Maximum and Minimum Modulus Principle for Bicomplex Holomorphic Functions

(1)

Maximum and Minimum Modulus Principle

for Bicomplex Holomorphic Functions

Mr. Anand Kumar,

†

Mr. Pravindra Kumar,



Mr. Pranav Dixit

 Department of Applied Science, Roorkee Engineering & Management Technology Institute, Shamli (INDIA)

†

Department of Electronics and Communication, Roorkee Engineering & Management Technology Institute,

Shamli (INDIA)



Department of Applied Science, I.M.S. Enginnering College, Ghaziabad (INDIA)

E-mail: anand12617_ibs@rediffmail.com, ait.pravs@gmail.com , pranav27.dixit@rediffmail.com,

Abstract – Bicomplex is the most recent mathematical tool to develop the theory of analysis. In complex analysis we can not give approximate region in which f Z  attains their max. or min. value on the

boundary. But in advance with bicomplex variable we can give the approximate region on the boundary on which f  attains max. or min. value. In this paper Maximum Modulus Principle and Minimum Modulus Principle are promoted for bicomplex holomorphic function which are highly applicable for analysis, and from this result we have seen that in complex analysis it is necessary that if f Z  is a non constant analytic in D(0,1) and f Z

 

K C 0,1

 

, then f Z

 

has a zero in D(0,1). But in bicomplex it is not necessary.

Index Term – bicomplex numbers, Maximum and Minimum Modulus Principle.

INTRODUCTION

In 1892, in search for special algebras, Corrado Segre [1] (1860-1924) published a paper in which he treated an infinite family of algebras whose elements are commutative generalization of complex numbers called bicomplex numbers, tricomplex numbers, etc. Maximum Modulus Principle and Minimum Modulus Principle are most useful results in complex analysis [2]. In this paper we are developing the concept of Maximum Modulus Princ- iple and Minimum Modulus Principle for bicomplex holomorphic functions for the purpose of analysis of bicomplex number. This paper is divided in to different sections. Section 1. gives the Certain Basics Theory of Bicomplex Numbers. In Section 2. we are giving the Certain Basics of Bicomplex Analysis. Section 3. has the Maximum and Minimum Modulus Principle for Complex variables. In section 4. we are giving Some Useful results on Bicomplex Function.

1 Certain Basics Theory of Bicomplex Numbers:

Segre defined a bicomplex number as  x₀ i₁x₁  i₂x₂  i₁i₂x₃, where x₀,x₁,x₂,x₃are real numbers,

1 2 2 2

1  i  

i and i₁i₂  i₂i₁. The set of bicomplex numbers is denoted as C2. In the theory of bicomplex

numbers, the sets of real numbers and complex numbers are denoted as C0 and C1 respectively. Thus

} , , , , :

{ ₀ ₁ ₁ ₂ ₂ ₁₂ ₃ ₀ ₁ ₂ ₃ ₀

2 x ix i x ii x x x x x C

C        Or equivalently C2{ :  z1i2z2 z1,z2C1}. 1.1 Idempotent Representation:

The idempotent representation of a bicomplex number plays a very important role in the development of the theory. Besides 0 and 1, there are precisely two non-trivial idempotent elements in

2

C, denoted by e₁and 2

e and

defined as 1 2 1 2

1 2

1 1

,

2 2

i i i i

e   e   ;e1e2 1and e e1 2 e e2 10.Every element of C₂ can be uniquely expressed as a complex combination of _e₁ and

2

e , viz. 



z1i2z2

 

 z1i1z2



e1



z1i1z2



e2. This

representation of a bicomplex number as a combination of complex multiples of e₁and e₂ is known as the Idempotent Representation of .Further, the complex coefficients





1 1 2

z i z and



z₁i z₁ ₂



are called the

Idempotent Components of the bicomplex number

2 2

1 i z

z 



(2)

1.2 Singular Elements:

An element z₁i z₂ ₂ is singular if and only if 2 2

1 2 0

z z  . The set of singular elements is denoted as O₂ and is characterized as O₂ { C₂: is th e co llectio n o f a llco m p lex m u ltip les o f e a n d e₁ ₂} 1.3 Norm:

The norm _ 

0 2 :

. C C of a bicomplex number is defined as follows: (where C₀ denote the set of all

non-negative real numbers). If   _z₁ _i₂_z₂ _C₂ then -





2 2 1 2

1 2

2 2 1 1 2 1 1 2

1 2

2

z i z z i z

z z

        

 

 

1 / 2

2 2 2 2

1 2 3 4

x x x x

 

 _    _

C2 is Banach space which is not Banach algebra because, in general,    2   holds instead of the standard condition, viz.      In this sense, C2,  , , , . is treated as a modified Banach

algebra.

1.4 Auxiliary Complex Spaces:

The Auxiliary complex spaces

A

₁and

A

₂are defined as follows:A₁ {z₁i z₁ ₂z z₁, ₂C₁}, 2 { 1 1 2 1, 2 1}

A  z i z z z C . Since each elements of _C₁ can be represented in the form _z₁ _i₁_z₂ and ,

2 1

1 i z

z  the elements in A₁ and A₂ are same as the elements in C₁. 1.5 Cartesian Set:

A cartesian set determined by X₁ and X2 in A1and A2 respectively is denoted as X1e X2and is defined as:

 

1 e 2 1 2 2 2: 1 2 2 1 1 2 2, 1 1, 2 2

X  X  z i z C z i z  w e w e w  X w X By the help of idempotent

representation we define some functions such as h₁:C₂  A₁, h₂:C₂  A₂, and H A: ₁eA₂C₂ as

follows:

1( 1 2 2) 1[( 1 1 2) 1 ( 1 1 2) 2]

h z i z h z i z e  z i z e (z1i z1 2)A1  z1i z2 2C2;

2( 1 2 2) 2[( 1 1 2) 1 ( 1 1 2) 2]

h z i z  h z i z e  z i z e (z1i z1 2)A2  z1i z2 2C2;

1 1 2 1 1 2 1 1 2 1 1 1 2 2 2

( , ) ( ) ( )

H z i z z i z  z i z e  z i z e C  (z1i z1 2,z1i z1 2) A1e A2 The functionsh h₁, ₂ restricted to a setXC₂, map X into the setsX₁,X₂ inA A1, 2respectively. The function H, restricted to a set(X₁_eX₂)(A₁_eA₂), maps(X1e X2)intoX1e X2C2,Thus -

1( ) 1, 2, 1 1

h X  X X C X A ,h X2( )X2, X C2, X2A2,

1 2 1 2 1 2 1 2 1 2 2

( ) e , ( e ) ( e ) , e

H X  X  X  X X  X  A  A X  X  C 1.6 Open Discus in C2:

An open discus with centre

2

2 C

i 



 

 and associated radii ₀

1

r and r₂0 is denoted as D(;r1,r2) and defined as: D( ; , r r₁ ₂){z₁i z₂ ₂C₂:z₁i z₂ ₂ w e_{1 1}w e₂ ₂, (w w₁, ₂)X₁_e X₂

Where X₁ {w₁ A₁: w₁( i₁)  r₁} and X2 {w2A2 : w2 (i1) r2}. 1.7 Closed Discus in C2:

A closed discus with centre   i₂ C₂and associated radii r₁0andr₂ 0is denoted as

) , ; ( r1 r2

D  and defined as: D( ; r r1, 2){z1i z2 2C2:z1i z2 2 w e1 1w e2 2, (w w1, 2)X1X2} Where X₁ {w₁ A₁: w₁  (  i₁)  r₁} and X₂ {w₂ A₂ : w₂  (  i₁)  r₂}. 1.8 C₂- Disc:

If both radii r₁ 0 and r₂ 0 are equal, say r1  r2  r, then discus is called a C2-Disc and is denoted as

) ; ( ) , ;

( r r D r

D    . This kind of disc is in fact the Cartesian set determined by two discs of C₁, viz. }

) (

:

{ 1 1 1 1

1 w A w i r

X        and X2 {w2 A2 : w2  (  i1)  r}.

1.9 Open and Closed Balls: The open ball and closed ball with centre







i₂



C₂ and radius r >0 are denoted by B(



,r) B(



,r)and B( , ) r respectively, and are defined as follows:







1 2 2 2 1 2 2 2 2



1 2

2 2 1 2 2 2 1

) (

: )

, (

) (

: )

, (

r i

z i z C z i z r B

r i

z i z C z i z r B

 

 

 



 

 

 





  

(3)

2.1 Holomorphic Functions of a Bicomplex Variable:

In 1928 and 1932, Michiji Futagawa originated the concept of holomorphic functions of a bicomplex variable, in a series of papers [3], [4]. In 1934, Dragoni [5] gave some basic results in the theory of bicomplex holomorphic functions. A full account of the updated theory can be hald from Price [6] A function f of a bicomplex variable z₁  i₂z₂ is said to be a holomorphicfunction in domain U if

f

satisfies the following properties: (a) f is defined on a domain U in C2.

(b) f (z₁  i₂z₂) C₂,  z₁  i₂z₂ C₂; (c) For each  a1  i2a2 in U there exists a discus

) , ; ( r₁ r₂

D  with r10 and r20 in U, and a power series such thatz1i2z2D(a1i2a2;r1,r2),

    



 

0 2 1 2 2 1 2 2 2

2

1 ) ( )[( ) ( )]

(

k

k k

k i z i z a i a

z i z

f   .

The class of all functions which are holomorphic in U denoted as H (U). 2.1.1 Statement 1: If : ₁ ₁

1 X C

fe  and fe₂:X2C1 are holomorphic functions on the domainsX1andX2

respectively, then the function f:X₁_eX₂C₂ defined

as:













1 2

1 2 2 e 1 1 2 1 e 1 1 2 2, 1 2 2 1 e 2

f z i z  f z i z e f z i z e  z i z X  X isC₂- holomorphic on the domain

2

1 X

X e .

2.1.2 Statement 2: Let X be a domain in C₂, and let

2 :X C

f  be a C2- holomorphic function on X. Then there exist holomorphic functions : ₁ ₁

1 X C

fe  and fe2 :X2 C1 with X1  h1(X ) and X2 h2(X), such

that:













1 2

1 2 2 e 1 1 2 1 e 1 1 2 2, 1 2 2

f z i z  f z i z e  f z i z e  z i z X ; We note here that X1 and X2 will also be domains of C1.

2.1.3 Statement 3: f

 

 has Zero iff

 

1 1

e

f  and fe₂

 

2 both have zero at



1 and



2 respectively.

3. Maximum and Minimum Modulus Principle (for complex variables): 3.1 Maximum Modulus Principle [2]:

Suppose that f is analytical in domain D and ‘

a

’ is a point in D such that f z  f a holds for allzD. Then f is a constant.

3.2 Minimum Modulus Principle [2]:

Suppose that f is analytical in domain D and ‘

a

’ is a point in D such that f  z  f  a holds for allzD. Then f is a constant.

4. Some Useful results on Bicomplex Function.

Theorem 1:- Let

 

1 1 1 2 2 2

e e

f   f  e  f  e is a bicomplex function. If

 

1 1

e

f  and

 

2 2

e

f  have maximum value, then f

 



will also take the maximum value.

Proof :- Let fe₁

 

1 M1 ; fe₂

 

2 M2 . Then by the definition of norm-

  1  2  1 2

2 2 _{1 2}

2 2

1 2 1 2 2 2 2

1 2

= M ( s a y ) ; s u c h a s 2 M

2 2

e e

f f M M

f    M M

 _  _ _ _

 

      

 

 

  ….. (1)

From equation (1) we see thatf

 

 take the maximum value M. Now we are considering that M is not the max. value of f

 

 and it attain another maximum value that is Mh, such as-

  M ; 0 ; M 0

f   h  h   . Then –

   

1 2

2 2

1 2

M 2

e e

f f

h

 

 _ 

  _ _

 

 

 

;

 









1 2

2 2 2 2 2 2

1 2 2 M = 2 M + 2 + 4 M = 2 M 2 2 M

e e

f   f   h h h  h h …...(2) Since the max. value of

 

1 1

e

(4)









2 2 2 2 2

1 2 = 2 M 2 2 M = 1 2 2 2 M

M  M  h h M  M  h h



2 M



0 2 M 0 o r 0 2 M ; M 0; 0 o r 0

h h   h  h   h   H en ce h h . This is a

contradiction of our assumption h0.Thus we can conclude that_f_{ }_ attains the maximum value if both

 

1 1

e

f  and

 

2 2

e

f  have maximum value.

Theorem 2:- Let

 

1 1 1 2 2 2

e e

f   f  e  f  e is a bicomplex function. If

 

1 1

e

f  and

 

2 2

e

f  have minimum value, then f

 

 will also take the minimum value.

Proof :- Let fe₁

 

1 m1 ; fe₂

 

2 m2 . Then by the definition of norm-

  1  2 

1 2

2 2 _{1 2}

2 2

1 2 1 2 2 2 2

1 2

= m ( s a y ) ; s u c h a s 2 m

2 2

e e

f f _m _m

f    m m

 _  _ _ _

 

      

 

 

  …. (3)

From equation (3) f

 

 take the minimum value

m

. Now we are considering that

m

is not the min. value of

 

f  and it attain another min. value that is

m -

h

, such as- f

 

  m h ; m  h 0 ; m0 Then-

 

1 2

2 2

1 2

m 2

e e

f f

h

 

 _ 

  _ _

 

 

 

;

       

1 2

2 2 2 2 2 2

1 2 2 m = 2 m + 2 4 m = 2 m 2 2 m

e e

f   f   h h  h  h h ..…(4) Since the min. value of

 

1 1

e

f  is

m

₁ and min. value

 

2 2

e

f  is

m

₂. So –









2 2 2 2 2

1 2 = 2 m 2 2 m = m1 2 2 2 m

m m  h h m  h h



2 m



0 2 m 0 o r 0; 2 m o r 0 m > 0

h h   h  h H e n c e h h  .This is a contradiction of our assumption m h 0 ; m0. Thus we can conclude that f   attains the minimum value if both

 

1 1

e

f  and

 

2 2

e

f  have minimum value.

Theorem 3: Let

 

1 1 1 2 2 2

e e

f  f  ef  ebicomplex function has max. value M.Then

 

1 1

e

f  and

 

2 2

e

f  both should

have the max. value.

Proof: Let f    M .By previous theorem f

 

 attains the maximum value if both

 

1 1

e

f  and

 

2 2

e

f  have maximum value. Let

 

1 1 1 ; 2 2 2 .

e e

f  M f  M

Claim: _{ } 2 2 1 2

1 2 _{= M}

2

M M

f     _

 

.

Now we consider three possible cases-

Case 1: Let

 

1 1 1 1 ; 1 1 0 ; 1 0 & 2 2 2

e e

f   M  h M  h  M  f   M

 

1

 

2

 





1 2

2 2 ₂ 1 2

2

1 2 1 1 2

2 2

e e

f f M h M

f   

 _  _ _ _ _

   

     

 

   

 

 

12 22 1



1 2 1



1 2 12 22 1 1 1 2 _M

2 2 2 2

h h M

M M M M h M

f   _    _  _   _ 

 

  …..(5)

This is a contradiction of our assumption M₁  0 h₁ ; 0M₁

Case 2: Let _{ } _{ }

1 1 1 & 2 2 2 2 ; 2 2 0 ; 2 0

e e

f   M f   M h M  h  M 

  1  2   

1 2

2 2 1 2

2 2

1 2 1 2 2

2 2

e e

f f M M h

f   

 _  _ _ _ _

   

 _ _  _ _

 

   

 

 

12 22 2



2 2 2



1 2 12 22 2 2 1 2 _M

2 2 2 2

h h M

M M M M h M

f             

 

  ….(6)

(5)

Case 3:

Let

 

1 1 1 1 ; 1 1 0 ; 1 0 & 2 2 2 2 ; 2 2 0 ; 2 0

e e

f   M h M  h  M  f   M h M  h  M 

  1   2      

1 2

2 2 1 2

2 2

1 2 1 1 2 2

2 2

e e

f f M h M h

f   

 _  _ _ _ _ _                  

 

12 22 1



1 2 1



2



2 2 2



1 2 12 22 1 1 2 2 1 2 _M

2 2 2 2 2 2

h h M h h M

M M M M h M h M

f               

 

  ..(7)

This is a contradiction of our assumption M₁  0 h₁ ; 0M₁ , M₂  0 h₂ ; 0M₂ . Thus we can say that if

 

f  has max. value then

 

1 1

e

f  and

 

2 2

e

f  both have max. value. Particularly-

    _{ }   _{ }     1 2 2 1 1 2 1 2 2 1 1 2

; if 0

2

= ; if 0

2

0 ; if 0

e

e e

M a x f

f

M a x f

M a x f f

f f                         ....(8)

Theorem 4: Let

 

1 1 1 2 2 2

e e

f  f  ef  e bicomplex function has min.value

m

, then  

1 1

e

f  andfe₂

 

2 both should

have the min. value.

Proof: Let _f _{ }_ __m. By previous theorem _f_{ }_ attains the minimum value if both

 

1 1

e

f  andfe₂

 

2 have minimum value.

Let

 

1 1 1 ; 2 2 2 .

e e

f   m f   m

Claim: _{ } ₂ ₂ 1 2

1 2 _{= m}

2

m m

f     _

 

.

Now we consider three possible cases-

Case 1: Let

 

1 1 1 1 ; 1 0 ; 1 0 & 2 2 2

e e

f   m h h  m  f   m

  1  2   

1 2

2 2 ₂ 1 2

2

1 2 1 1 2

2 2

e e

f f m h m

f   

 _  _ _ _ _                   ;

 





1 2 2 2

1 1 1

1 2 2 _m

2 2

h h m

m m

f        

  ……..(9) This is a contradiction of our assumption h₁0.

Case 2: Let

 

1 1 1& 2 2 2 2 ; 2 0 , 2 0

e e

f   m f   m  h h  m 

  1  2   

1 2

2 2 ₂ 1 2

2

1 2 1 2 2

2 2

e e

f f m m h

f   

 _  _ _ _ _                  ;

 

12 22 2



2 2 2



1 2 _m

2 2

h h m

m m

f  _    _ 

  ....(10) This is a contradiction of our assumption h₂ 0.

Case 3: Let

_{ }

1 1 1+ 1 ; 1 0 , 1 0 & 2 2 2 2 ; 2 0 , 2 0

e e

f  m h h  m  f   m h h  m 

 

1

 

2

 









1 2

2 2 ₂ ₂ 1 2

1 2 1 1 2 2

2 2

e e

f f m h m h

f   

 _  _ _ _ _ _                  

 









1 2 2 2

1 1 1 2 2 2

1 2 2 2 _m

2 2 2

h h m h h m

m m

f         

  …..(11) This is a contradiction of our assumption h₁0 , h₂ 0. Thus we can say that if f

 

 has min. value then

 

1 1

e

(6)

 

  _{ }

 

   

1

2

1

1 2

1

2

1

1 2

; if 0

2

= ; if 0

2

0 ; if 0

e

e e

M in f

f

M in f

M in f f

f f



 



 

 

  

 

   

  

...(12)

Theorem 5: (MAXIMUM-MODULUS PRINCIPLE)

Suppose thatf

 

 is a holomorphic function in a closed discus the D and “a” point inside the discus D is such that f

 

  f

 

a holds   D, then f  will be constant.

Proof: Let

 

1 1 1 2 2 2

e e

f   f  e  f  e . Such as-  1 1e 2e2. Suppose is the interior point of bicomplex

discus D( ; , r r₁ ₂). Hence  ₁, ₂are the interior point of complex discD₁₁,r₁&D₂₂,r₂respectively. We know from the theorem 3 that if f

 

 has max. value then  

1 1

e

f  andfe₂

 

2 both have the max. value. Now

if

 

1 1

e

f  takes max. value inside point ₁of D₁



₁,r₁



then  

1 1

e

f  will be constant. Similarly we can show that

 

2 2

e

f  will be constant. Sof

 

 is constant becausefe₁

 

1 andfe2 2

both are constant.

Corollary 1: If f

 

 is a non constant holomorphic function, thenf

 

 attains its max. value on the boundary of discus D( ; , r r₁ ₂).

Theorem 6: (MINIMUM-MODULUS PRINCIPLE)

Suppose that f

 

 is a holomorphic function in a closed discus D and “a” point inside the discus D is such that f

 

  f

 

a holds  _ _D, then f

 

 will be constant.

Proof: Let

 

1 1 1 2 2 2

e e

f   f  e  f  e , such as-   _{1 1}e _{2 2}e . Suppose is the interior point of bicomplex discus D( ; , ) r r₁ ₂ . Hence  1, 2 are the interior point of complex discD11,r1&D22,r2respectively. We know

from the theorem 4 that if _f_{ }_ has min. value then _{ }

1 1

e

f  andfe₂

 

2 both have the min. value. Now

if _{ }

1 1

e

f  takes min. value inside point1of D11,r1 thenfe1 1

will be constant. Similarly we can show that

 

2 2

e

f  will be constant. Sof

 

 is constant because

 

1 1

e

f  and  

2 2

e

f  both are constant.

 

 is a non constant holomorphic function, thenf

 

 attains its min. value on the boundary of

discus D( ; , r r₁ ₂).

 

 is a non constat holomorphic function in a closed discus D. Then f  attains its max. value on the intersection of the boundary of boundary of the discus D a r r( ; ,₁ ₂) and the boundary of the closed ball 2 2 1 2

1 2

, 2 r r Ba _  _ 

 _ _ 

 

.

Proof: By the definition of boundary of discus- D a



a e_{1 1}a e r r_{2 2}; ,₁ ₂



we can define three sets as

1 2 3 ;

S  S  S









1 1 1 2 2: 1 1 1& 2 2 2 , S =2 ξ=ξ1e +1 ξ2e :2 ξ1-a1 = r &1 ξ2-a2 < r2

W h e r e S   e  e  a  r  a  r





2 2 1 2

1 2 3 1 1 2 2: 1 1 1& 2 2 2 ;

2

r

r r

S  e e  a r  a r B a

 _ _ _ 

 

        _ _

 _ _ 

 

, B_r is the set of boundary points of the

closed ball 2 2 1 2 1 2

; 2 r r Ba _  _ 

 _ _ 

 

. By theorem (1)      

1 1 1 2 2 2

e e

f   f  e f  e has maximum value if

 

1 1

e

f  and

 

2 2

e

f  have maximum value and these function have maximum value if ₁and

2

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construct S₃which is the subset of _; 12 22 1 2

2 r r B a

 _ _ _ 

 _ _ 

 _ _ 

 

. So f  attains the maximum value on the

intersection of the boundary of boundary of the discus D a r r( ; ,₁ ₂) and the boundary of the closed ball

1 2

2 2

1 2

, 2 r r B a

 _ _ _ 

 _ _ 

 _ _ 

 

.

Corollary 4: If f_{ } is a non constat holomorphic function in a closed discus D. Then _f_{ }_ attains its min. value on the intersection set of the boundary of boundary of the discus D a r r( ; ,₁ ₂) and the boundary of the closed ball _, 12 22 1 2

2 r r Ba _  _ 

 _ _ 

 

.

Proof: We can prove this corollary in similar way as corollary 3.

Example 1: f

 

 is a non constant analytic in D(0;1,1). f

 

 K B_r

 

0,1 D_r

 

0,1 .Then discuss the case f

 

 has zero inD(0;1,1).

Proof: Since

 

1 1 1 2 2 2

e e

f   f  e  f  e is holomorphic iff  

1 1

e

f  ,

 

2 2

e

f  both holomorphic in (0,1) ; 1, 2

i

D i (closed unit disc) and it is non constant then there are three possibilities. 1.

 

1 1

e

f  is non constant and

 

2 2

e

f  is constant.

2.

 

1 1

e

f  is constant and

 

2 2

e

f  is non constant.

3.

 

1 1

e

f  is non constant and

 

2 2

e

f  is non constant.

If

 

i

e i

f  does not have a zero. Then

 

1

i

e i

f 

is holomorphic in Di(0,1). So its maximum occurs on the

boundary. Since both the maximum and the minimum of

  

1, 2



i

e i

f  i are the same because

 

C 0,1

 

i

e i i

f  K  , (Ci0 ,1 is a unit circle) which means fei

 

i is constant.

Now we shall discuss the three cases: 1.

 

1 1

e

f  has one zero,  

2 2

e

f  has no zero in D2(0,1). So there is no zero of f

 

 in D(0;1,1).

2. _{ }

1 1

e

f  has no zero in D1(0,1), fe₂ 2 has one zero. So there is no zero of f

 

 in D(0;1,1).

3.  

1 1

e

f  has zero at 1 in D1(0,1), and fe2 2

has zero at ₂in D₂(0,1). Thenf   has zero at  1e12e2

in D(0;1,1).

Thus we conclude that f

 

 has a zero inside the discus D(0;1,1) if    1, 2

i

e i

f  i both are non constant.

Example 2: If

 

2

f   is a holomorphic in a closed discus D(0;1,1), f

 

 1   Br

 

0,1 Dr

 

0,1

here f

 

 is a non constant analytic function and it has zero at origin



 0



inside the D(0;1,1).

Example 3 : If f

 

 _{1 1}e e₂is a holomorphic function because

 

1 1 1

e

f  ,  

2 2 1

e

f   both are

holomorphic. f

 

 1   Br

 

0,1Dr

 

0,1. Thenf  has no zero in D(0;1,1). We also note thatf

 

 has

no zero because

 

2 2 1

e

f



 has nowhere zero or constant.

REFERENCES

[1] C. Segre “Le rappresentazioni reale delle forme complesse’e Gli Enti Iperalgebrici”, Math. Ann., 40 (1892), 413-467. [2] W. Rudin “ Real and Complex Analysis” , McGraw-Hill, 1987.

[3] M. Futagawa “On the theory of functions of quaternary variable-I” Tohoku Math. J., 29 (1928), 175-222. [4] M. Futagawa “On the theory of functions of quaternary variable-II”. Tohoku Math. J., 35 (1932), 69-120.

[5] G.S.Dragoni “Sulle funzioni olomorfe di una variable bicomplessa” Reale Acad.d’Italia Mem. Class Sci. Fic. Mat. Nat., 5(1934), 597-665.

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Mr. Anand Kumar - received his M.Sc degree from M. J. P. Rohilkhand Univ., Bareilly (U.P). He has done M. Phil from Dr. B.R.Ambedkar Univ., Agra (U.P). He has 3 years of teaching experience. His research interest area is bicomplex number. He is working as Lecturer in REMTech, Shamli (U.P).

Mr. Pravindra Kumar – received his B.Tech Degree in Electronics and Communication Engineering from

Ideal Institute of Technology, Ghaziabad (U.P). He has done M.Tech. in Digital Communication (D.C) from Ambedkar Institute of Technology, Delhi (I.P.Univ., Delhi). He has 3 years of teaching experience. His teaching and research interests are is the wireless communication and digital signal processing. He has published four papers in International Journal and one paper in IEEE. He is working as an Assistant Professor in REMTech, Shamli (U.P).

Mr. Pranav Dixit- received his M.Sc degree from C.C.S.Univ., Meerut (U.P). He has done M.Phil from Dr.