On the bivariate Baskakov-Durrmeyer type operators

* Grażyna Krech (gkrech@up.krakow.pl), Renata Malejki, Institute of Mathematics, Pedagogical University of Cracow.

FUNDAMENTAL SCIENCES

1-NP/2016

NAUKI PODSTAWOWE

GRAŻYNA KRECH*, RENATA MALEJKI*

ON THE BIVARIATE BASKAKOV-DURRMEYER TYPE OPERATORS

O OPERATORACH DWÓCH ZMIENNYCH TYPU BASKAKOWA-DURRMEYERA

A b s t r a c t

In this paper we introduce some linear positive operators of the Baskakov-Durrmeyer type in the space of continuous functions of two variables. The theorems on convergence and the degree of approximation are established.

Keywords: Baskakov-Durrmeyer type operators, linear operators, approximation order S t r e s z c z e n i e

W artykule definiuje się dodatnie operatory liniowe typu Baskakowa-Durrmeyera w przestrze- ni ciągłych funkcji dwóch zmiennych. Formułuje się i dowodzi twierdzenia dotyczące zbież- ności oraz rzędu zbieżności.

Słowa kluczowe: operatory typu Baskakowa-Durrmeyera, operatory liniowe, rząd aproksy- macji

DOI: 10.4467/2353737XCT.16.142.5753

1. Introduction

In recent years, several researchers have studied various modifications of the Baskakov- -Durrmeyer operators. The approximation properties of these operators in many different spaces were considered, for example, in [4, 8, 10, 11, 18, 19].

A large amount of literature is available on approximation of function of one variable, but the corresponding problem for bivariate functions has received less attention. The bivariate Bernstein operator was first introduced by Dhingas [3] and it was also considered by Lorentz [9] and Stancu [14]. Recently, some positive linear operators for function of two variables and their approximation properties were investigated in a series of research articles (e.g.

[2, 5, 6, 7, 12, 13, 15, 17, 20, 21].

In this paper, we will introduce the Baskakov-Durrmeyer type operators in the space of continuous functions of two variables. This is an extension of the paper [10] for a bivariate case.

Let 0 2  

0 0

= [0, ) =

+ ¥ and + +× +. We denote by C( )²+ the space of all real-valued functions continuous on +2 and by C_B(²+) ‒ the space of functions continuous and bounded on +2. The norm on C_B(²+) is defined by

f _C f x y

B( ) x y ( , )

2 = 2 ( , ) .

⁺ sup∈+

Let

W x e k

i n a x

k x

n ka ax

x i

k

i k i k

, 1 n k

=0

( ) = ( )

!(1 ) ,

−+ −

∑

^_^{ }_^ ₊ +

where a∈⁺₀, ( )n₀=1, ( )n_i =n n( +1) (… n i+ −1), i³1.

We consider the class of operators M_{n m}^{α β}_,^{, , ,a b} given by the formula

M f x y mn W x W y

n ma b k

k l n ka m lb ,, , ,

, =0 , ,

( ; , ) = ( ) ( ) 1

( 1)

1 (

α β

α β

∑

¥ _{Γ + + Γ +ll}

e ^ns ns ^ke ^mz mz ^l f s z dsdz +

×

∫ ∫

^{− + − +}

1)

( ) ( ) ( , )

0 0

¥ ¥ α β

for ( , )x y ∈ ,²+ where m n, ∈, ,a b∈ α β >₀⁺, , −1. It is clear that the operator M_{n m}^{α β}_,^{, , ,a b} is linear and positive on +2. In this paper we study some approximation properties of

M_{n m}^{α β}_,^{, , ,a b} in the space of continuous functions of two variables on a compact set. We find the order of this approximation using full and partial modulus of continuity.

Observe that if f s z( , ) = ( ) ( )f s f z_{1 2} , then

M_{n m}^{α β}_,^{, , ,a b}( ; , ) =f x y M_n^α,a( ; )f x M_{1 m}^β,b( ; ),f y₂ (1.1)

where

M f x n W x

k e ns f s ds

n a

k n ka ns k

α α

α

, 1

=0 , 0 1

( ; ) = ( ) 1

( 1) ( ) ( ) .

¥ ¥

∑

_{Γ + +}

∫

^{− +}

Some properties of the operator M_n^α,a, in particular, an estimation of the rate of convergence, were studied in [10].

Let ( , )x y ∈²+ and

e s z^{i j,} ( , ) =s z^{i j}, φ^{i j}_{x y}^,_, ( , ) = (s z s x z y− ) (ⁱ − ) ,^j i j, = 0 1,2,4, , ( , )s z ∈∈₊².

Now, we give some lemmas which will be useful in the future proofs of the main results.

The following lemmas are simple consequences of the above definitions and the results obtained in [10, Lemma 2.2, Lemma 2.3].

Lemma 1. Let m n, ∈, ,a b∈₀⁺, ,α β >−1. For ( , )x y ∈²₊ we get

M_{n m}^{α β}_,^{, , ,a b}(e^0,0; , ) = 1,x y (1.2)

M e x y

n x ax

n x

n ma b

,, , , ( 1,0; , ) = 1

(1 ),

α β α +

+ + + (1.3)

M e x y

m y by

m y

n ma b

,, , , ( 0,1; , ) = 1

(1 ),

α β β +

+ + + (1.4)

M e x y

n

x x

n x a x

n ma b n

,, , , 2,0

2

2 2 2 2

( ; , ) = ( 1)( 2) 2( 2) 2

(1

α β α+ α+ α

+ + +

+ +

++

+ + + +

+

x ax

n x

ax

n x

) 2

(1 )

2( 2) (1 ) ,

2 2

2

α (1.5)

M e x y

m

y y

m y b y

n ma b m

,, , , 0,2

2

2 2 2 2

( ; , ) = ( 1)( 2) 2( 2) 2

(1

α β β+ β+ β

+ + +

+ +

++

+ + + +

+

y by

m y

by

m y

) 2

(1 )

2( 2) (1 )

2 2

2

β .

(1.6)

Lemma 2. Let m n, ∈, ,a b∈₀⁺, ,α β >−1. For ( , )x y ∈²₊ we get

M x y

n

ax

n x

n ma b

,, , , ( 1,0x y; , ) = 1

(1 ),

α β φ α

, + +

+

M x y

m

by

m y

n ma b

,, , , ( 0,1x y; , ) = 1

(1 ),

α β φ β

, + +

+

M x y

nm

by nm y

n ma b ,, , , x y

1,1,

( ; , ) = ( 1)( 1) ( 1) (1 )

( 1)

α β φ α+ β+ + α+ β

+ + + aax nm x

abxy

nm x y

(1 ) (1 )(1 ),

+ +

+ +

M x y

n

x x n

a x

n x

n ma b x y ,, , , 2,0

2

2 2 2

2 2

( ; , ) = ( 1)( 2) 2

(1 )

α β φ α α

, + +

+ +

+ + + 22( 2)

(1 ) ,

2

α + +

ax

n x

M x y

m

y y m

b y

m y

n ma b ,, , , x y0,2

2

2 2 2

2 2

( ; , ) = ( 1)( 2) 2

(1 )

α β φ β β

, + + + + +

+ + 22( 2) (1 ).

2

β + +

by

m y

Theorem 1. For each f C∈ _B( ,²₊) we have M_{n m}^{a b} f f

C_B C_B

,, , ,

( ) ( )

( ) ₂ 2

α β

₊ £ +

for all n m, ∈ .

Proof. Using the definition M_{n m}^{α β}_,^{, , ,a b}, we obtain

M f x y mn W x W y

n ma b k

k l n ka m lb ,, , ,

, =0 , ,

( ; , ) ( ) ( ) 1

( 1)

1 (

α β

α β

≤

∑

^∞ _{Γ + + Γ +ll}

e ns e mz f s z dsdz f

ns k mz l

s z

+

×

∫ ∫

^{− + − +}

∈ ₊

1)

( ) ( ) ( , )

0 0

2

∞ ∞

≤

α β

sup

( , )  (( , ) ( ) ( ) 1

( 1)

1

( 1)

, =0 , ,

0 0

s z mn W x W y

k l

e

k l n ka m lb

∞

∞ ∞

∑

∫ ∫

+ + + +

× ⁻

Γ α Γβ

nns k mz l

s z n ma b

ns e mz dsdz f s z M e

( ) ( )

= ( , ) (

2 ,, , , 0

α β

α β

+ − +

∈ ₊ ( , ) sup

,,

( , )sup ,

0; , ) ( , ) x y 2 f s z f

s z

= =

∈₊

which gives the result. 

Theorem 2 [22]. Let I₁ and I₂ be compact intervals of the real line. Let n m, ∈  and T_{n m,} : (C I I₁× ₂)→C I I( ₁× ₂) be linear positive operators. If

n m Tn m ei j ei j i j

,

, ,

( ) = , , { 0,0 1 0 0 1 }

→ ∈

lim¥ _, ( ) ( ),( , ),( , ) and

n m Tn m e e e e

,

2,0 0,2 2,0 0,2

( ) =

→ + +

lim¥ _, ,

uniformly on I₁ × I₂, then the sequence (T_n,m f) converges to f uniformly on I₁ × I₂, for any f C I I∈ (₁× ₂).

Let A,B > 0. Throughout the rest of this paper we will denote ²AB= [0, ] [0, ]A× B. Theorem 3. Let ( , )x y ∈ ²_AB are fixed. If f C∈ (²_AB), then

n m Mn ma b f x y f x y

, ,, , , ( ; , ) = ( , )

lim→¥ ^{α β} .

Moreover, this convergence is uniform on 2AB. Proof. Using (1.2)‒(1.6), we have

n m Mn ma b ei j x y ei j x y i j

, ,, , , ( , ; , ) = , ( , ), , { 0,0 1 0

→ ∈

lim¥ ^{α β} ( ) ( ),( , ),,( , )0 1 } and

n m Mn ma b e e x y e x y e x y

, ,, , , ( 2,0 0,2; , ) = 2,0( , ) 0,2( , )

→ + +

lim¥ ^{α β}

uniformly on ²_AB. Applying Theorem 2, the proof of the theorem is completed. 

2. Local approximation results

In this section we will investigate the degree of approximation for functions of two variables by operators M_{n m}^{α β}_,^{, , ,a b} in terms of the modulus of continuity on a compact set.

Let f C∈ (²_AB) and δ > .0 The full continuity modulus of the function f is defined as (see [1], [16])

ω δ

δ

( ; ) = ( , ) ( , )

( ( , ),( , ))2 ( )2² 2

f f s z f x y

s x z y

s z x y _AB

− + −

∈

−

£



sup

and its partial continuity moduli are given by

ω δ

δ (1)

|0 |

( ; ) =f f s z( , ) f x z( , ) ,

s xz B

−

−

£ £sup£

ω δ

δ (2)

|0 |

( ; ) =f f s z( , ) f s y( , ) .

z ys A

−

−

£ £sup£

It is known that limδ_→0ω( ; ) = 0 ( ; )f δ , ω f δ₁ ≤ω( ; )f δ₂ for 0<δ₁≤δ₂ and for any λ>0 ( ; ) (1, ω f λδ ≤ +λ ω) ( ; )f δ. The same properties are satisfied by partial continuity moduli. The details of the modulus of continuity for the bivariate case can be found in [1].

Theorem 4. Let f C∈ (²_AB).For x y( , )∈²_AB, we have M_{n m}^{α β}_,^{, , ,a b}( ; , )f x y − f x y( , ) £2 ( ; ),ω f δ where

δ α α α

β

= ( 1)( 2) 2

(1 )

2( 2) (1 ) (

2

2 2 2

2 2 2

+ +

+ +

+ +



 + +

+

+ +

n

x x n

a x

n x

ax

n x

11)( 2) 2

(1 )

2( 2) (1 )

2

2 2 2

2 2 2

β+ β 1/2

+ +

+ + + +

+





m

y y m

b y

m y

by

m y .

Proof. Let δ > 0. If (s x− )²+ −(z y)² £ δ, then f s z( , )−f x y( , ) £ ω( ; ).f δ If (s x− )²+ −(z y)²> δ, then

(s x− )²+ −₂(z y)² (s x− )²+ −(z y)² 1.

δ > δ >

Therefore, we obtain

f s z f x y f s x z y

s x z y

( , ) ( , ) ; ( ) ( )

1 ( ) ( )

2 2

2 2

−

(

− + −

)

+ − + −













£

£

ω

δ ω ff;δ 1 (s x)² ₂(z y)² f; .

δ ω δ

( )

^{ + − + −}

 



( )

£

The operator M_{n m}^{α β}_,^{, , ,a b} is positive and linear, so

M f x y f x y M f f x y x y

f

n ma b

n ma b ,, , ,

,, , ,

( ; , ) ( , ) ( , ) ; ,

( ; )

α β α β

ω δ

− ^£

(

−

)

£ MM_{n m}^{α β}_,^{, , ,a b}(e^0,0; , ) 1x y ₂M_{n m}^{α β}_,^{, , ,a b}( ^2,0_{x y} ^0,2_{x y}; , )x y

δ φ φ

+ _, + _,



 

.

From Lemma 2 we obtain

M f x y f x y M f f x y x y

f

n ma b

n ma b ,, , ,

,, , ,

( ; , ) ( , ) ( , ) ; ,

( ; )

α β α β

ω δ

− ^£

(

−

)

£ 11 1 ( 1)( 2) 2

(1 )

2( 2)

2 2

2 2 2

2 2 2

+ + +

+ +

+ +









 + +

δ

α α α

n

x x n

a x

n x

ax n ((1 ) ( 1)( 2) 2

(1 )

2( 2) (1 )

2

2 2 2

2 2 2

+

+ + + + + +

+ + +

+





x

m

y y m

b y

m y

by

m y

β β β







,

which ends the proof. 

Theorem 5. If f C∈ (²_AB), then for all ( , )x y ∈  ,²_AB we have M f x y f x y

n x x a x

n x

n ma b ,, , ,

2 2 2

2

( ; , ) ( , )

1 ( 1)( 2) 2

(1 )

α β

α α

−

+ + +

+ + +

+ +

£ 22( 2)

(1 ) ; 1

1 ( 1)( 2) 2

(1)

2

α ω

β β

+ +



 

 

 



+ + + +

+ + +

ax

n x f

n

m y y bb y

m y

by

m y f

m

2 2

2 (2)

(1 )

2( 2)

(1 ) ; 1 .

+ + +

+



 

 

 



β ω

Proof. Let f C∈ (²_AB). Observe that

M f x y f x y mn W x W y

n ma b k

k l n ka m lb ,, , ,

, =0 , ,

( ; , ) ( , ) ( ) ( ) 1

(

α β

− α

∑

+ +

≤ ^∞

Γ 11)

1

( 1)

( ) ( ) ( , ) ( , )

0 0

Γ β

α β

+ +

×

∫ ∫

^{− + − +} −

l e ^ns ns ^ke ^mz mz ^l f s z f x z dsdz

∞ ∞

++ + + + +

×

∑

∫ ∫

⁻

mn W x W y

k l

e n

k l n ka m lb ns

, =0 , ,

0 0

( ) ( ) 1

( 1)

1

( 1)

(

∞

∞ ∞

Γ α Γ β

ss e mz f x z f x y dsdz J J

k mz l

) ( ) ( , ) ( , )

= _{1 2}.

α+ − β+ −

+

Using the properties of the modulus of continuity and (1.5), we have

J mn W x W y

k l

e

k l n ka m lb ns

1

, =0 , ,

0 0

= ( ) ( ) 1

( 1)

1

( 1)

(

∞

∞ ∞

∑

∫ ∫

+ + + +

× ⁻

Γα Γβ

nns e mz f s z f x z dsdz

f M

k mz l

n n n m

) ( ) ( , ) ( , )

( ; ) 1 1

(1) 2 ,,

α β

ω δ α

δ

+ − + −

≤ + ^ββ φ

α α

, , 2,0

2 2 2

( ; , )

1 ( 1)( 2) 2

(1

a b x y x y

n x x a x

n

,













+ + +

+ + +

≤ +

xx

ax

n x f

n )

2( 2)

(1 ) ; 1 ,

2 + + (1)

+



 

 

 



α ω

where δ_n

= 1 . Similarly, we obtainn

J m y y b y

m y

by

m y

2 2 2 2

1 ( 1)( 2) 2 2

(1 )

2( 2) (1 )

£ + + +

+ + +

+ + +

+



 



β β β

ω⁽⁽²⁾ f; 1 . m



 



Hence, the proof is completed. 

Now, we consider the mixed modulus of smoothness and the modulus of smoothness (see [16]). Let δ_j > 0, j = 1,2.

The mixed modulus of smoothness is defined as

ω δ δ

δ δ

mix( ; , ) =_{1 2} ( , )

( , ),( , )

| | , | |

1 2 2

f f s z f

x y s z

s x z y

∈ AB

− − −



£sup £ (( , )x z −f s y( , )+ f x y( , ) and the modulus of smoothness of the first and the second order are given by

ω δ δ

δ δ

1 1 2

( , ),(0 , 0, )

( ; , ) = ( , )

1 22

f f x h y k

x y x h y kh k + + ∈ AB

+ +



£ £sup£ £ −−f x y( , ) ,

ω δ δ

δ δ

2 1 2

( , ),( 2 , 2 )0 , 0

( ; , ) = ( 2 ,

1 2 2

f f x h y

x y x h y kh k + + ∈ AB

+



£ £sup£ £ ++2 ) 2 (k − f x h y k+ , + )+ f x y( , ) , respectively.

Theorem 6. Let f C∈ (²_AB) and

H f x y M f x y f

n x ax

n x

n ma b

n ma b ,, , ,

,, , ,

( ; , ) = ( ; , ) 1

(1 ), 1

α β α β α β

− +

+ + +

+ m

m y by

m y f x y

+ + +



 

 + (1 ) ( , ).

There exists a positive constant C such that, for all ( , )x y ∈  ,²_AB we have

H g x y g x y C n

g

u m

g

n ma b v

C _AB C _AB

,, , , 2

2 ( )

2

2 ( )

( ; , ) ( , ) 1 1

2 2

α β − ∂

∂ + ∂

£ ∂

 

++ ∂

∂ ∂













1 ²

( ² )

nm g u v _C

AB

for any function g, such that g g x

g y

g x y i

i i

i

, ∂ , i,

∂

∂

∂

∂

∂ ∂

2 ( = 1,2) belong to C(²_AB).

Proof. Let ( , )x y ∈  .²_AB Observe that g s z g x y s x g x y

x z y g x y y s u g u

x s

( , ) ( , ) = ( ) ( , ) ( ) ( , )

( ) ² ( ,

− − ∂

∂ + − ∂

∂

+ − ∂

∫

_u ^{yy du} y ^{z v g x v}_v ^{dv g u v}_{u v} ^dvdu z

x s

y

) ( ) ( , ) z ( , ) .

2

2 2

2

∂ + − ∂

∂ + ∂

∫ ∫ ∫

∂ ∂

We have

H_{n m}^{α β}_,^{, , ,a b}(e^{0 0,} ; , ) = 1,x y H_{n m}^{α β}_,^{, , ,a b}(φ^1,0_{x y,} ; , ) =x y H_{n m}^{α β}_,^{, ,a,,b}(φ^0,1_{x y,} ; , ) = 0.x y Let

ξ^{i j}_g

x

s i

y

s z s u g u y z

u du z v g x v

, ( , ) = − ∂

∂



 

 − ∂

∫

^{( ) 2 ( , )2 }_^

∫

^{( ) 2}∂^{( , )}_vv^{2 dv}_^{j, ,i j=0 1,} and

ξ_g

x s

y

s z z g u v u v dudv

( , )= ∂ .

∫ ∫

²∂ ∂^{( , )} Hence

H_{n m}^{α β}_,^{, , ,a b}( ; , )g x y g x y− ( , ) =H_{n m}^{α β}_,^{, , ,a b}(ξ^{1 0}_g^, ; ,x y H)+ _{n m}^{α β}_,^{, ,a,,b}(ξ^{0 1}_g^,; ,x y H)+ _{n m}^{α β}_,^{, , ,a b}(ξ_g; , .x y)

Using the definition of H_{n m}^{α β}_,^{, , ,a b}, we can write H_{n m}^{a b} _g x y M_{n m}^{a b} _g x y

x

n x ax

n ,, , , 1 0

,, , , 1 0 1

(

; , = ; ,

α β α β

α

ξ ξ

( ^, ) ( ^, )

−

++ +

11 ) 2

2 ,, , , 1

1

(1 )

( , )

∫

+  ^{+ + +} + −

 

∂

∂

x

n ma b g

n x ax

n x u g u y

u du

M

α

α β ξ

≤ ( ^,,0 )

1

(1 ) 2

; , 1

(1 )

( , ) x y

n x ax

n x u g u y

x

n x ax

n x

+ + + +

+ −



 

∂

∂

++ +

∫

^α + ^α _uu ^du

g u y

u M x y

u v n ma b

x y

AB

2

( , ) 2

2 ,, , , 2,0

1 2

( , ) ; ,

1

2

≤

∈

∂

∂ +



sup ^{α β} (φ_, )

22

( , ) 1

(1 ) 1

( , ) 2

2

2

1 2

2 u v 2

C

AB

g u y

u n

ax

n x

C n g u

∈

∂

∂

+ + +



 



∂

∂



sup α

≤

((²AB)

and similarly, we get

H x y C

m g

n ma b v

g

C AB

,, , , 0 1

2 2

2 ( )

; , 1 ,

2

α β (ξ ^, )£ ∂

∂ _

H x y C

nm g

n ma b u v

g

C AB

,, , ,

3 2

( )

; , 1 ,

2

α β (ξ ) £ ∂

∂ ∂ 

where C₁,C₂,C₃ are positive constants. Hence H g x y g x y C

n g

x m

g

n ma b v

C _AB C _AB

,, , , 2

2 ( )

2

2 ( )

( ; , ) ( , ) 1 1

2 2

α β − ∂

∂ + ∂

£ ∂

 

++ ∂

∂ ∂













1 ²

( ² )

nm g u v _C

AB

for some C > 0 and the theorem is proved. 

Theorem 7. If f C∈ (²_AB), then

M f x y f x y C f

n m f

n ma b n m

mix ,, , ,

( ; , ) ( , ) 2 ; 1 , 1 ; 1 , 1

α β − ω  ω

 

 + 

£  









+ + +

+



 

 + + +



 





 



ω₁ ; 1 α 1 β

1 , 1 1

f 1 n

ax x m

by y





, where C > 0, ( , )x y ∈  .²_AB

Proof. Let f C∈ (²_AB) and δ_j > 0, j = 1,2. We shall use the Steklov function of second order defined by

f_{δ δ} x y f x s s y z z

δ δ δ δ

δ δ

1 2

2 2 1 1

( , ) = 16 2 ( , )

12 22 02

0 2

0 2

0

2 1 2 1 2

∫ ∫ ∫ ∫

^{+ + + +}

−− f x( +2(s s y₁+ ₂), +2(z z ds ds dz dz₁+ ₂)) _{1 2 1 2}. Observe that

f_{δ δ} x y f x y ω f δ δ

1 2( , )− ( , )£ ₂( ; , )_{1 2} and

f x y f s y z z dsd

x x

u u δ δ

δ δ δ δ

δ δ

1 2

2 2 1 1

( , ) = 32 ( , )

12 22 02

0

2 2 2

1 2

∫ ∫ ∫ ∫

^{+ + + + uudz dz}

f s y z z dsdud

x x

u u

1 2

12 22 02

0

2 1 2

4 ^{2 2 1 1} ( , 2( ))

−_{δ δ}

∫ ∫ ∫ ∫

^{δ δ +δ +δ} + + ^{zz dz} f x s s w ds ds d

y y

v v

1 2

12

22 2 2

0 2

0

2 1 2 1 2

= 32 ^{2 2 1 1} ( , )

δ δ

δ δ δ δ

+ +

∫ ∫ ∫ ∫

^{+ + wwdv}

f x s s w ds ds dwdv

y y

v

− ₁²⁴₂²

∫ ∫ ∫ ∫

⁺ v⁺ 0 ⁽ +² +^{2 , )}

2 0

2 1 2 1 2

2 2 1 1

δ δ

δ δ δ δ

== 32 ( , )

4

12

22 02 2 0

2 2

2 2 2 2

2 2 1 1

δ δ

δ δ δ δ

∫ ∫ ∫ ∫

^{+ + + +}

−

y y

x

x f u s v z duds dvdz

δδ δ

δ δ δ δ

12 22 02

0

2 2 2 2 2

2 2 1

1 ( 2 , 2 )

∫ ∫ ∫ ∫

y^{+ + + +} y

x

x f u s v z duds dvdz

Hence

∂

∂

[

+ + +

− +

2

∫ ∫

2 12

22 02 0

2 1 1 2

1

1 2

2 2

( , ) = 32 ( , )

2 2

x f x y f x y z z

f x

δ δ

δ δ

δ δ δ

δ ,, ( , )

4 (

1 2 1 2 1 2

12 22 02

0

2 22

y z z f x y z z dz dz

f x

 + +

 

 + + + 



−_{δ δ}

∫ ∫

^{δ δ}

^[

^{++ + +}

− + + + + + +

]

2 , 2( ))

2 ( , 2( )) ( , 2( ))

1 1 2

1 1 2 1 2 1

δ δ

y z z

f x y z z f x y z z dz ddz₂

and ∂

∂



 

 +

2

2 12 2 1 2

12 2 1 2

12

1 2( , ) 8 ; 2,

2

1 ; , 9

x fδ δ x y f f

δ ω δ δ

δ ω δ δ

≤ ( ) ≤δ ωω₂(f; ,δ δ_{1 2}). .

Similarly, we get

∂

∂

2

2 _{1 2}( , ) 922 2 ; ,1 2 , y f_{δ δ} x y f

δ ω δ δ

£ ( )

∂

∂ ∂² ∈

1 2 1 2 2

1 2( , ) 9 ; , , ( , ) .

x y fδ δ x y £δ δ ω_mix(f δ δ ) x y _AB From the above and by Theorem 6, we obtain

M f x y f x y

H f f x y H

n ma b n ma b

n m ,, , ,

,, , ,

,,

( ; , ) ( , )

1 2 ; ,

α β

α β δ δ α β

−

(

−

)

⁺

£ ^{,, ,} ( ; , ) ( , ) ( , ) ( , )

1 (

1 2 1 2 1 2

a b f x y f x y f x y f x y

f n x ax

n

δ δ δ δ δ δ

α

− + −

+ + + +

11 ), 1

(1 ) ( , ) ( ; , ) ( ; ,

2 1 2 1

+

+ + + +



 

 − +

x m y by

m y f x y

C f f

β

ω δ δ ω δ

£ _mix δδ ω α β

2) 1 ; 1

(1 ), 1

(1 ) ,

+ + +

+

+ + +



 











 f 

n

ax

n x m

by

m y

where C is a positive constant. This completes the proof. 

The authors would like to thank the referees for their helpful remarks which improved the exposition of the paper.

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