Integral transforms defined by a new fractional class of analytic function in a complex Banach space

0.13 MB·

by  Rabha W. Ibrahim

Integral transforms defined by a new fractional class
of analytic function in a complex Banach space
6
1
0
2
Rabha W. Ibrahim2, Adem Kilicman1,∗ and Zainab E. Abdulnaby1,3

n
a 1 Department of Mathematics, Universiti Putra Malaysia,
J

3 43400 UPM Serdang, Selangor, Malaysia
1
2 Faculty of Computer Science and Information Technology

]
A University Malaya, 50603, Malaysia
F 3 Department of Mathematics, College of science
.
h Al-Mustansiriyah University, Baghdad, Iraq
t
a
∗ corresponding author: Email:
m
[
January 14, 2016

1
v
2
4 Abstract
1
3 In this effort, we definea new class of fractional analytic functions containing
0
functional parameters in the open unit disk. By employing this class, we intro-
.
1 duce two types of fractional operators, differential and integral. The fractional
0
differential operator is considered to be in the sense of Ruscheweyh differential
6
1 operator, while the fractional integral operator is in the sense of Noor integral.
:
v The boundedness and compactness in a complex Banach space are discussed.
i Other studies are illustrated in the sequel.
X
r
a Keywords: Analyticfunctions, Hadamardproduct,Fox- Wright function,norm
Banach space.
1 Introduction
Fractional calculus is a major branch of analysis (real and complex) that deals
withthepossibilityofcaptivatingrealnumberpowersorcomplexnumberpowers
of operators (differential and integral). It has increased substantial admiration
and significance throughout the past four decades, in line for mainly to its es-
tablished requests and applications in various apparently different and extensive
fields and areas of science, medicine and engineering. It does certainly deliver
several possibly advantageous apparatuses for explaining and solving differen-
tial, integral and differ-integral equations, and numerous other difficulties and
1
problems connecting special functions of mathematical physics as well as their
generalizations, modification and extensions in one and more variables (real and
complex) (see [6], [11]). The utensils employed contain numerous standard and
contemporarynonlinearanalysismethodsinrealandcomplex,suchasfixedpoint
theory, boundedness and compactness techniques. It is beneficial to investiga-
torsandresearchers, inpureandappliedmathematics. Theclassical integraland
derivativeareunderstandingandemployingwithnormal,ordinaryandsimulated
methods. FractionalCalculusisafieldofmathematicalstudiesthatproducesout
oftheclassical definitionsofthecalculusintegralandderivative operatorsincon-
siderable the similar technique fractional advocates is an extension of advocates
with integer value.
The theory of geometric function concerns with a special class of analytic func-
tions,whicharedefinedintheopenunitdisk;suchassee[10],theKoebefunction
of first order
z
f(z) = ,
(1−z)
and of second order
z
f(z) = .
(1−z)2
The fractional type of analytic functions is suggested and studied in [10] as
follows:
zα
f(z) = ,
(1−z)α
with α = n+m, n,m ∈ N, where α = 1, in the case n = 1.
m
In this effort, we define a new class of fractional analytic functions F in unit disk
U := {z ∈ C :|z| < 1}, with two functional parametric power as follows:
zµ ∞ (α)
F(z) = = zµ nzµn , z ∈ U
(1−zµ)α (n)!
(cid:18)n=0 (cid:19)
X
∞
(α)
= zµ 1+ nzµn
(n)!
(cid:18) n=1 (cid:19)
X
∞
(α)
= zµ+ n−1 ,zµn (1.1)
(n−1)!
n=2
X
where α ≥ 1 and µ ≥ 1, not that the latter takes it value from the following
relation;
n+m
µ := m,n ∈ N.
m
Hence, we obtain the formal of fractional analytic function:
∞
(α)
F(z) = z+ n−1 zµn,
(n−1)!
n=2
X
2
where µ = 1, when n= 1. Let A be the class of all analytic functions F in unit
µ
disk U and take the form
∞
F(z) := z+ α zµn (1.2)
n
n=2
X
(α)
(µ ≥ 1,|α |≤ n−1 ;α ≥ 1,n ∈ N\{0,1}).
n
(n−1)!
And let X be the class of all normalized analytic functions F in unit disk U
µ
taking the formal
∞
F(z) := z− α zµn, (1.3)
n
n=2
X
(α)
(µ ≥ 1,|α |≤ n−1 ;α ≥ 1,n ∈ N\{0,1}).
n
(n−1)!
For two functions F ∈ A , j = 1,2, give by
j µ
∞
F (z) = z+ α zµn, (j = 1,2),
j n,j
n=2
X
the convolution (or Hadamard) product, denoting by F ∗ F and taking the
1 2
formal
∞
F ∗F (z) = z+ α α zµn,
1 2 n,1 n,2
n=2
X
And
(F ∗F )′(z) = F ∗F′(z), (|z| < 1).
1 2 1 2
Weproceedtodefineanewoperator Dβ,µ :A → A bytheconvolution product
µ µ
of two functions
zµ
Dβ,µF(z) = ∗F(z)
(1−zµ)β+1
∞
(β+1)
= z+ n−1α zµn. (1.4)
n
(n−1)!
n=2
X
Note that
D0,µF(z) = F(z), (z ∈ U).
We expected thatthe operator Dβ,µ is closer to similarities than theRuscheweyh
differential operator (see [9]).
Next, by using (1.4) we aim to define a new integral operator denote by I :
β,µ
A → A as follows: define the functional
µ µ
zµ
F = , (z ∈ U,µ ≥ 1,β ≥ 1)
β (1−zµ)β+1
3
such that
zµ
F (z)∗F−1(z) = . (1.5)
β β 1−zµ
Consequently, we receive the integral operator I defined by
β,µ
I F(z) = F−1(z)∗F(z)
β,µ β
zµ −1
= ∗F(z)
(1−zµ)β+1
(cid:20) (cid:21)
∞
(n−1)!
= z+ α zµn. (1.6)
n
(β +1)
n−1
n=2
X
and it is clear that
I F(z) = F(z), |z| < 1.
0,µ
Forα ≥ 1andµ ≥ 1,thentheintegraloperatorI isclosedtotheNoorIntegral
1,µ
(see [7]) of the n-th order of function F ∈ A . Corresponding to (1.7), we have
µ
the following conclusion:
∞
µ(n)!
zI F′(z) = z+ α zµn (1.7)
β,µ n
(β+1)
n−1
n=2
X
or
∞
Γ(n+1)Γ(β +1)
zI F′(z) = z+ µ α zµn. (1.8)
β,µ n
Γ(n+β)
n=2
X
Inthe following section, weaim to study someproperties of theintegral operator
I .
β,µ
2 Geometric properties of I
β,µ
In this section, we study the geometrical properties of the integral operator I
β,µ
of analytic functions F in class A . First of all, we need the following lemma,
µ
which due to Duren.
Lemma 1. ( Duren [3]) Let the function f(z) ∈ A be a starlike, then |a | ≤ n
n
for all n ≥ 2. And if the function f(z)∈ A is convex, then |a |≤ 1 for all n ≥ 2.
n
Theorem 1. If α≥ 1 and µ ≥ 1 achieving the inequality
α(α+1)...(α+n−2) < n!
then F ∈ S∗, and
µ
I F(z) ≤ Γ(β +1)r2µ Ψ rµ|(3,1)(1,1) ,
β,µ 2 1 (β+2,1)
for all r < 1. (cid:12) (cid:12) h i
(cid:12) (cid:12)
4
Proof. Directly by the assumption of the theorem, we obtain
|α | < n
n
consequently, in view of Lemma 1, this implies that F is starlike, where
α(α+1) α(α+1)...(α+n−2)
α = 1, α = α, α = , ...,α = .
1 2 3 n
2! (n−1)!
We proceed to show that the integral operator I is bounded by a special
β,µ
function.
∞
(n−1)!
I F(z) = z+ α zµn
β,µ n
(β +1)
(cid:12) n=2 n−1 (cid:12)
(cid:12) (cid:12) (cid:12) X (cid:12)
(cid:12) (cid:12) (cid:12) ∞ Γ(n+3)Γ(cid:12)(n+1)rµn
≤ (cid:12)Γ(β+1)r2µ (cid:12) , |z| < r,|α | < n
n
Γ(β +n+2) n!
n=0
X
= Γ(β+1)r2µ Ψ rµ|(3,1)(1,1) ,
2 1 (β+2,1)
h i
where Ψ is the well known Fox-Wright function.
2 1
Similarly, we have the following result:
Theorem 2. If α≥ 1 and µ ≥ 1 achieving the inequality
α(α+1)...(α+n−2) <(n−1)!
then F ∈ C , and
µ
I F(z) ≤ Γ(β+1)r2µ Ψ rµ|(2,1)(1,1)
β,µ 2 1 (β+2,1)
for all r < 1. (cid:12) (cid:12) h i
(cid:12) (cid:12)
Proof. By the hypotheses of the theorem, we obtain
|α |< 1
n
consequently, in view of Lemma 1, this yields F is convex. We proceed to show
that the integral operator I is bounded by a special function.
β,µ
∞
(n−1)!
I F(z) = z+ α zµn
β,µ n
(β +1)
(cid:12) n=2 n−1 (cid:12)
(cid:12) (cid:12) (cid:12) X (cid:12)
(cid:12) (cid:12) (cid:12) ∞ Γ(n+2)Γ(cid:12)(n+1)rµn
≤ (cid:12)Γ(β+1)r2µ (cid:12) , |z| < r,|α | < n
n
Γ(β +n+2) n!
n=0
X
= Γ(β+1)r2µ Ψ rµ|(2,1)(1,1) , r < 1.
2 1 (β+2,1)
h i
5
3 Class of uniformly convex functions
Let E be a Banach space and E† its dual. For any A ∈ E†, we interest the
set W(A) := {w ∈ E : A(w) 6= 0} and let the set γ(A) := {w ∈E :E \W(A)}.
If A 6= 0 then W(A) is dense in E and W(A) ∩ Bˆ is dense in Bˆ, where Bˆ :=
{w ∈ E :||w|| = 1}. Let define B be a complex Banach space and H(B,C) be
a family of all functions f : B → C, such that f(w)| = 0, this means that
w=0
these functions are holomorphic in B and have the Fre´chet derivative f′(w) for
all points w ∈ B.
Recall that : Let Υ and Ξ be two Banach spaces, and Ω ⊂ Υ be an open
subset of V. A function φ : Ω → Ξ is called Fre´chet differentiable at x ∈ Ω if
there exists a bounded linear operator Λ: Υ → Ξ such that
kφ(x+h)−φ(x)−Λhk
Ξ
lim = 0.
h→0 khkΥ
If f ∈ H(B,C), then the function f is defined as the form:
∞
f(w) = P (w). (3.1)
n
n=1
X
Remark 1. We note that, the series P :E → C are
n
1- Uniformly convergent on some neighborhood V of the origin.
2- Continuous and homogeneous polynomials of degree n.
In unit disk U, let denote the family CV of functions by
∞
F(z) = z+ α zµn, (z ∈U), (3.2)
n
n=2
X
which are convex in U = {z ∈C : |z| < 1}.
In [2], Goodman considered geometrically defined the class UCV of uniformly
convex functions, which is subclass of the class CV of convex functions in U. A
function f ∈ CV, if normalized by f(0) = f′(0)−1 = 0 and has the property
that for every (positive oriented) circular arc γ contained in U, with center ζ also
in U the image arc f(γ) is a convex arc.
Theorem A.(Goodman [2]) Let the function f(z) = z + a z2 + a z3 + ···
2 3
analytic in class UCV if and only if
f′′(z)
ℜ (z−ζ) +1 ≥ 0, (z,ζ) ∈ U×U.
f′(z)
(cid:26) (cid:27)
6
Lemma 2. (Goodman [2]) If f ∈ UCV, then
1
|a | ≤ , n ≥ 2.
n
n
The following result is due to Rønning in [8] and Ma and Minda (see [5]).
Theorem B. Let the function f analytic and belongs in UCV if and only if
zf′′(z) zf′′(z)
ℜ +1 ≥ , (|z| < 1).
f′(z) f′(z)
(cid:26) (cid:27) (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3.1 The class of UCV un(cid:12)iform(cid:12)ly convex function
A
µ
Let A∈ E∗, A 6= 0. For any f ∈ H(B,C) of the form
∞
F(w) = A(w)+ P (w), w ∈ B (3.3)
n
n=2
X
for any a ∈ W(A)∩Bˆ we put
F(za)
F (z) = , µ ≥ 1,z ∈ U. (3.4)
a
A(a)
It is clear that
∞
P (a)
F (z) = z+ n zµn, |z| < 1 (3.5)
a
A(a)
n=2
X
In additional, it is easy to obtain
(n)
F (za)(a,··· ,a)
F(n)(z) = a , n∈ N,|z| < 1. (3.6)
a A(a)
Let UCV denote the family of all functions F ∈H(B,C) of the form (3.3) such
Aµ
that, for any a ∈W(A)∩Bˆthe function F belongs to the class UCV. From the
a
following results, we investigate some properties of the functions F in the class
UCV.
Theorem 3. (Bounded coefficient) If the function F is belong in UCV and
Aµ
a ∈ Bˆ. Then
1
P (a) ≤ A(a) , n ≥ 2
n
n
(cid:12) (cid:12) (cid:12) (cid:12)
Proof. Assume that, the fu(cid:12)nction(cid:12)F ∈ U(cid:12) CV (cid:12) , if a ∈ W(A)∩Bˆ, then F ∈ UCV
Aµ a
and from here we have (3.7). In another side, if a ∈ γ(A) ∩ Bˆ, clearly that
a = lim a , where a ∈ W(A),m ∈ N. There exists r ∈ R+ such that
m→∞ m m m
7
am ∈ W(A)∩Bˆ,m ∈ N, it is clear that (r ,m > 0) is bounded for the origin is
rm m
an interior point of B. For am ∈ W(A)∩Bˆ, m ∈ N, we obtain
rm
a 1 a
m m
P ( ) ≤ A( ) , n≥ 2
n
r n r
m m
(cid:12) (cid:12) (cid:12) (cid:12)
consequence (cid:12) (cid:12) (cid:12) (cid:12)
rn−1
P (a ) ≤ m A(a ) , n ≥ 2
n m m
n
lastly by letting m → ∞(cid:12) , we ge(cid:12)t P (a) =(cid:12) 0. (cid:12)
(cid:12) (cid:12) n (cid:12) (cid:12)
Corollary 1. AllthefunctionsF whichbelonginUCV arevanishonγ(A)∩B.
Aµ
Corollary 2. If F ∈ UCV , then
Aµ
1
||P || ≤ ||A||, n ≥ 2.
n
n
Theorem 4. (Sufficient condition) Ifthe functionF belongs to theclass UCV
Aµ
and F′(w) 6=0, for all w ∈ B. Then
F′′(w)(w,w) F′′(w)(w,w)
ℜ 1+ ≥ , w ∈ W(A)∩B. (3.7)
F′(w)(w) F′(w)(w)
(cid:26) (cid:27) (cid:12) (cid:12)
(cid:12) (cid:12)
Proof. Let w ∈ W(A) ∩ B, w 6= 0.(cid:12)(cid:12)Then a = w(cid:12)(cid:12) ∈ W(A) ∩Bˆ and thus the
||w||
F ∈ UCV. By using Theorem B, we get
a
zF′′(z) zF′′(z)
ℜ 1+ a ≥ a , z ∈ U. (3.8)
F′(z) F′(z)
(cid:26) a (cid:27) (cid:12) a (cid:12)
(cid:12) (cid:12)
then, by recall the equality (cid:12) (cid:12)
(cid:12) (cid:12)
zF′′(z) F′′(za)(za,za)
a =
F′(z) F′(za)(za)
a
then, we have
zF′′(z) zF′′(za)(za,za) F′′(za)(za,za)
a = ≤ 1+
F′(z) F′(za)(za) F′(za)(za)
(cid:12) a (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
By putting za = ||w||, we obtain (3.7).
Corollary 3. For F ∈ H(B,C), F′(w)| = A and F′(w) 6= 0, for all w ∈ B. If
w=0
zF′′(w)(w,w) zF′′(w)(w,w)
ℜ 1+ ≥ , w ∈ W(A)∩B. (3.9)
F′(w)(w) F′(w)(w)
(cid:26) (cid:27) (cid:12) (cid:12)
(cid:12) (cid:12)
Then F ∈UCV (cid:12) (cid:12)
Aµ (cid:12) (cid:12)
8
Proof. Let a ∈ W(A)∩Bˆ. Then F′(z) = F′(za)(a) 6= 0, |z| < 1 and
a
zF′′(z) F′′(za)(za,za)
a = , |z| < 1.
F′(z) F′(za)
a
From (3.9), we get F ∈ UCV, for all a ∈ W(A)∩Bˆ, hence F ∈ UCV .
a Aµ
4 Quasi Hadamard-product
In this section, we set up some certain results which dealing with the quasi-
Hadamard product of functions F(w) defined by (3.3) in the class UCV .
Aµ
Let define
∞
F (w) = A(w)+ P (w)zµn j = {1,2,···l}. (4.1)
j n,j
n=2
X
with
∞
P (a )
F (z) = z+ n j zµn j = {1,2,···l},z ∈ U. (4.2)
aj A(a )
j
n=2
X
Also, let define the quasi-Hadamard product of two functions F(w) and G(w) in
class UCV by
Aµ
∞
F(w)∗G(w) := A(w)+ P(w)Φ(w),
n=2
X
where G(w) := A(w)+ ∞ Φ(w) w ∈ B.
n=2
Theorem 5. Let the fuPnction F given by (4.1) be in the class ∈ UCV for
j Aµ
every j = 1,2,···l; and let the function G defined by
i
∞
G (w) = A(w)+ Φ (w)zµn,i =1,2,··· ,s.
i n,i
n=2
X
Then the quasi Hadamard product of more two functions F ∗F ∗···∗F ∗G ∗
1 2 l 1
G ,···∗G (z) belongs to the class UCVl+s.
2 s Aµ
Proof. Let
∞ l s
H(w) = A(w)+ P (w) Φ (w) zµn.
n,j n,i
 
nX=2jY=1 Yi=1 
We aim to show that  
9
∞ l s l s
nl+s P (a ) Φ (a ) ≤ A(a ) A(a ).
n j n i j i
 
nX=2 jY=1 Yi=1  jY=1 Yi=1
Since F ∈ UCV , then from Theorem 3, we obtain (4.3) and (4.4)
j Aµ  
∞
nP (a ) ≤ A(a )
n j j
n=2
X
for every j = 1,2,··· ,l. then we have
A(a )
j
P (a )≤ (4.3)
n j
n
for every j = 1,,2,··· ,s. In a similar way, for G ∈ UCV we get
i Aµ
∞
nΦ (a )≤ A(a ).
n i i
n=2
X
Therefore
A(a )
i
Φ (a ) ≤ (4.4)
n i
n
forevery i= 1,2,··· ,s. By (4.3)and(4.4), for j =1,2,··· ,l andi= 1,2,··· ,s,
we attain
∞ l s
nl+s P (a ) Φ (a )
n j n i
  
nX=2 jY=1 Yi=1 
 
l s
 
≤ nl+s n−sn−l A(a ) A(a ) ,
j i
  
 jY=1 Yi=1 
 
l s
 
≤ A(a ) A(a ) .
j i
 
jY=1 Yi=1 
Hence H(w) ∈UCVl+s 
Aµ
Corollary 4. Let the function F (w) = A(w)+ ∞ P (w)zµn given by (4.1)
j n=2 n,j
be in the class ∈ UCV for every j = 1,2,···l. Then the Hadamard product
Aµ P
F ∗F ,···∗F (z) belongs to the class UCVl .
1 2 l Aµ
Corollary 5. Let the function G (w) = A(w) + ∞ Φ (w)zµn, defined by
i n=2 n,i
(4.1) be in the class ∈ UCV for every i = 1,2,···s. Then the Hadamard
Aµ P
product G ∗G ,···∗G (z) belongs to the class UCVs .
1 2 s Aµ
10