Keywords and Phrases: Bicomplex Numbers, Entire Bicomplex Dirichlet series, Modified Banach ... C and the sets of complex and real numbers are denoted by.
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A Class of Entire Function Defined By Bicomplex Dirichlet Series Dr. Jogendra Kumar Assistant Professor- Mathematics Govt. Degree College, Raza Nagar Swar, Rampur(UP)-244924, India ________________________________________________________________________________________________________ Abstract: We have studied Entireness of Bicomplex Dirichlet Series in [6] and Banach Algebra of Entire Bicomplex Dirichlet
Series in [2]. In this paper we have studied a class H f ()
nn n e n : Sup n 1
n 1
n
is bounded
of Entire
Functions represented by Bicomplex Dirichlet Series. H has a Modified Banach algebra structure. H is a Modified Banach – algebra which is neither a division algebra nor a B* algebra. Invertible and quasi invertible elements in H have been investigated. Keywords and Phrases: Bicomplex Numbers, Entire Bicomplex Dirichlet series, Modified Banach – Algebra, Invertible and Qausi invertible elements. 2000 AMS Subject Classification: Primary-32A30, 30G35, Secondary-30B50, 30D10. ________________________________________________________________________________________________________ 1. INTRODUCTION Throughout this paper, the set of Bicomplex numbers is denoted by
C 2 and the sets of complex and real numbers are denoted by
C1 and C 0 , respectively. For details of the theory of bicomplex numbers, we refer to [3], [4] and [5]. The set of Bicomplex Numbers defined as:
C2 {x1 i1x 2 i 2 x 3 i1i 2 x 4 : x1, x 2 , x 3 , x 4 C0 , i1 i 2 and i12 i 22 1, i1 i 2 i 2 i1} We shall use the notations C ( i1 ) and C ( i 2 ) for the following sets:
C ( i1 ) {u i1v : u, v C0 } C ( i 2 ) { i 2 : , C0 }
1.1 Idempotent Elements: Besides 0 and 1, there are exactly two non – trivial idempotent elements in C2 , denoted as
e1
1 i1i 2 1 i1i 2 and e 2 2 2
Note that e1 e 2
e1
and
e2
and defined as
1 and e1e2 e2e1 0.
1.2 Cartesian idempotent set: Cartesian idempotent set X determined by
X1 and X 2
is denoted as
X1 e X 2 and is defined as
X X1 e X 2 C2 : 1 e1 2 e2 , 1 , 2 X1 X 2 .
C2 C(i1 ) e C(i1 ) C(i1 ) e1 C(i1 )e 2 { C2 : 1 e1 2 e2 , (1 , 2 ) C(i1 ) C(i1 )} C2 C(i 2 ) e C(i 2 ) C(i 2 ) e1 C(i 2 )e 2 { C2 : 1 e1 2 e 2 , (1 , 2 ) C(i 2 ) C(i 2 )} 1.3 Idempotent Representation of Bicomplex Numbers There are two idempotent representation of a Bicomplex Number
x1 i1x 2 i 2 x 3 i1i 2 x 4 , C(i1 ) -idempotent
representation and C(i 2 ) -idempotent representation
C(i1 ) -idempotent representation ( x1 i1x 2 ) i 2 ( x 3 i1x 4 ) z1 i 2 z 2 (z1 i1z 2 ) e1 (z1 i1z 2 ) e 2
[( x1 x 4 ) i1 ( x 2 x 3 )] e1 [( x1 x 4 ) i1 ( x 2 x 3 )] e 2 1 e1 2 e 2 C(i 2 ) -idempotent representation
(x1 i2 x3 ) i1(x 2 i2 x 4 ) w1 i1w 2 (w1 i 2 w 2 )e1 (w1 i2 w 2 )e2 [ (x1 x 4 ) i2 (x 2 x3 )] e1 [( x1 x 4 ) i2 (x 2 x3 ) ] e2 1 e1 2 e2
Note 1.1: Out of the two idempotent representation, we use C(i1 ) -idempotent representation. All the results may also be proved with the help of C(i 2 ) -idempotent representation technique. 1.4 Norm The norm in
C2
is defined as
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12
2 12
1 2 2 2 = 2
2
2
2
2 1/ 2
= x1 x 2 x 3 x 4
C 2 becomes a modified Banach algebra, in the sense that . 2
.
… (1.1)
Singular Elements Non zero singular elements exist in C2 . In fact, a Bicomplex number Set of all singular elements in
C2
z1 z 2i 2
is singular if and only if
z12 z 22 0 .
is denoted as O 2 .
Definition 1.1: Quasi Invertible Elements: If A is an algebra, then an element x of A is said to be left quasi – invertible, if y A , s.t.
y x y x y.x 0 right quasi – invertible, if y A , s.t. x y x y x .y 0 and quasi invertible if y A , s.t. y x y x y.x 0 ; and x y x y x.y 0 . Definition 1.2: Gel’fand Algebra: A commutative Banach Algebra with a unity element of norm one is called Gel’fand algebra. Definition 1.3: Banach *-algebra A Banach *-algebra ‘A’ is a Banach algebra over C1 such that The map * : A → A, called involution, having the following properties: 1. (x*)* = x x A 2. (x + y)* = x* + y* ; (xy)* = y* x*
x, y A
(x) x for every λ in C1 and every x in A; here, denotes the complex conjugate of λ. 4. ||x*|| = ||x|| x A 3.
Definition 1.4:B*-algebra A B*-algebra ‘A’ is a Banach algebra over C1 such that The map *: A → A satisfying the following properties: 1.
x (x ) x x A
2. (x + y)* = x* + y* ; (xy)* = y* x*
3. (x) 4.
x, y A
x C1 and x A ;
denotes the complex conjugate of λ.
x x x x x A
2. Class H of Entire Bicomplex Dirichlet Series : We consider a Class of Bicomplex Dirichlet Series defined as
H f () n e n : Sup n n n n 1 n 1
Note 2.1: As,
f ()
n 1
is bounded
1 n e H nn
Hence H is non – empty.
Note 2.2: Note that every member of H is an Entire Bicomplex Dirichlet series
Let
f () n e n H n 1
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is bounded
Sup n n n n 1
nn n
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is bounded
nN
K 0 such that n n n K , n N Now, n N i.e.
K nn
n
1 n
n
1 n
K n n
1 n n
1 n n
1 n K
1 n n
1 n n (n )
1 n K
n
0 as n
f () n e n is an Entire Bicomplex Direichlet Series n 1
Note 2.3:
n
1 n
f () n1 n e n n 1 1 n 1 n n n
n1n
1 n
n
0 as n .
Hence,
f () n1 n e n is an entire Bicomplex Dirichlet series. n 1
But,
f () n1 n e n H
n 1
Sup n
n
n
Sup n n n1n
Sup n n n
Sup n
is not bounded.
Hence H is not the whole class of Entire functions represented by Bicomplex Dirichlet series.
α n e n H ,
Note 2.4: If
then the series
n 1
α
( n)n k
en ξ
obtained after k–times term – by – term integration of
n 1
α n e n is also belongs to H. n 1
Note 2.5: If
αn e n ξ H , then kth derivative defined by n 1
but belongs to H iff
Sup n k n α n n 1
Note 2.6:
1 2
ξ C2 , 2
2
2
2
2
1
2
2
1
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2
( n) k α n e n ξ
is also an entire Bicomplex Dirichlet series
n 1
1
2
1
1 Proof:
2
is bounded.
2
2
2 2
2
1
2
2
2
2 1
2
1
2
2
2
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1
2
The Uniqueness theorem Theorem 2.1: Two Bicomplex Dirichlet Series
f () n e n
g() n e n are
and
both
absolutely
convergent
in
the
region
n 1
n 1
{ C2 : Re(1 ) a and Re(2 ) b} such that f () g () then n n for all n. Proof: Let n n n and let h () f () g () 0 . To prove that n 0 n N we assume that n 0 for some n and obtain a contradiction. Let
N
n 0 .
be the smallest number for which
Re(1 ) a and Re(2 ) b ,
Now for with
0 h()
n e n N e N n e n
nN
Hence
n N 1
N eN
N N
n e n
n N 1
N eN N
n N 1
1
n N 1
1
1
n N 1
n e( n N )
n N 1
and
1
( n N ) Re( )
e ( n N ) Re(
2
)
1
1
1
1
e n ( Re(
eN N 1 e
2 n e( n N )
n e ( n N ) Re( e N ( Re(
2 ( n N ) 2 e 1 n e n N 1
e 2
2
n N 1
n e( n N )
1
Now, e
1 ( n N ) ( n N ) 1 e n n e n N 1 n N 1
n en
)
) a )
) a )
n N 1
2
n e( n N )
n N 1
2
n e ( n N ) Re(
eN a eN e n a e n
Re( 2 ) b
2
Re(1 ) a
2
)
eN a eN e n a e N 1
Re( 1 ) a
eN a en a
n N 1
eN b en b
Re( 1 ) a
Re( 2 ) b
eN eN N N 1 e N a 1 n e n a N 1 e N b 2 n e n b n N 1 n N 1 e e 1 2 1 2 Re( ) b N Re( ) a n n N a b N N N a b N 1 N 1 n n n N 1 n N 1 1 2 As Re( ) and Re( ) we get N 0 , which is a contradiction.
2.1 Algebraic Structure for H (i) Addition
Let
f () α n e n ξ , g() n 1
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n
n 1
n n Sup n Sup Sup
n
n
n
n 1
n
n
Sup n 1
n
n 1
Sup n
n
n
n
n 1
f () α n e n ξ , g()
Sup n
Sup n
n 1 n
n
n
n 1
n e n ξ H n 1 n
and Sup n are bounded. is bounded.
n
n 1
n
As,
n n n
n
n 1
αn n e n ξ H
Sup n n n n 1
f () g()
Now we check
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n
n 1
n
αn n e n ξ H .
f () g()
n 1
(ii) Scalar Multiplication
Let
a C1 and f () αn e n H n 1
We check
Sup n n 1
(a. αn ) e n H
a. f ()
n
n 1
a . n
a Sup n
Sup n n a n n 1
As,
f ()
Sup n n 1
n 1
α n e n H n 1
n
n
a Sup n n n 1
n
n
a. f ()
is bounded is also bounded n
(a αn ) e n H n 1
(iii) Weighted Hadamard Product
Let
f () α n e
nξ
,
g()
n e
f () g() n e
nξ
Where,
n 1
As, Let
f () α n e
n 1
Sup n n n n 1
nξ
,
g()
H
n n n (n n )
n e
nξ
n 1
is bounded and Sup n
n
n 1
K1, K2 0 s.t. n n n K1 Now,
nξ
n 1
n 1
Then,
and n
n
n
H
is bounded
n K 2 n N
n n n n n n n (n n ) (n n n )(n n n ) 2 n nn n n n 2 n n n n n n
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n n n 2 n n n n n n 2 K1K 2
is bounded
Sup n n n n 1
n e n ξ H .
f () g() Theorem 2.2: I()
Proof: As, Sup n n 1
Sup n n n 1
n
I() Let
n
n 1
n n en ξ , being the identity element of H.
n 1 n
n
Sup n n 1 n n
n
n 1
is bounded
n n e n ξ H
f ()
n 1
αn en ξ be any element of H n 1
I() f ()
Now,
Similar way, Hence
n 1
n 1
n n (n n n )en ξ nen ξ f ()
f ( ) I ( ) f ( )
n nen ξ be the identity of H.
I()
n 1
Theorem 2.3: ( H, +) is an Abelian group. Proof: (i) Closure Property
f () g()
n 1
n 1
n 1
(αn n ) en ξ H , f () αn en ξ , g() nen ξ H
(ii) Associativity
f ( ) ( g ( ) h ( ) ) ( f ( ) g ( ) ) h ( ) , C 2 is associative under addition.
f (), g (), h () H
(iii) Existence of Identity
O()
0 en ξ be the additive identity of H n 1
Obviously,
0 enξ H
O()
n 1
Now,
f ()
α n e n ξ H n 1
O() f () and,
n 1
n 1
(0 αn ) en ξ αn en ξ f ()
f () O()
(αn 0) en ξ αn e n ξ f () n 1
n 1
(iv) Existence of Inverse
f ()
α n e n ξ H n 1
f ()
( α n ) e n ξ H n 1
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0 en ξ O()
f () ( f () ) ( f () ) f ()
n 1
0 e n ξ O() n 1
(v) Commutativity Let
f ()
αn en ξ , g() n 1
n e n ξ H n 1
n 1
n 1
αn n en ξ n αn en ξ g() f ()
f () g()
As, α n , n C 2
α n n n α n
( C 2 is commutative under addition)
Theorem 2.4: ( H, +, ) is a Commutative ring with identity element. Proof: 1. (H, +) is an Abelian group (By Theorem 2.3) 2. (i)
n n (αnn ) enξ H , f (), g() H
f () g()
n 1
(ii) Let
αn en ξ , g()
f ()
n 1
n en ξ and h() n 1
n e n ξ H n 1
f ( ) ( g ( ) h ( ) ) α n e n ξ n e n ξ n e n ξ n 1 n 1 n 1 α n e λ n ξ n n n n e n ξ n 1 n 1
n n α n n n n n e n ξ n 1
n n n n αnn n en ξ n 1
n n αnn en ξ n en ξ n 1
n 1
α n e n ξ n e n ξ n e n ξ ( f ( ) g ( ) ) h ( ) n 1 n 1 n 1 (iii)
I()
n n en ξ be the identity of H. n 1
(iv) Let
f ()
αn en ξ , g() n 1
f () g()
n en ξ H n 1
n n αnn en ξ n n nαn en ξ g() f () n 1
n 1
As, α n , n C 2
α nn n α n
( C 2 is commutative under multiplication)
(v) Distributions Law
n ξ n ξ α e e n n n e n ξ n 1 n 1 n 1 ( α n n ) e n ξ n e n ξ n 1 n 1
(a) f () ( g() h()
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n n ( α n n ) n e n ξ n 1
n 1
n 1
n n αn n en ξ n n n n en ξ f ( ) h ( ) g ( ) h ( ) n ξ n ξ e α e n n e n ξ n n 1 n 1 n 1 n e n ξ ( α n n ) e n ξ n 1 n 1
(b) h() f () ( g()
n n n ( αn n ) en ξ n 1
n 1
n 1
n n n α n e n ξ n n n n e n ξ h ( ) f ( ) h ( ) g ( ) Hence the theorem. Theorem 2.5 : H is not an integral domain.
n n e1 e n ξ , g() n n e 2 e n ξ
f ()
Proof: Let
n 1
n 1
f (), g () H and f () O() , g () O() Obviously
But,
f () g() n n e1 e n ξ n n e 2 e n ξ n n n n e1 n n e 2 e n ξ n 1
n 1
n 1
n n e1e 2 e n ξ n n 0 e n ξ 0 e n ξ O () . n 1
n 1
n 1
Hence, H is not an integral domain. Theorem 2.6: ( H( C1 ), +, . ) is a Linear space. Proof: 1. ( H, +) is an Abelian group (By Theorem 2.3) 2. (I)
a. f ()
(a . α n ) e n ξ H n 1
a C1 AND f ()
α n e n ξ H . n 1
(ii) Let
a, b C1 and f () αn e n ξ H n 1
Now,
(a b). f ()
(a b) . αn en ξ n 1
(a . α n b . α n ) e n ξ n 1
n 1
n 1
n 1
n 1
(a . αn ) en ξ (b . αn ) en ξ
(a . αn ) en ξ (b . αn ) en ξ
a. f () b. f ()
(iii) Let
a, b C1 and f () αn eλ n ξ H n 1
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a . (b . f () ) a. b . α n e n ξ n 1
Now,
ab . αn en ξ n 1
(ab) . αn en ξ (ab). f () n 1
a C1 and f () αn en ξ , g() n en ξ H .
(iv) Let
n 1
n 1
a. f () g () a. α n e n ξ n e n ξ n 1 n 1 a. f () g() a. (α n n ) e n ξ n 1
Now,
a. (αn n ) e n ξ n 1
(a . α n a . n ) e n ξ n 1
(a . αn ) en ξ (a .n ) en ξ a. f () a.g () n 1
n en ξ H and 1 C1 be the identity of C1 .
f ()
(v) Let
n 1
n 1
Now, 1. f ()
n 1
n 1
(1.αn ) en ξ αn e n ξ f () Hence the theorem.
2.1.1 Norm in H: Norm in H is naturally defined as
f () H Sup n n n n 1
, where f () α
n 1
n
e n ξ
f () H exist.
Obviously,
Theorem 2.7: ( H( C1 ), +, .,
H
) is a Normed Linear space.
Proof: Since, ( H( C1 ), +, .) is a Linear space
(By Theorem 2.6)
f () is defined as
The norm of
f () H Sup n n n n 1
Since,
f () n e n H
n 1
Sup n n n 1
n
is bounded
f () H exist. Now we will check other properties of norm (i)
f ( ) H 0
(ii) f ()
H
0
Sup n n n n 1
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n 0 n 1
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n 0 n 1 nn 0 n 1 n 0 n 1 f () O ( )
(iii) Let,
f () n e
n
and
n 1
g() n e n n 1
(n n ) en
f () g() H
n 1
Sup n
Sup n
n
n 1
n n
n
n 1
H
Sup n
n
n
n 1
n
f () H g() H
f () n e n H
(iv) Let a C1 and
a. f ()
Now,
H
n 1
( a. n ) e n
n 1
a Sup n
H
Sup n
a. n
n
n 1
n
n
n 1
Theorem 2.8: ( H( C1 ), +, .,
H
Sup n
n
n 1
a f ()
a n
H
) is a Banach space.
f p () be a Cauchy sequence in H.
Proof: Let
where, f p ()
p, n e n . n 1
Since, f p () be a Cauchy sequence in H Hence given
0 , p 0 N , such that
f p () f q () Hence,
, p, q p0
H
p, q p 0
n 1
n 1
p, n e n q , n e n
p, n q, n en n 1
n
Hence,
H
Sup n n 1 n
H
n
p, n q , n
p, n q , n n 1
p, q p 0 and n 1
n n p, n q , n p, n q , n …………………….(1) Thus, the sequence p, n , is a bicomplex Cauchy sequence. Hence, by using the completeness of C 2 , it converges to a bicomplex number ,say n ,
n 1 .
We define
f () n e n n 1
Making q in (1) for a fixed n, we get IJRAR1903412
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Sup n n p, n n n 1
f p () f ()
Now,
H
p, n n
n
n 1
f p () f () Also,
Sup n
Sup n n n n 1
n p, n p, n Sup n n n p, n Sup n n p,n K
n
n 1
n 1
n 1
f () n en H n 1
Hence, H is complete. Hence, H is a Banach Space. Theorem 2.9: H is a Commutative Algebra with identity element. Proof: By Theorem (2.4), ( H, +, ) is a Commutative ring with identity element. By Theorem (2.6), ( H( C1 ), +, . ) is a Linear space.
a C1
Now, Let
and
f ()
n 1
n 1
αn en ξ , g() n en ξ H
a.( f () g() ) a. n n ( n n ) e n ξ n 1 n n (a n ) ( n ) e n ξ (a n ) e n ξ n e n ξ a.f () g() n 1 n 1 n 1
Similarly,
a.( f () g() ) f () a.g()
Hence the theorem.
Theorem 2.10: H is a Commutative Modified Banach Algebra with identity. Proof: By Theorem (2.9), H is a Commutative Algebra with identity element. By Theorem 2.8 ( H( C1 ), +, .,
H
) is a Banach space
Also,
f () g() n n (n n ) e n ξ n 1
f () g() H Sup n n n n ( n n ) Sup n n n n n n
2 Sup n 2 Sup n
n 1
2 Sup n n n n n n 1
n
n 1
n
n
n 1
2 f ()
H
n
n
n 1
n Sup n n
n
n
n 1
g ()
n
Hence the Theorem.
H
Corollary 2.1: H is a Modified Gelfand Algebra. Proof: The identity element of H is given by
I() n n e n ξ n 1
Now,
I()
T
Sup n n n n n 1
1.
Hence, the result.
In literature, three types of conjugations are defined on C 2 (cf. [1]): Let
ξ 1ξ e1 2 ξ e 2 be a bicomplex number, then
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The i1 – conjugation of
ξ 1ξ e1 2 ξ e 2 is * 1 e 2 2 e1
The i 2 – conjugation of
ξ 1ξ e1 2 ξ e 2 is # = 1 e 2 + 2 e1
The
j – conjugation of ξ 1ξ e1 2 ξ e 2 is ' 1 e1 2 e 2
We now proceed to provide H with a Modified Banach*–algebraic structure with i1 , i 2 and
j conjugation as involution.
Corollary 2.2: H is a modified Banach –algebra. Note 2.7: H is not a B* Algebra As, H is a modified Banach –algebra Therefore, H is not a B* Algebra. But, we have the following relation
f () f * () f () f * ()
H
2 f () H f * ()
H
2 ( f () H )2
H
2 f ( ) H f ( ) H 2 ( f ( ) H ) 2
Note 2.8: H is not division algebra.
f () n e n ξ H such that
Theorem 2.11: Let
n 1
1 Sup n n n n 1
is bounded.
The inverse of
α n O2 n 1 . Then f () is invertible if and only if
f () n e
nξ
, in case it exist, is given by
n 1
n 2n n ξ f () e . n n 1 1
Proof: Let
f () n en ξ H is invertible in H. n 1
g() n e n ξ H , such that n 1
f ( ) g ( ) g ( ) f ( ) I ( ) Let, f () g () I()
n n (n n ) en ξ n n en ξ n 1
n 1
n (n n ) n n
n
n 2n n
n
n 1 n 1
Since,
g() n en ξ H
Sup n n 1
n 1
n
n
is bounded
n 2 n Sup n n is bounded n 1 n 1 Sup n n is bounded n n 1 n 1 Conversely, let Sup n is bounded . n n 1
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© 2018 IJRAR August 2018, Volume 5, Issue 3 Define a sequence
n , such that ,
n
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n 2n n
Now, we define a series
g() n en ξ n 1
As,
Sup n n 1
n
Sup n n n 1
n
is bounded
Sup n n n n 1
n 2 n n
n 1 Sup n n n 1
g () H
Now,
f () g() n n (n n ) en ξ I() n 1
Similarly,
g ( ) f ( ) I ( )
f () is invertible.
f () n e n ξ H such that n n αn 1 O2 n 1 . Then f () is quasi invertible if and only
Theorem 2.12: Let
n 1
if
n n Sup n n n 1 n n 1
is bounded.
The quasi inverse of
f () n e
nξ
, in case it exist, is given by
n 1
Proof: Let
n e n ξ . n 1 (n n 1)
g()
n
f () is quasi invertible
i.e.
g() n e n ξ H such that n 1
f ( ) g ( ) f ( ) g ( ) f ( ) g ( ) O ( ) f ( ) g ( ) f ( ) g ( )
n 1
n 1
n 1
n 1
n e n ξ n e n ξ n e n ξ n e n ξ
( n n ) e n 1
n ξ
n n ( n n ) e nξ n 1
( n n ) e n ξ (n n nn ) e n ξ n 1
n 1
n n n n n n
(n n n 1) n n n n n n n 1
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© 2018 IJRAR August 2018, Volume 5, Issue 3
www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
Since,
g() n en ξ H
Sup n n 1
n 1
n
n
is bounded
Sup n n n n is bounded n n 1 n 1 n n n Sup is bounded n n 1 n 1 n n n n Conversely, let Sup is bounded . n n 1 n 1 n Define a sequence n , such that , n n n n n 1 Now, we define a series
g() n en ξ n 1
As,
Sup n n n 1
n
Sup n n n n 1
Sup n n 1
is bounded
n
n n n n 1
n n n Sup n n 1 n n 1
g () H Now,
n 1
n 1
n 1
f () g() f () g() n e n ξ n e n ξ n n ( n n ) e n ξ ( n n n n n n ) e n ξ n 1
( n (1 n n n ) n e n ξ n 1
(1 n n n ) n n ξ n e n ( n 1 ) n 1 n
0 e n ξ n 1
O ( ) ACKNOWLEDGMENTS I am heartily thankful to Mr. Sukhdev Singh, Assistant Professor-Mathematics, Lovely Professional University, Punjab and Dr. Mamta Nigam, Assistant Professor-Mathematics,University of Delhi for their encouragement and support during the preparation of this paper. REFERENCE [1] Jogendra Kumar, 2016 “Conjugation of Bicomplex Matrix”. “Journal of Sci. and Tech. Res. (JSTR) Vol. 1(1): 21-25 [2] Jogendra Kumar, 2014 “Banach Algebra of Entire Bicomplex Dirichlet Series”. “Inter. J. of Math. Sci. & Engg. Appls. (IJMSEA), 8 (III): 309-314. [3] M.E. Luna-Elizarraras, M. Shapiro, D. C. Struppa, A. Vajiac, 2015 “Bicomplex Holomorphic Functions:The Algebra, Geometry and Analysis of Bicomplex Numbers” Springer International Publishing [4]G.B. Price, 1991 “An introduction to Multicomplex spaces and functions” Marcel Dekker , Inc. [5] Rajiv K. Srivastava, 2008 “Certain Topological Aspects of Bicomplex Space”. Bull. Pure & Appl. Math,: 222-234. [6] Rajiv K. Srivastava and Jogendra Kumar, 2011“On Entireness of Bicomplex Dirichlet Series” Inter. J. of Math. Sci. & Engg. Appls. (IJMSEA), 5 (II),:221-228.
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