IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-3008, p-ISSN:2319-7676. Volume 10, Issue 2 Ver. II (Mar-Apr. 2014), PP 97-105 www.iosrjournals.org

The Best Growth and Approximation of Entire Functions of Two Complex Variables in Banach Spaces Dr.Deepti Goyal, Dr.D.P.Singh RKGIT, Ghaziabad ,DIT School of engineering Gr.Noida

Abstract: In this Paper we are studding the polynomial approximation of entire functions of two complex

) order of entire variables in Banach spaces; concept is depend on index-pair. The characterizations of ( functions of two complex variables have been studied in terms of approximation errors. The results can be extended to m-variables but to reduce the mechanical labour we have considered only two variables. Key Words: Approximation error, order, type, entire function, index-pair.

I.

Introduction:

) ∑ Let ( { ( )} be a function of the complex variables z1 and z 2 , regular ) represents an entire function of for | | If and can be taken arbitrarily large, then ( ) as complex variables z1 and z 2 . Following Bose and Sharma [1] we define the maximum modulus of ( ( The order

)

|

| (

|

)|

 of the entire function (

) is defined as [1]; (

(

) )

Bose and Sharma [1], obtained the following characterizations for order of entire functions of two complex variables. Theorem 1.1. The entire function (

)

(

{

(

)

(|

|

)

) is equal to  . is finite and then the order  of ( Let denote the space of functions ( | | | | { } Such that ‖ ‖

) analytic in the unit bi-disc ( )

Where ( functions ( ‖ ‖

) 2 ∫ ∫ | ( )| 3 ) analytic in and satisfying the condition 2

∫|

|

∫|

|

)} is of finite order if and only if

| (

)|

3

and let

denote the space of

Set ‖ ‖

)| {| ( } and are Banach spaces for In analogy with spaces of functions of one variable, we call and the Hardy and Bergman spaces respectively. ) analytic in Following the Vakarchuk and Zhir [4] we say that the function ( belongs to the space ( ) where and if ‖ ‖

8∫ ∫ *( ‖ ‖

The space (

)(

)+

.

8*(

) is a Banach space for

/

)(

( )+. and

) /

9 (

)

9

otherwise it is a Frechet space. Further, we have

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97 | Page

The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach .

/

.

(1.1)

) Let Y is a Banach space and let by elements of ( ) be the best approximation of a function ( the space that consists of algebraic polynomials of degree in two complex variables. *‖ + ‖ ( ) Recently, Ganti and Srivastava [2] characterized the order and type in terms of the approximation errors ( ( ) But their results leave to study a big class of entire functions such as slow ( )) and ) order introduced growth and fast growth. To bridge this gap in this chapter we pick up the concept of ( by Juneja et al. [3] and consider it for entire functions of two variables. Roughly speaking, this concept is a modification of the classical definition of order obtained by replacing logarithms by iterated logarithms, where the degrees of iteration are determined by p and q. ) order of entire functions of two To the best of our knowledge, characterizations for the ( complex variables in Banach spaces have not been obtained so far. We define the ( p, q)  order of an ) by entire function ( (

(

)

)

(

Where p and q are integers such that

.

Notations: We are using the following notations in this paper. , , , (i) ( , ) ( , , Provided that ⁄

(ii)

)

0(

.

(

/1

0(

)

0(

)

.

/1

(

0(

)

.

/1

(

.

(1.3)

)

)-

)

, -

.

)

(

/1

(

) ) )

II. Basic results: In this section we have given some lemmas as basic results, which have been used in the sequel. Lemma 2.1. If ( ( )

) (

∑ { ( ) be defined by (1.3) then ( )

Where

,

(

)

,

-

)}

be an entire function and for a pair of integers

( (

{(

2

))

)

}

|

| 3

(

)

4.

( (

))

(

( ( (

) ) ))

{ And {

(

(

)

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)

(

)

98 | Page

The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach III. Theorem3.1: If ( ( )

Main Results: In this section we prove our main results.

) ∑ { ( be defined by (1.3) then ( ) (

)} be an entire function and for a pair of integers . ( ,

)

,

)/ where - 2(

)

-{

0

3 (

(

))1

Proof. We prove the above result in two steps, first we consider the space ( ) ( ) be of ( ) order ( and .Let ( ( ) such that there exists a natural number (

|

|

,

6

-

,

2

-

Now first we show that ( entire function. If ( (

)

{(

)

}3

. (

)

(

(

)/ If (

)

. (

)

(3.1)

}

) ) From (2.1), for any

)

7

(3.2)

)

then (

)/ since (

)

or else (

) is not an

) is nonnegative then we have

)

We denote the partial sum of the Taylor series of a function ( (

)

) by

∑∑

and (

( 8∫ ∫ *(

)(

Where ∑ ∑

|

∑ Since (

.

)+

/

.∑ ∑

∑ | are bounded and ))

( )‖

|

|

(

))

| /

9

|

(

| ,

∑ ∑ | and therefore (3.3) becomes )

6∫ {(

.

/

}

(

)

)

|

7{ ∑

|

∑ |

| }

Where )

6∫ {(

.

/

}

(

)

7 )

6∫ {(

.

/

(

}

)

7

6∫ {(

) 2∑

.

)

/

}

(

)

7

Therefore (

(

))

(

) (

where is a constant and ( In view of (3.2), we have ∑

∑ |

|

)(

)

|

| 3

denotes the beta function. (

,

8

-

{

,

-

(

)}

,

-

{

,

-

(

)}

(

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)

)

9 (

( )6

(3.4)

)

7 99 | Page

The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach Using the above inequality in (3.4) we get ( )) ( ( ) (

) (

,

6

-

,

{

-

(

)}

(

)

7

)

(

)

The result for has been obtained by Ganti and Srivastava [2]. ) ( ) Now consider for ( ( )) ( ( ) ( ) (

*(

6

)+

(

)

7

)

Or (

(

))

(

) (

*(

6

)+

(

(

)

(

)

)

7

)

7. Or (

(

(

))

)

(

)

*(

8

)+

(

)

9

Or (

(

))

(

)

(

)

(

Or (

)

(

) 0

{

)

}

)

(

))

(

)]}

(

)+ (

*(

{

(

)

(

)}1

Since )

{ [(

â &#x201E;(

)

Now proceeding to limits, we get (

(

(

) [

) (

(

Or . (

)

))]

(

)/ [

(

(

)

) (

))]

(

)

Or . (

)/

(

)

8. Or (

)

. ( )/ ) ( ) Now for ( ( )) (

( ) ( ) we have from (3.5) that ( ) ( )

(

)

( ,

6

-

,

{

-

(

)}

(

)

)

7

Or ( (

( )

)) 6

( ,

-

{

) ,

-

(

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(

) )}

(

)

7 100 | Page

The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach Taking (3.6) into account, we get [ Or

(

( ,

-

[

))](

)

,

-

{

,

-

(

))](

)

{

,

-

(

Or

,

(

-

)

(

)}

(

)

(

)}

(

)

)

, -[

(

( Proceeding to limits, we obtain (

(

))](

) ,

) , -0

-(

)

(

))1(

(

(

)

)

9. Combining other results for (

)

(

(

)

)

(

) with (3.8) we get

. (

)/

(

To prove reverse inequality consider (eq.2.4 [2]) which gives | | ( ) ( ) Or |

|

(

)

(

(

))](

Now using Lemma (2.1), since (

(

) ,

-

(

, -[

)

(

))

) ](

the inequality for ( )/ .

)

))

(

)

( and for

(

(

Again (3.6) taking into account in above inequality, we obtain , ( ) , -[

)

)

gives

it gives

( ) ( ) ) )/ Hence combining above results we get ( . ( This is the proof of first step. ) Now we consider the space ( we have ( )) ‖ (

∫ ∫ *( { Where ∑ ∑

)(

)+

.

/

:∑ ∑

|

(

)

( )‖

| ;

(

)

} |

∑ Since (

|

∑ | are bounded and (

))

6∫ {(

|

∑ ∑ | and therefore (3.11) becomes )

.

/

}

(

)

7{ ∑

| , |

|

∑ |

| }

Where www.iosrjournals.org

101 | Page

The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach )

8∫ {(

.

/

}

(

)

9 .

)

8∫ {(

/

(

}

)

.

)

9 8∫ {(

/

(

}

)

9

Therefore (

(

))

(

) (

is constant and (

Where

){ ∑

∑ |

| }

(

) is Euler’s integral of the first kind. By using (3.2) we have (

∑ |

|

)

,

8

-

,

{

-

(

)}

(

( ,

( )6

-

,

{

using this inequality in (3.12), we get ( )) ( (

-

(

) (

)}

(

)}

(

)

7

)

) (

,

6

-

,

{

)

9

)

-

(

)

7

)

(

)

) ( ) the result has been proved by Ganti and Srivastava [2] . For ( ) ( )we have from (3.13) that Now ( (

(

(

))

(

) (

)

(

8

)

(

)

9

)

Or (

(

))

(

)

(

)

(

(

8

)

(

)

9

)

Or (

(

))

(

(

) )

8

(

)

(

)

(

)

9

)

9

Since 0(

)

.

(

/1

((

)( )(

) ( . )

.

/) /)

and {

[(

)

(

(

)]}

)

(

)

Therefore from above inequality, we get (

(

))

(

)

8

(

)

(

Or

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102 | Page

The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach (

(

(

))

(

8

)

(

)

9

)

Or [

(

(

))]

(

Or (

(

)

(

) , -[

(

(

))]

(

) , -[

)

(

)

) (

(

Or . (

(

)

Proceeding to limits, we get (

)

))]

(

)/ , -[

(

)

) (

(

))]

(

)

Or . (

)/

(

)

Or ( ) . ( )/ ( ) ( ) from (3.13) we get ( ) ( )

) Now consider the case ( ( )) (

(

)

(

6

,

(

(

-

,

{

-

(

)}

(

)

7

)

Using (3.14), we obtain ))

(

)

,

6

-*

,

-(

)+

(

(

)

)

7

14. Or [

(

(

))]

(

,

)

Or

,

(

-

) , -[

Proceeding to limits, immediately we get (

)

-

(

)/

))] (

To prove reverse inequality taking (3.12) into account this gives | | ( ) ( ) Or ( )) ( | | Again using (3.14), we get , -[

( (

(

)}

(

)

)

( (

) (

-

) (

(

. (

,

{

(

))

)

(

( ))]

(

(

)

, -|

)

)

) |

(

)

In view of lemma (2.1), we obtain (

)

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The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach Or ( Since ( gives

)

)

(

)

( gives (

the inequality for (

)

)

(

)

)

(

(

.

)/ and for

it

)

Combining (3.15), (3.16), (3.17) and (3.18) the proof of second step is immediate. Now consider the third step. Let and Since . where

(

(

)/

)

(

6 4(

(

(

))

and the condition (3.1) is already proved for the space ( , ) {( }

and , -[

(

(

))](

) hence

)

,

-

)

{(

, -[

Now let

)

)57

}

(

(

(

))](

)

)

Since (

)

(

)

.

Therefore .

(

8∫ ∫ *(

)/ [|

)(

| (

)+

) (

.

/

9

)]

16. where ,

(

) ,

-

, -[

{( (

- Hence we get ) } (

))](

) ,

- {( , - [|

)

}

|](

(3.20)

)

In view of (3.19), (3.20) and Lemma 2.1, we get the required result. This is complete proof of the theorem. ) ∑ Theorem 3.2. If ( { ( )} be an entire function, then for a pair of ) ) ) order ( ) if and only if ( ) integers ( the function ( is of ( )) ( ( where , ) {( } ( ) ( ) , -{ ( )] } [ ( ) ) ∑ Proof. Let ( { ( )} be an entire transcendental function. Since is entire, we have |(

| and and (

therefore )

(

)

( ) ( ⁄

where K is a constant independent of

))

) we have (

and

)

) (

(

(

)

(

)

In the case of space

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The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach (

(

))

(

)

(

)

From (3.23) we have ,

( )

,

-

)

{(

-{

(

(

-

(

[

) ,

,

}

-

{ (

)] }

{( [

)

)

)

}

(

( ⁄

))] }

(3.25)

using (3.24) we prove inequality (3.25) for the case For the reverse inequality, we have (

)

( )

,

∑ |

|

using (3.2) we have

(

(

)

( )

8

-

,

{

-

(

)

}

Or

(

)

(

(

)

,

( )6

-

,

{

-

(

)

}

(

)

)

9 )

7

18. which gives

,

(

-

)

)

{(

, -[

(

}

)](

)

Proceeding to limits we get (

)

( )

(

)

In the consequence of Theorem 3.1 with (3.25) and (3.26) we obtain the result immediately. Now to prove sufficiency, assume that the condition (3.21) is satisfied. Then it follows that 6

(

)

7

(

)

It gives 6

(

( )| This relation and the estimate| is an entire transcendental function. Hence the proof is completed.

(

)

7

(

)

) yield the relation (3.22). It follows that

(

)

REFERENCES [1] [2]

Bose, S.K. and Sharma, D., Integral functions of two complex variables, Compositio Math. 15 (1963), 210-226. Ganti, R. and Srivastava, G.S., Approximation of entire functions of two complex variables in Banach spaces, J. Inequalities in Pure & Applied Mathematics, 7, Issue 2, (2006), 1-11.

[3]

Juneja, O.P., Kapoor, G.P. and Bajpai, S.K., On the

[4]

Angew. Math. 282 (1976), 53-67. Vakarchuk, S.B. and Zhir, S.I., On some problems of polynomial approximation of entire transcendental functions, Ukarainian Mathematical Journal, (54) (9) (2002), 1393-1401.

( p, q)  order and lower ( p, q)  order of an entire function, J. Reine

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