 2 βh2(d) − h2(c) 2 f(a, c) − x Z a y Z c w₁(t)w2(s) f (t, s)dsdt. (3.2)

Using a similar method, we have

I2= x Z a d Z y h1(t) − h 1 −α 2 h1(a) + α 2h1(b) i × h2(s) − β 2h2(c) + 1 −β 2 h2(d) dsdtf(t, s) =hh1(x) − 1 −α 2 h1(a) − α 2h1(b) i βh2(d) − h2(c) 2 f(x, d) −hh1(x) − 1 −α 2 h1(a) − α 2h1(b) i h2(y) − β 2h2(c) − 1 −β 2 h2(d) f(x, y) + αh1(b) − h1(a) 2 βh2(d) − h2(c) 2 f(a, d) − αh1(b) − h1(a) 2 h₂(y) −β 2h2(c) − 1 −β 2 h₂(d) f(a, y) − x Z a d Z y w₁(t)w2(s) f (t, s)dsdt, (3.3)

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I₃= b Z x y Z c h₁(t) −hα 2h1(a) + 1 −α 2 h₁(b)i × h₂(s) − 1 −β 2 h₂(c) +β 2h2(d) d_sd_tf(t, s) = αh1(b) − h1(a) 2 h₂(y) − 1 −β 2 h₂(c) −β 2h2(d) f(b, y) + αh1(b) − h1(a) 2 βh2(d) − h2(c) 2 f(b, c) −hh₁(x) −α 2h1(a) − 1 −α 2 h₁(b)i h₂(y) − 1 −β 2 h₂(c) −β 2h2(d) f(x, y) −hh1(x) − α 2h1(a) − 1 −α 2 h1(b) i βh2(d) − h2(c) 2 f(x, c) − b Z x y Z c w1(t)w2(s) f (t, s)dsdt, (3.4) and I4= b Z x d Z y h1(t) − h_α 2h1(a) + 1 −α 2 h1(b) i × h2(s) − β 2h2(c) + 1 −β 2 h2(d) dsdtf(t, s) = αh1(b) − h1(a) 2 βh2(d) − h2(c) 2 f(b, d) − αh1(b) − h1(a) 2 h2(y) − β 2h2(c) − 1 −β 2 h2(d) f(b, y) −hh₁(x) −α 2h1(a) − 1 −α 2 h₁(b)i βh2(d) − h2(c) 2 f(x, d) +hh₁(x) −α 2h1(a) − 1 −α 2 h₁(b)i h₂(y) −β 2h2(c) − 1 −β 2 h₂(d) f(x, y) − b Z x d Z y w₁(t)w2(s) f (t, s)dsdt. (3.5)

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Adding (3.2)-(3.5), we have b Z a d Z c q(t)p(s)dsdtf(t, s) =   b Z a w(t)dt     d Z c g(t)dt   (1 − α) (1 − β ) f (x, y) + (1 − α) β f(x, c) + f (x, d) 2 +α (1 − β ) f(a, y) + f (b, y) 2 + αβ f(a, c) + f (a, d) + f (b, c) + f (b, d) 4 − b Z a d Z c w1(t)w2(s) f (t, s)dsdt.

On the other hand, using Lemma 2.2, we have b Z a d Z c q(t)p(s)dsdtf(t, s) ≤ sup t∈[a,b] |q(t)| sup s∈[c,d] |p(s)| b _ a d _ c ( f ) = max{h1(x) − [(1 − α 2)h1(a) + α 2h1(b)], [ α 2h1(a) + (1 − α 2)h1(b)] − h1(x), α 2[h1(b) − h1(a)]} × max{h₂(y) − [(1 −β 2)h2(c) + β 2h2(d)], [ β 2h2(c) + (1 − β 2)h2(d)] − h2(y), β 2[h2(d) − h2(c)]} b _ a d _ c ( f ) = max 1 − α 2 [h1(b) − h1(a)] + h₁(x) −h1(a) + h1(b) 2 ,α 2[h1(b) − h1(a)] × max 1 − β 2 [h2(d) − h2(c)] + h2(y) − h₂(c) + h2(d) 2 ,β 2 [h2(d) − h2(c)] b _ a d _ c ( f ) = max    1 − α 2 b Z a w₁(t)dt + h₁(x) −h1(a) + h1(b) 2 ,α 2 b Z a w₁(t)dt    × max    1 − β 2 d Z c w2(t)dt + h2(y) − h2(c) + h2(d) 2 ,β 2 d Z c w2(t)dt    b _ a d _ c ( f ) = KL b _ a d _ c ( f ) .

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This completes the proof of inequality (3.1). Assume (α, β ) ∈0,1₂ × 0,1₂ . Suppose (3.1) holds with a constant A = A1.A2, A1, A2> 0, that is,

  b Z a w1(t)dt     d Z c w2(t)dt  [(1 − α) (1 − β ) f (x, y) + (1 − α) β f(x, c) + f (x, d) 2 + α (1 − β ) f(a, y) + f (b, y) 2 +αβ f(a, c) + f (a, d) + f (b, c) + f (b, d) 4 − b Z a d Z c w1(t)w2(s) f (t, s)dsdt ≤  A1 b Z a w1(t)dt + h1(x) − h₁(a) + h1(b) 2   ×  A2 d Z c w2(t)dt + h2(y) − h₂(c) + h2(d) 2   b _ a d _ c ( f ) . (3.6) If we choose f : Q → R with f(t, s) =    1, if (t, s) =h₁h1(a)+h1(b) 2 , h2 _h 2(c)+h2(d) 2 , 0, if (t, s) ∈ [a, b] × [c, d] \nh₁h1(a)+h1(b) 2 , h2 _h 2(c)+h2(d) 2 o ,

then f is of bounded variation on Q. For (x, y) =h₁h1(a)+h1(b)

2 , h2 _h 2(c)+h2(d) 2 , we have β f(x, c) + f (x, d) 2 = 0, f(a, y) + f (b, y) 2 = 0, f(a, c) + f (a, d) + f (b, c) + f (b, d) 4 = 0 b Z a d Z c w₁(t)w2f(t, s)dsdt = 0, and _ Q ( f ) = 4.

Putting this equalities in (3.6), we get   b Z a w₁(t)dt     d Z c w₂(t)dt  (1 − α) (1 − β ) ≤ 4   b Z a w₁(t)dt     d Z c w₂(t)dt  A₁A₂.

It follows that A ≥(1−α)(1−β )₄ which implies (1−α)(1−β )₄ is the best possible.

The sharpness of inequality (3.1) for α, β ∈2₃, 1 can be easily proved by choosing the function f f(t, s) =    1, if (t, s) = (b, d) , 0, if (t, s) ∈ [a, b] × [c, d] \ {(b, d)} . This completes the proof.

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Remark 3.2. If we choose w(t) ≡ g(s) ≡ 1 (h₁(t) = t and h₂(s) = s), and α = β = 0, then the inequality (3.1) reduces the inequality (2.5).

Remark 3.3. If we choose w(t) ≡ g(s) ≡ 1 (h1(t) = t and h2(s) = s), α = β =1₃, x =a+b₂ and

y= c+d₂ in inequality (3.1), then we have the Simpson’s inequality

f(b, d) + f (b, c) + f (a, d) + f (a, c) 36 +f a, c+d 2 + f a+b 2 , c + f b, c+d 2 + f a+b 2 , d 9 +4 9f a + b 2 , c+ d 2 − 1 (b − a) (d − c) b Z a d Z c f(t, s)dsdt ≤ 1 9 b _ a d _ c ( f ),

which is proved by Jawarneh and Noorani in [18].

Remark 3.4. If we choose w1(t) ≡ w2(s) ≡ 1 (h1(t) = t and h2(s) = s), α = 1, β = 0, x = a+b₂

and y = c+d₂ in inequality (3.1), then we have

f a,c+d₂ + f b,c+d₂ 2 − 1 (b − a) (d − c) b Z a d Z c f(t, s)dsdt ≤ 1 4 _ Q ( f ),

which is given by Budak and Sarikaya in [10]. The constant 1₄ is the best possible.

Remark 3.5. If we choose w1(t) ≡ w2(s) ≡ 1 (h1(t) = t and h2(s) = s), α = 0, β = 1, x = a+b₂

and y = c+d₂ in inequality (3.1), then we have

f a+b₂ , c + f a+b₂ , d 2 − 1 (b − a) (d − c) b Z a d Z c f(t, s)dsdt ≤ 1 4 _ Q ( f ),

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Remark 3.6. If we choose w1(t) ≡ w2(s) ≡ 1 (h1(t) = t and h2(s) = s), α = β = 1₂, x = a+b₂

and y = c+d₂ in the inequality (3.1), then we have

1 4 f (a, c) + f (a, d) + f (b, c) + f (b, d) 4 + 1 2 f a,c+ d 2 + f b,c+ d 2 + f a + b 2 , c + f a + b 2 , d + f a + b 2 , c+ d 2 − 1 (b − a) (d − c) b Z a d Z c f(t, s)dsdt ≤ 1 16 _ Q ( f ), (3.7)

which is given by Budak and Sarikaya in [6]. The constant ₁₆1 is the best possible.

Corollary 3.7. (Weighted Ostrowski Inequality) Under the assumption Theorem 3.1, if we choose α = β = 0, for all (x, y) ∈ [a, b] × [c, d], then we have the following weighted Ostrowski inequality   b Z a w₁(t)dt     d Z c w₂(t)dt  f(x, y) − b Z a d Z c f(t, s)dsdt ≤   1 2 b Z a w₁(t)dt + h₁(x) −h1(a) + h1(b) 2     1 2 d Z c w₂(t)dt + h₂(y) −h2(c) + h2(d) 2   b _ a d _ c ( f ).

Corollary 3.8. (Weighted Trapezoid Inequality) Under the assumption Theorem 3.1, if we choose α = β = 1, then we have the following weighted trapezoid inequality;

f(b, d) + f (b, c) + f (a, d) + f (a, c) 4   b Z a w₁(t)dt     d Z c w₂(t)dt  − b Z a d Z c f(t, s)dsdt ≤ 1 4   b Z a w₁(t)dt     d Z c w₂(t)dt   b _ a d _ c ( f ).

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Corollary 3.9. (Weighted Simpson’s Inequality) Under assumption Theorem 3.1, if we choose α = β = 1₃, x = h−1₁ _h 1(a)+h1(b) 2 and y= h−1₂ h2(c)+h2(d) 2

, then we have the weighted Simp-son’s inequality   b Z a w1(t)dt     d Z c w2(t)dt   f (b, d) + f (b, c) + f (a, d) + f (a, c) 36 + f a, c+d 2 + f a+b 2 , c + f b, c+d 2 + f a+b 2 , d 9 + 4 9f a + b 2 , c+ d 2 − b Z a d Z c f(t, s)dsdt ≤1 9   b Z a w1(t)dt     d Z c w2(t)dt   b _ a d _ c ( f ).

4. Applications for qubature formulae

Now, we apply the results presented previous section to qubature formulae.

Let us consider the arbitrary division In: a = x0< x1< ... < xn= b, Jm: c = y0< y1< ... <

y_m= d, l₁i := xi+1− xi, and l₂j:= yj+1− yj, υ (l1) := max l1i i= 0, ..., n − 1 , υ (l2) := max n l₂j j= 0, ..., m − 1 o , υ (W1) := max W1i i= 0, ..., n − 1 , W₁i:= x_i+1 Z xi w₁(u)du = h1(xi+1) − h1(xi), υ (W2) := max n W₂j j= 0, ..., m − 1 o , W₂j:= yj+1 Z yj w2(u)du = h2(yj+1) − h2(yj).

Let us have α, β , w1, h1, w2, and h2defined as in Theorem 3.1. Let ai₁= h−1₁ ((1 −α₂)h1(xi) + α 2h1(xi+1)), b i 1= h −1 1 α2h1(xi) + 1 − α 2 h1(xi+1) , c j 1= h −1 2 1 −β₂ h2(yj+1) +β₂h2(yj+1) and d₁j= h−1₂ β₂h2(yj) + 1 −β₂h2(yj+1) , ξi∈ai₁, bi₁ , (i = 0, ..., n − 1) and ηj∈ h c₁j, d₁ji ( j = 0, ..., m − 1).

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Define the sum A( f , w1, h1, w2, h2, In, Jm) = n−1

∑

i=0 m−1

∑

j=0 (1 − α) (1 − β ) f ξi, ηj + (1 − α) β f(ξi, yj) + f (ξi, yj+1) 2 + α (1 − β ) f(xi, ηj) + f (xi+1, ηj) 2 +αβ f(xi, yj) + f (xi, yj+1) + f (xi+1, yj) + f (xi+1, yj+1) 4 W₁iW₂j.

Theorem 4.1. Let f defined as in Theorem 3.1 and let

b Z a d Z c w₁(t)w2(s) f (t, s)dsdt = A( f , w1, h1, w2, h2, In, Jm) + R( f , w1, h1, w2, h2, In, Jm).

The remainder term R( f , w₁, h₁, w₂, h₂, I_n, J_m) satisfies

|R( f , w1, h1, w2, h2, In, Jm)| ≤ n−1

∑

i=0 m−1

∑

j=0 KiLj xi+1 _ xi yj+1 _ yj ( f ) ≤ M1N1 b _ a d _ c ( f ) ≤ M2N2 b _ a d _ c ( f ) ≤ M3N3 b _ a d _ c ( f ), (4.1) where K_i=          1−α 2 W i 1+ h1(ξi) − h1(xi)+h1(xi+1) 2 , if0 ≤ α ≤ 1 2, maxn1−α₂ W₁i+ h1(ξi) − h1(xi)+h1(xi+1) 2 , α 2W i 1 o , if 1₂ < α <2₃ α 2W i 1, if 2 3 ≤ α ≤ 1, (i = 0, ..., n − 1) , M₁=            max i=0,...,n−1 n 1−α 2 W1i+ h1(ξi) − h1(xi)+h1(xi+1) 2 o , if0 ≤ α ≤1₂, max i=0,...,n−1 n max n 1−α 2 υ (W1) + h1(ξi) − h1(xi)+h1(xi+1) 2 , α 2υ (W1) oo , if 1₂< α < 2₃, α 2υ (W1), if 2 3≤ α ≤ 1, M₂=            1−α 2 υ (W1) +_{i=0,...,n−1}max h1(ξi) − h1(xi)+h1(xi+1) 2 , if0 ≤ α ≤ 1 2, max i=0,...,n−1 n maxn1−α₂ υ (W1)dt + h1(ξi) − h1(xi)+h1(xi+1) 2 , α 2υ (W1) oo , if 1₂< α < 2₃, α 2υ (W1), if 2 3≤ α ≤ 1, M₃=    (1 − α) υ(W1) if 0 ≤ α ≤2₃, α 2υ (W1), if 2 3≤ α ≤ 1,

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and similarly L_i=          1−β 2 W j 2 + h2(ηj) − h2(yj)+h2(yj+1) 2 , if0 ≤ β ≤ 1 2, max n 1−β 2 W j 2+ h2(ηj) − h2(yj)+h2(yj+1) 2 , β 2W j 2 o , if 1₂< β < 2₃, β 2W j 2, if 23≤ β ≤ 1. ( j = 0, ..., m − 1) , N₁=            max j=0,...,m−1 n 1−β 2 W j 2+ h2(ηj) − h2(yj)+h2(yj+1) 2 o , if0 ≤ β ≤1₂, max j=0,...,m−1 n max n 1−β 2 υ (W2) + h2(ηj) − h2(yj)+h2(yj+1) 2 , β 2υ (W2) oo , if 1₂ < β < 2₃, β 2υ (W2), if 2 3 ≤ β ≤ 1, N2=            1−β 2 υ (W2) +_{j=0,...,m−1}max n h2(ηj) − h2(yj)+h2(yj+1) 2 o , if0 ≤ β ≤1₂, max j=0,...,m−1 n maxn1−β₂ υ (W2) + h2(ηj) − h2(yj)+h2(yj+1) 2 , β 2υ (W2) oo , if 1₂ < β < 2₃, β 2υ (W2), if 2 3 ≤ β ≤ 1, N₃=    (1 − β ) υ(W2), if 0 ≤ β ≤ 2₃, β 2υ (W2), if 2 3≤ β ≤ 1.

Proof. Applying Theorem 3.1 to the bidimentional interval [xi, xi+1] ×yj, yj+1 , we have

(1 − α) (1 − β ) f ξi, ηj + (1 − α) β f(ξi, yj) + f (ξi, yj+1) 2 + α (1 − β ) f(xi, ηj) + f (xi+1, ηj) 2 +αβ f(xi, yj) + f (xi, yj+1) + f (xi+1, yj) + f (xi+1, yj+1) 4 W₁iW₂j − xi+1 Z xi yj+1 Z yj w1(t)w2(s) f (t, s)dsdt ≤ KiLj xi+1 _ xi yj+1 _ yj ( f ) (4.2)

for any ξi∈ai₁, bi₁ , (i = 0, ..., n − 1) and ηj∈

h

c₁j, d₁ji( j = 0, ..., m − 1) . Summing inequality (4.2) over i from 0 to n − 1 and j from 0 to m − 1 and using the generalized triangle inequality,

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we get |R( f , w1, h1, w2, h2, In, Jm)| ≤ n−1

∑

i=0 m−1

∑

j=0 K_iL_j xi+1 _ xi yj+1 _ yj ( f ) ≤ max i=0,...,n−1Ki max j=0,...,m−1Lj n−1

∑

i=0 m−1

∑

j=0 xi+1 _ xi yj+1 _ yj ( f ) = M1N1 b _ a d _ c ( f ) ≤ M2N2 b _ a d _ c ( f ).

This completes the proof of the first three inequalities in (4.1). In inequality (4.3), we observe that h1(ξi) − h1(xi)+h1(xi+1) 2 ≤ 1−α 2 W1i. Hence, we have max i=0,...,n−1 h1(ξi) − h1(xi) + h1(xi+1) 2 ≤ 1 − α 2 υ (W1). Similarly, we obtain max j=0,...,m−1 h2(ηj) − h₂(yj) + h2(yj+1) 2 ≤1 − β 2 υ (W2). These show that M2≤ M3and N2≤ N3. This completes the proof.

Remark 4.2. If we choose α = β = 0, w1(t) ≡ 1, h1(t) = t on [a, b] and w2(s) ≡ 1, h2(s) = s

on [c, d] in Theorem 4.1, then inequalities (4.1) reduce to the inequality (4.2) in [5].

Remark 4.3. If we choose α = β = 1₃, w(t) ≡ 1, h1(t) = t on [a, b] and g(s) ≡ 1, h2(s) = s on

[c, d] , ξi= xix₂i+1 (i = 0, ..., n − 1) and ηj=

yj+yj+1

2 ( j = 0, ..., m − 1) in Theorem 4.1, then we

have the Simpson’s sum

A_S( f , In, Jm) = 4 9 n−1

∑

i=0 m−1

∑

j=0 f xi+ xi+1 2 , yj+ yj+1 2 l₁il₂j +1 9 n−1

∑

i=0 m−1

∑

j=0 f xi+ xi+1 2 , yj + f xi+ xi+1 2 , yj+1 l₁il₂j +1 9 n−1

∑

i=0 m−1

∑

j=0 f xi, y_j+ yj+1 2 + f xi+1, y_j+ yj+1 2 l₁il₂j + 1 36 n−1

∑

i=0 m−1

∑

j=0 f (xi, yj) + f (xi, yj+1) + f (xi+1, yj) + f (xi+1, yj+1) l1il j 2 with b Z a d Z c f(t, s)dsdt = A_S( f , In, Jm) + RS( f , In, Jm)

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and the remainder term RS( f , In, Jm) satisfies |RS( f , In, Jm)| ≤ 1 9υ (l1)υ(l2) b _ a d _ c ( f ),

which was given by Budak and Sarikaya in [10].

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