Volume 2, Number 2, 2019, Pages 515-526 https://doi.org/10.34198/ejms.2219.515526
Received: August 4, 2019; Accepted: September 12, 2019 2010 Mathematics Subject Classification: 30C45, 30C50.
Keywords and phrases:multivalent function, extreme points, integral representation, linear operator.
Copyright © 2019 Asraa Abdul Jaleel Husien and Qassim Ali Shakir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution,
On a Certain Subclass for Multivalent Analytic Functions with a Fixed Point Involving Linear Operator
Asraa Abdul Jaleel Husien1 and Qassim Ali Shakir2
1 Technical Institute, Diwaniya, Al-Furat Al-Awsat Technical University, Iraq e-mail: [email protected]
2College of Computer Science and Information Technology, University of Al-Qadisiyah, Iraq e-mail: [email protected]
Abstract
In the present paper, we study a subclass for multivalent analytic functions with a fixed point w defined in the unit disk U involving linear operator. Also, we obtain coefficient estimates, extreme points, integral representation and radii of starlikeness and convexity.
1. Introduction
Denote by A
(
p, w)
the class of functions f of the form:( ) ( ) ∑∞ ( ) ( )
=
+ − + ∈
+
−
=
1
,
n
p n p
n
p a z w p
w z z
f N (1.1)
which are analytic in the unit disk U =
{
z∈C: z <1}
and w is a fixed point in U.Let S
(
p, w)
denote subclass of A(
p, w)
containing of functions of the form:( ) ( ) ∑∞ ( ) ( )
= + +
+ − ≥ ∈
−
−
=
1
. ,
0
n
p p n
p n
p an z w a p
w z z
f N (1.2)
For the functions f ∈S
(
p, w)
given by (1.2) and g∈S(
p,w)
defined by( ) ( ) ∑∞ ( ) ( )
= + +
+ − ≥ ∈
−
−
=
1
, ,
0
n
p n p n p
n
p b z w b p
w z z
g N
we define the Hadamard product of f and g by
( )( ) ( ) ∑∞ ( )
=
+ +
+ −
−
−
=
∗
1
.
n
p p n
n p
p an b z w
w z z g f
For a∈R,c∈R\Z0−, with Z0− =
{
0, −1, −2,...}
, 0 ≤δ<1, p∈N, τ >−p,∈R β
α, with α+β− p <1 and f ∈S
(
p,w)
. The linear operator Lαp,,βw,,δτ(
a,c)
:(
p w)
S(
p w)
S , → , (see [3]) is defined by
( ) ( ) ( ) ∑∞ ( ) ( )
=
+ + δτ
β
α = − + ϕ α β δ τ −
1 ,
, ,
, , , , , , , , , ,
n
p n p
n p
w
p a c f z z w a c n p a z w
L (1.3)
where
( ) ( ) ( ) ( ) ( )
( ) (
1) (
1)
! .1 , 1
, , , , ,
, a p p n
p p
p p c
n c
a
n n
n
n n
n n
β + α
− + +
+ τ β + δ
− + α
−
= + τ
δ β α
ϕ (1.4)
Now, we define the class Swp
(
λ,η,µ, τ,α,β,δ)
consisting the functions(
p w)
S
f ∈ , such that
( )( ( ) ( )) ( )( ( ) ( ))
( )( (
,) ( )) ( )( (
,) ( ))
1,, 2
,
, ,
, , ,
, , ,
, ,
, , ,
, ,
, <
µ ″
− η
′″ +
− λ
− ″
′″ −
−
δτ β τ α
δ β α
δτ β τ α
δ β α
z f c a z
f c a w
z
z f c a p
z f c a w
z
w p w
p
w p w
p
L L
L L
(1.5)
where 0≤ λ<1,0< η≤1,0≤µ <1, p∈N and p >2.
We note other studies of various other classes with different results, like, Ghanim and Darus [2], Najafzadeh and Rahimi [4], Shenan [5], Atshan and Wanas [1] and Wanas [6].
2. Main Results
In the first theorem, we find sharp coefficient estimates for the class
(
λ,η,µ, τ,α,β,δ)
.wp
S
Theorem 2.1. Let f ∈S
(
p, w)
. Then f ∈Swp(
λ, η,µ, τ, α,β, δ)
if and only if( )( ) ( ( ) ) ( )
∑
∞= + + − +η−µ+λ + − ϕ α β δ τ +
1
, , , , , , , 2 1
n
p
an
p n c
a p
n n
p n p n
(
−1) (
η−µ+λ(
−2) )
,≤ p p p (2.1)
where 0≤ λ<1,0 <η≤1,0≤µ <1 and ϕ
(
a,c,α,β, δ, τ,n, p)
is given by (1.4).The result is sharp for the function f given by
( ) (
z z w)
pf = −
( ) ( ( ) )
( )( ) ( ( ) ) ( ) (
z w)
n pp n c
a p
n n
p n p n
p p
p +
τ − δ β α ϕ
− + λ + µ
− η +
− + +
− λ + µ
− η
− −
, , , , , , , 2 1
2 1
(
n ≥1)
. (2.2) Proof. Suppose that the inequality (2.1) holds true and(
z −w)
∈∂U, where ∂U denotes the boundary of U. Then, we find from (1.5) that(
z −w)(
Lαp,,βw,,δτ(
a,c) ( ))
f z ′″ −(
p−2)(
Lαp,,βw,,δτ(
a,c) ( ))
f z ″(
−)( ( ) ( ))
′″ +(
η−µ)( ( ) ( ))
″λ
− z w Lαp,,βw,,δτ a,c f z Lαp,,βw,,δτ a,c f z
( )( ) ( ) ( )
∑
∞=
− + − +
τ δ β α ϕ
− + +
−
=
1
, 2
, , , , , , 1
n
p n p
n z w
a p n c
a p
n p n n
(
−1) (
η−µ+λ(
−2) )(
−)
−2− p p p z w p
∑
∞( )( )
=
− + +
−
1
1
n
p n p n
( )
(
η−µ+λ + −2) (
ϕ , ,α,β,δ,τ, ,)
+(
−)
+ −2× n p a c n p an p z w n p
( )( ) ( ) ( )
∑
∞=
− + − +
τ δ β α ϕ
− + +
≤
1
, 2
, , , , , , 1
n
p p n
n z w
a p n c
a p
n p n n
(
−1) (
η−µ+λ(
−2) )
− −2− p p p z w p
( )( )
∑
∞=
− + +
+
1
1
n
p n p n
( )
(
η−µ+λ + −2) (
ϕ , ,α,β, δ,τ, ,)
+ − + −2× n p a c n p an p z w n p
( )( ) ( ( ) ) ( )
∑
∞= + + − +η−µ+λ + − ϕ α β δ τ +
=
1
, , , , , , , 2 1
n
p
an
p n c
a p
n n
p n p n
(
−1) (
η−µ+λ(
−2) )
≤0.− p p p
Hence, by maximum modulus theorem, we conclude f ∈Swp
(
λ, η,µ, τ, α,β, δ)
. Conversely, suppose that f ∈Swp(
λ,η, µ,τ,α,β, δ)
. Then from (1.3), we have( )( ( ) ( )) ( )( ( ) ( ))
(
−)( ( ) ( ))
′″+(
η−µ)( ( ) ( ))
″λ
− ″
′″ −
−
δτ β τ α
δ β α
δτ β τ α
δ β α
z f c a z
f c a w
z
z f c a p
z f c a w
z
w p w
p
w p w
p
, ,
, 2
,
, ,
, , ,
, , ,
, ,
, , ,
, , ,
L L
L L
( )( ) ( ) ( )
( ) ( ( ) )( ) ( )( )
( )
( ) ( ) ( )
21 2 1
2
, , , , , , , 2
1 2
1
, , , , , , , 1
− + +
∞
=
−
∞
=
− + +
− τ
δ β α ϕ
− + λ + µ
− η
×
− + +
−
−
− λ + µ
− η
−
− τ
δ β α ϕ
− + +
=
∑
∑
p p n
n n
p n
p p n
n
w z a p n c
a p
n
p n p n w
z p
p p
w z a p n c
a p
n p n n
.
<1
So, we obtain
( )( ) ( ) ( )
( ) ( ( ) )( ) ( )( )
( )
( ) ( ) ( )
. 1
, , , , , , , 2
1 2
1
, , , , , , , 1 Re
2 1
2 1
2
<
− τ
δ β α ϕ
− + λ + µ
− η
×
− + +
−
−
− λ + µ
− η
−
− τ
δ β α ϕ
− + +
− + +
∞
=
−
∞
=
− + +
∑
∑
p n p
n n
p n
p n p
n
w z a p n c
a p
n
p n p n w
z p
p p
w z a p n c
a p
n p n n
By letting
(
z −w)
→1−, through real values, we have( )( ) ( ( ) ) ( )
∑
∞= + + − +η−µ+λ + − ϕ α β δ τ +
1
, , , , , , , 2 1
n
p
an
p n c
a p
n n
p n p n
(
−1) (
η−µ+λ(
−2) )
.≤ p p p
Corollary 2.1. Let f ∈Swp
(
λ,η,µ, τ,α,β,δ)
. Then( ) ( ( ) )
(
n p)(
n p) (
n(
n p) ) (
a c n p)
p p
an p p
, , , , , , , 2 1
2 1
τ δ β α ϕ
− + λ + µ
− η +
− + +
− λ + µ
− η
≤ −
+
(
n ≥1)
. In the next result, we discuss extreme points for the class Swp(
λ,η,µ,τ, α,β, δ)
. Theorem 2.2. Let fp( ) (
z = z−w)
p and( ) ( )
pp
n z z w
f + = −
( ) ( ( ) )
( )( ) ( ( ) ) ( ) (
z w)
n pp n c
a p
n n
p n p n
p p
p +
τ − δ β α ϕ
− + λ + µ
− η +
− + +
− λ + µ
− η
− −
, , , , , , , 2 1
2 1
(
n ≥1)
. Then f ∈Swp(
λ, η,µ, τ, α,β, δ)
if and only if it can be expressed in the form( ) ∑∞ ( )
= γ + +
=
0
,
n
p n p
n f z
z
f (2.3)
where
∑
∞= +
+ ≥ γ =
γ
0
. 1 ,
0
n p n p
n
Proof. Let the f of the form (2.3). Then
( ) ( )
( )
−
γ + γ
=
∑
∞= + p
n p n p
pf z z w
z f
1
( ) ( ( ) )
( )( ) ( ( ) ) ( ) ( )
− τ
δ β α ϕ
− + λ + µ
− η +
− + +
− λ + µ
− η
− − z w n+p
p n c
a p
n n
p n p n
p p
p
, , , , , , , 2 1
2 1
( ) ( ) ( ( ) )
( )( ) ( ( ) ) ( )
∑
∞= + + − +η−µ+λ + − ϕ α β δ τ
− λ + µ
− η
− −
−
=
1 1 2 , , , , , , ,
2 1
n p
p n c
a p
n n
p n p n
p p
w p z
( )
n p.p
n+ z−w + γ
× Now,
( )( ) ( ( ) ) ( )
( ) ( ( ) )
∑
∞= − η−µ +λ −
τ δ β α ϕ
− + λ + µ
− η +
− + +
1 1 2
, , , , , , , 2 1
n p p p
p n c
a p
n n
p n p n
( ) ( ( ) )
(
n p)(
n p) (
n(
n p) ) (
a c n p)
n pp p
p γ +
τ δ β α ϕ
− + λ + µ
− η +
− + +
− λ + µ
− η
× −
, , , , , , , 2 1
2 1
∑
∞=
+ = −γ ≤ γ
=
1
. 1 1
n
p p
n
Thus f ∈Swp
(
λ,η,µ, τ,α,β,δ)
.Conversely, let f ∈Swp
(
λ,η,µ,τ,α,β,δ)
. It follows from Corollary 2.1 that( ) ( ( ) )
( )( ) ( ( ) ) ( ) (
1)
., , , , , , , 2 1
2
1 ≥
τ δ β α ϕ
− + λ + µ
− η +
− + +
− λ + µ
− η
≤ −
+ n
p n c
a p
n n
p n p n
p p
an p p
Setting
( )( ) ( ( ) ) ( )
( ) ( ( ) )
n pp
n a
p p
p
p n c
a p
n n
p n p
n +
+ − η−µ+λ −
τ δ β α ϕ
− + λ + µ
− η +
− +
= +
γ 1 2
, , , , , , , 2 1
(
n≥1)
and
∑
∞= γ +
−
= γ
1
, 1
n p n
p we have
( ) ( ) ∑∞ ( )
=
+ − +
−
−
=
1 n
p p n
p an z w
w z z f
( ) ( ) ( ( ) )
( )( ) ( ( ) ) ( )
∑
∞= + + − +η−µ+λ + − ϕ α β δ τ
− λ + µ
− η
− −
−
=
1 1 2 , , , , , , ,
2 1
n p
p n c
a p
n n
p n p n
p p
w p z
( )
n pp
n+ z−w + γ
×
( ) (( )
n p( ))
n pn
p
p z w f z
w
z + +
∞
=
γ
−
−
−
−
=
∑
1
( ) ∑ ( )
∑
∞= + +
∞
= + − + γ
− γ
=
1 1
1
n
p n p p n
n p
n z w f z
( ) ∑∞ ( ) ∑ ( )
=
∞
= + +
+
+ = γ
γ + γ
=
1 0
,
n n
p n p n p
n p n p
pf z f z f z
that is the required representation.
In the following theorem, we establish integral representation for functions belongs to the class Swp
(
λ, η,µ, τ,α,β,δ)
.Theorem 2.3. Let f ∈Swp
(
λ,η,µ,τ,α,β,δ)
. Then( ) ( ) ( ) ( )
( ) ( ( ) )
∫ ∫
∫
η−−µ ψ −λψ+ − τ =
δ β α
z z z
w
p dt dt dt
t w
t
p z t
f c a
0 0 2 3
0 1
1 1
, 1 ,
,
, ,
1 exp 2
, L
where ψ
( )
z <1, z∈U.Proof. By putting
( )( ( ) ( ))
( ( ) ( ))
Q( )
zz f c a
z f c a w
z
w p
w p
″ =
− ′″
δτ β α
δτ β α
, ,
, ,
, ,
, ,
, ,
L L
in (1.5), we have
( ) ( )
1,2 <
µ
− η + λ
+
− z Q
p z Q
or equivalently
( ) ( )
2( )
,( ( )
1,)
.U z z
z z Q
p z
Q = ψ ψ < ∈
µ
− η + λ
+
−
So
( ( ) ( ))
( ( ) ( ))
( ) ( )
( ) (
1( ) )
, 2 ,,
, ,
, ,
, ,
, ,
z w
z
p z z
f c a
z f c a
w p
w p
λψ
−
−
− + ψ µ
−
= η
″
′″
δτ β α
δτ β α
L L
after integration, we get
(( ( ) ( )) ) ( ) ( )
( ) ( ( ) )
∫
η−−µ ψ −λψ+ −″ =
δτ β α
w z
p dt
t w
t
p z t
f c a
0 1
1 1
, 1 ,
,
, .
1 , 2
log L Therefore
( ( ) ( )) ( ) ( )
( ) (
1( ) )
.exp 2 ,
0 1
1 1
, 1 ,
,
,
λψ
−
−
− + ψ µ
−
= η
″
∫
δτ β α
w z
p dt
t w
t
p z t
f c a L
By integration once again, we have
( ( ) ( )) ( ) ( )
( ) (
1( ) )
.exp 2
, 2
0 0 1
1 1
, 1 ,
,
, dt dt
t w
t
p z t
f c
a z z
w
pβ δτ ′ =
∫
∫
η−−µ ψ −λψ+ − Lα
Also, after integration, we conclude that
( ) ( ) ( ) ( )
( ) ( ( ) )
30 2
0 0 1
1 1
, 1 ,
,
, 1
exp 2
, dt dt dt
t w
t
p z t
f c
a z z z
w
pβδτ =
∫ ∫
∫
η−−µ ψ −λψ+ − Lα
and this the required result.
Theorem 2.4. If f ∈Swp
(
λ,η,µ,τ,α,β,δ)
, then f is starlike of order θ(
0≤ θ< p)
in the disk z −w <r1, where( )( )( ) ( ( ) ) ( )
(
1)( ) ( (
2) )
., , , , , , , 2 inf 1
1 1
n
n p p n p p
p n c
a p
n n
p n p p r n
− λ + µ
− η θ
− +
−
τ δ β α ϕ
− + λ + µ
− η +
− + θ
−
= +
The result is sharp for the function f given by (2.2).
Proof. It is sufficient to show that
( ) ( )
( )
′ − ≤ −θ− p p
z f
z f w
z for z−w < r1. (2.4)
But
( ) ( )
( )
( )
( ) ( )
. 1
1 1
1 1
∑
∑
∑
∑
∞
= +
∞
= +
∞
= + +
∞
= + +
−
−
−
≤
−
−
−
−
−
=
′ −
−
n p n n n
p n n
n
p p n
p n n
p p n
n
w z a
w z na
w z a w
z
w z na z p
f z f w z
Thus (2.4) will be satisfied if
, 1
1
1 ≤ −θ
−
−
−
∑
∑
∞
= +
∞
= +
p w z a
w z na
n p n n n
n p
n
or if
( )
( )
∑
∞= + − ≤
θ
− θ
− +
1
, 1
n
n p
n z w
p a p
n (2.5)
with the aid of (2.1), (2.5) is true if
( )
(
p)
z w np
n −
θ
− θ
− +
( )( ) ( ( ) ) ( )
(
1) ( (
2) )
,, , , , , , , 2 1
− λ + µ
− η
−
τ δ β α ϕ
− + λ + µ
− η +
− +
≤ +
p p
p
p n c
a p
n n
p n p n
or equivalently w z −
( )( )( ) ( ( ) ) ( )
( )( ) ( ( ) )
n
p p
n p p
p n c
a p
n n
p n p p n
1
2 1
, , , , , , , 2 1
− λ + µ
− η θ
− +
−
τ δ β α ϕ
− + λ + µ
− η +
− + θ
−
≤ +
(
n ≥1)
, which follows the result.Theorem 2.5. If f ∈Swp
(
λ,η,µ,τ,α,β,δ)
, then f is convex of order θ(
0≤ θ< p)
in the disk z−w <r2, where( )( ) ( ( ) ) ( )
(
1)( ) ( (
2) )
., , , , , , , 2 inf 1
1 2
n
n p n p p
p n c
a p
n n
p n r p
− λ + µ
− η θ
− +
−
τ δ β α ϕ
− + λ + µ
− η +
− + θ
= −
The result is sharp for the function f given by (2.2).
Proof. It is sufficient to show that
( ) ( )
( )
+ − ≤ −θ′
− ′′
p z p
f z f w
z 1 for z−w < r2. (2.6)
But
( ) ( )
( )
( ) ( )
( ) ∑ ( ) ( )
∑
∞
=
− + +
−
∞
=
− + +
− +
−
−
− +
−
=
−
′ +
− ′′
1
1 1
1
1
1
n
p n p
n p
n
p p n
n
w z a p n w
z p
w z a p n n z p
f z f w z
( )
( )
.
1 1
∑
∑
∞
= +
∞
= +
− +
−
− +
≤
n
p n n n
n p
n
w z a p n p
w z a p n n
Thus (2.6) will be satisfied if
( )
( )
,
1
1 ≤ −θ
− +
−
− +
∑
∑
∞
= +
∞
=
+
p w z a p n p
w z a p n n
n
p n n n
n p
n
or if
( )( )
( )
∑
∞= + − ≤
θ
− θ
− + +
1
, 1
n
n p
n z w
p a p
p n p
n (2.7)
with the aid of (2.1), (2.7) is true if
( )( )
(
p)
z w np p n p
n −
θ
− θ
− + +
( )( ) ( ( ) ) ( )
(
1) ( (
2 2) )
, , , , , , , ,1
− λ + µ
− η
−
τ δ β α ϕ
− + λ + µ
− η +
− +
≤ +
p p
p
p n c
a p
n n
p n p n
or equivalently
( )( ) ( ( ) ) ( )
(
p)(
n p) ( (
p) )
np n c
a p
n n
p n w p
z
1
2 1
, , , , , , , 2 1
− λ + µ
− η θ
− +
−
τ δ β α ϕ
− + λ + µ
− η +
− + θ
≤ −
−
(
n ≥1)
, which follows the result.References
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https://doi.org/10.22436/jmcs.08.04.01
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