Volume 10, Number 2, 2022, Pages 271-288 https://doi.org/10.34198/ejms.10222.271288
Received: June 20, 2022; Accepted: July 2, 2022 2020 Mathematics Subject Classification: 30C45.
Keywords and phrases: holomorphic function, multivalent functions, coefficient inequality, radii of starlikeness and convexity, closure theorem, extreme points.
*
New Class of Multivalent Functions with Negative Coefficients Ali Mohammed Ramadhan1 and Najah Ali Jiben Al-Ziadi2,*
1 Department of Mathematics, College of Education, University of Al-Qadisiyah, Diwaniya, Iraq e-mail: [email protected]
2 Department of Mathematics, College of Education, University of Al-Qadisiyah, Diwaniya, Iraq e-mail: [email protected]
Abstract
In the present paper, we define a new class ( , , , , ) of multivalent functions which are holomorphic in the unit disk ∆ = ∈ℂ∶ | | < 1 . A necessary and sufficient condition for functions to be in the class ( , , , , ) is obtained. Also, we get some geometric properties like radii of starlikeness, convexity and close-to-convexity, closure theorems, extreme points, integral means inequalities and integral operators.
1. Introduction
Let be symbolize the function class of the form:
( ) = +
∞
( ∈ ∆; ≥ + 1; ∈ℕ= 1,2, … ), (1.1)
which are holomorphic and multivalent in the open unit disk ∆ = ∈ℂ∶ | | < 1 . Let be symbolize the function subclass of containing of functions of the form:
( ) = −
∞
( ∈ ∆; ≥ 0; ≥ + 1; ∈ℕ= 1,2, … ). (1.2)
For function ( ) ∈ , given by (1.2), and ℎ( ) ∈ given by ℎ( ) = − &
∞
( ∈ ∆; & ≥ 0; ≥ + 1; ∈ℕ= 1,2, … ), (1.3)
the convolution (or Hadamard product) of ( ) and ℎ( ) is defined by ( ∗ℎ)( ) = − &
∞
= (ℎ∗ )( ). (1.4)
A function ( ) ∈ is called multivalent starlike of order * (0 ≤ * < ), if ( ) satisfies the condition:
,- .
′( )
( ) / > * ( ∈ ∆; 0 ≤ * < ; ∈ℕ= 1,2, … ). (1.5) Also, a function ( ) ∈ is called multivalent convex of order *(0 ≤ * < ), if ( ) satisfies the condition:
,- .1 +
′′( )
′( ) / > * ( ∈ ∆; 0 ≤ * < ; ∈ℕ= 1,2, … ). (1.6) Symbolize by 3∗( , *) the class of multivalent starlike functions of order *. Symbolize by 4 ( , *) the class of multivalent convex functions of order *, which were studied by Owa [11]. It is observed that
( ) ∈ 4 ( , *) if and only if 56
′(5)∈ 3∗( , *).
A function ( ) ∈ is called multivalent close to convex of order *(0 ≤ * < ), if ( ) satisfies the condition:
,- .
′( )
7 / > * ( ∈ ∆; 0 ≤ * < ; ∈ℕ= 1,2, … ). (1.7) We recall the principle of subordination between two holomorphic functions ( ) and ℎ( ) in 9. It is known that ( ) is subordinate to ℎ( ), written as ( ) ≺ℎ( ), ∈ 9, if there is a ;( ) holomorphic in 9, with ;(0) = 0 and |;( )| < 1, ∈ 9, such that
( ) =ℎ<;( )=. Moreover, ( ) ≺ℎ( ) is equivalent to (0) =ℎ(0) and (9) ⊂ℎ(9), if ( ) is univalent in 9.
Definition 1.1. Let ( ) ∈ given by (1.2), is said to be in the class ( , , , , ) if and only if satisfies the following inequality:
,- ? ′( ) + @ ′′( )
′( ) + (1 − ) ( )A > B
′( ) + @ ′′( )
′( ) + (1 − ) ( ) − B + , (1.8) where ∈ ∆, 0 ≤ < , ≥ 0, 0 ≤ ≤ 1, ≥ + 1 and ∈ℕ= 1,2, … .
We note that by customizing the parameter , , , we get the following various subclasses as studied by different researchers:
1) If = 1, the class ( , , , , ) reduces to the class D4E( , , ) which is introduced by Khairnar and More [8].
2) If = 1 and = 0, the class ( , , , , ) reduces to the class 4 ( , ) which is studied by Owa [11] and Sălăgean et al. [12].
3) If = 1 and = 1, the class ( , , , , ) reduces to the class D4I( , ) which is studied by Bharati et al. [6].
4) If = 1, = 1 and = 0, the class ( , , , , ) reduces to the class 4( ) which is studied by Silverman [13].
5) If = 0, = 1 and = 0, the class ( , , , , ) reduces to the class J(1, ) which is studied by Lashin [9].
Lemma 1.1 [3]. Let K = L + MN be complex number. Then ,-(K) ≥ if and only if
|K − ( + )| ≤ |K + ( − )|, where ≥ 0.
Lemma 1.2 [3]. Let K = L + MN and , be real numbers. Then ,-(K) ≥ |K − | + if and only if ,-OK<1 + -PQ= − -PQR ≥ .
Lemma 1.3 [10]. If and ℎ are holomorphic in ∆ with ≺ ℎ, then T U (V-PQ)UW X ≤
@Y Z
T Uℎ(V-PQ)UW X,
@Y
Z
where [ > 0, = V-PQ and (0 < V < 1).
Some of the following properties studied for other classes in [1, 2, 4, 5, 7, 14].
2. Coefficient Inequality
From the following theorem, we get the necessary and sufficient condition for the function ( ) to be in the class ( , , , , ).
Theorem 2.1. Let ( ) be in the form (1.2). Then ( ) is in the class ( , , , , ) if and only if
[ @( + 1) − ( + )(1 + − )] ≤
∞
@( + 1) − ( + )(1 + − ), (2.1)
where ∈ ∆, 0 ≤ < , ≥ 0, 0 ≤ ≤ 1, ≥ + 1 and ∈ℕ= 1,2, … . The result is sharp for the function ( ) given by
( ) = − @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ) , ( ≥ + 1; ∈ℕ). (2.2) Proof. Let ( ) ∈ ( , , , , ). Then ( ) satisfies the inequality (1.8). By using Lemma 1.2, the inequality (1.8) is equivalent to
,- ?( ′( ) + @ ′′( )) (1 + -PQ)
′( ) + (1 − ) ( ) − -PQA ≥
(0 ≤ < , ≥ 0, 0 ≤ ≤ 1, ≥ + 1, ∈ℕ and − ^ < X ≤ ^). Or equivalently,
,- ?( ′( ) + @ ′′( ))(1 + -PQ)
′( ) + (1 − ) ( ) − -PQ< ′( ) + (1 − ) ( )=
′( ) + (1 − ) ( ) A ≥ . (2.3) Let
_( ) = ( ′( ) + @ ′′( ) )(1 + -PQ) − -PQ< ′( ) + (1 − ) ( )=.
`( ) = ′( ) + (1 − ) ( ).
Then by Lemma 1.1, (2.3) is equivalent to
|_( ) + ( − )`( )| ≥ |_( ) − ( + )`( )| abV 0 ≤ < . Now
|_( ) + ( − )`( )|
=
c c
cd −
∞
+ ( − 1) − ( − 1)
∞
e <1 + -PQ=
− -PQd −
∞
+ (1 − ) − (1 − )
∞
e
+( − ) d −
∞
+ (1 − ) − (1 − )
∞
e c c c
= c
c d −
∞
+ ( − 1) − ( − 1)
∞
e <1 + -PQ=
+< − − -PQ=d −
∞
+ (1 − ) − (1 − )
∞
ec c
= cc
@<1 + -PQ= + < − − -PQ=( 1 + − )
− @<1 + -PQ=
∞
− < − − -PQ=(1 + − )
∞ cc
≥ < @(1 + ) + ( − − )(1 + − )=| |
− < @(1 + ) + ( − − )(1 + − )=
∞
| | .
Similarly,
|_( ) − ( + )`( )|
= c
c d −
∞
+ ( − 1) − ( − 1)
∞
e <1 + -PQ=
−< + + -PQ=d −
∞
+ (1 − ) − (1 − )
∞
ec c
≤ <( + + )(1 + − ) − @(1 + )=| |
+ < @(1 + ) − ( + + )(1 + − )=
∞
| | .
Therefore
|_( ) + ( − )`( )| − |_( ) − ( + )`( )|
≥ @( + 1) − ( + )(1 + − )
− [ @( + 1) − ( + )(1 + − )] ≥ 0.
∞
Hence
[ @( + 1) − ( + )(1 + − )] ≤
∞
@( + 1) − ( + )(1 + − ).
Conversely, by considering (2.1), we must show that ,- ?( ′( ) + @ ′′( )) (1 + -PQ)
′( ) + (1 − ) ( ) −( -PQ+ )( ′( ) + (1 − ) ( ))
′( ) + (1 − ) ( ) A ≥ 0. (2.4) Upon choosing the values of on the positive real axis where 0 ≤ = V < 1, ,-<−-PQ= ≥ −U-PQU = −1 and letting V → 17, we conclude to (2.4) by using (2.1) in left hand of (2.4).
Corollary 2.1. Let ( ) be in the class ( , , , , ). Then
≤ @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ), (2.5) where 0 ≤ < , ≥ 0, 0 ≤ ≤ 1, ≥ + 1 and ∈ℕ= 1,2, … .
3. Radii of Starlikeness, Convexity and Close-to-Convexity
In the next theorems, we will find the radii of starlikeness, convexity and close-to- convexity for the functions in the class ( , , , , ).
Theorem 3.1. Let ( ) ∈ ( , , , , ). Then the function ( ) is multivalent starlike of order *(0 ≤ * < ) in the disk | | < , , where
, = inf i( − *)< @( + 1) − ( + )(1 + − )=
( − *)< @( + 1) − ( + )(1 + − )=j
lmnk
, ( ≥ + 1; ∈ℕ).
The result is sharp for the function ( ) given by (2.2). Proof. It is sufficient to prove
B ′( )
( ) − B ≤ − * (0 ≤ * < ), for | | < , , we get
B ′( ) ( ) − B ≤
∑∞ ( − ) | | 7 1 − ∑∞ | | 7 .
Thus
B ′( )
( ) − B ≤ − * , if
− *
− * | | 7 ≤ 1. (3.1)
∞
Therefore, by using Theorem 2.1, (3.1) will be true if
− *
− * | | 7 ≤ @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ).
Hence
| | ≤ i( − *)< @( + 1) − ( + )(1 + − )=
( − *)< @( + 1) − ( + )(1 + − )=j
lmnk
, ( ≥ + 1; ∈ℕ ).
Setting | | = , , we obtain the desired result.
Theorem 3.2. Let ( ) ∈ ( , , , , ). Then the function ( ) is multivalent convex of order *(0 ≤ * < ) in the disk | | < ,@, where
,@= inf i ( − *)< @( + 1) − ( + )(1 + − )=
( − *)< @( + 1) − ( + )(1 + − )=j
lmnk
, ( ≥ + 1; ∈ℕ ).
The result is sharp for the function ( ) given by (2.2). Proof. It is sufficient to show that
B1 + ′′( )
′( ) − B ≤ − * (0 ≤ * < ), for | | < ,@, we get
B1 + ′′( )
′( ) − B ≤
∑∞ ( − ) | | 7
− ∑∞ | | 7 .
Thus
B1 + ′′( )
′( ) − B ≤ − *,
if
( − *)
( − *) | | 7 ≤ 1. (3.2)
∞
Therefore, by using Theorem 2.1, (3.2) will be true if ( − *)
( − *) | | 7 ≤ @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ), and hence
| | ≤ i ( − *)< @( + 1) − ( + )(1 + − )=
( − *)< @( + 1) − ( + )(1 + − )=j
lmnk
, ( ≥ + 1; ∈ℕ ).
Setting | | = ,@, we get the desired result.
Theorem 3.3. Let ( ) ∈ ( , , , , ). Then the function ( ) is multivalent close to convex of order *(0 ≤ * < ) in the disk | | < ,q, where
,q= inf i( − *)< @( + 1) − ( + )(1 + − )=
< @( + 1) − ( + )(1 + − )= j
lmnk
, ( ≥ + 1; ∈ℕ ).
The result is sharp for the function ( ) given by (2.2). Proof. It is sufficient to show that
B ′( )
7 − B ≤ − * (0 ≤ * < ), for | | < ,q, we get
B ′( )
7 − B ≤ | | 7 .
∞
Thus
B ′( )
7 − B ≤ − *, if
| | 7
− * ≤ 1. (3.3)
∞
Therefore, by using Theorem 2.1, (3.3) will be true if
− * | | 7 ≤ @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ), and hence
| | ≤ i( − *)< @( + 1) − ( + )(1 + − )=
< @( + 1) − ( + )(1 + − )= j
lmnk
, ( ≥ + 1; ∈ℕ ).
The result is sharp for the function ( ) given by (2.2).
4. Closure Theorems
Theorem 4.1. Let the functions r defined as
r( ) = − ,r
∞
, < ,r ≥ 0, ≥ + 1, ∈ℕ, L = 1,2, … , s=, ( 4.1 )
be in the class ( , , , , ) for every L = 1,2, … , s.
Then the function t ( ) defined as
t ( ) = − -
∞
, (- ≥ 0, ≥ + 1, ∈ℕ ),
also belongs to the class ( , , , , ), where - =1
s ,r
u r
, ( ≥ + 1, ∈ℕ).
Proof. Since r∈ ( , , , , ) it follows from Theorem 2.1 that [ @( + 1) − ( + )(1 + − )] ,r≤
∞
@( + 1) − ( + )(1 + − ),
for every L = 1,2, … , s. Therefore
[ @( + 1) − ( + )(1 + − )]-
∞
= [ @( + 1) − ( + )(1 + − )] d1
s ,r
u r
e
∞
=1 s
u r
d [ @( + 1) − ( + )(1 + − )] ,r
∞
e
≤ @( + 1) − ( + )(1 + − ).
Using Theorem 2.1, it follows that t ( ) ∈ ( , , , , ).
Theorem 4.2. Let the function r defined by (4.1) be in the class ( , , , , ) for every L = 1,2, … , s. Then the function t@( ) defined as
t@( ) = vr u r
r( ) is also in the class ( , , , , ), where
vr= 1
u r
, (vr≥ 0).
Proof. Using Theorem 2.1, for every L = 1,2, … , s, we get [ @( + 1) − ( + )(1 + − )] ,r≤
∞
@( + 1) − ( + )(1 + − ).
But
t@( ) = vr
u
r r( ) = vr
u r
d − ,r
∞
e = − d vr ,r
u r
e
∞
.
Therefore
[ @( + 1) − ( + )(1 + − )] d vr ,r
u r
e
∞
= vr
u r
d [ @( + 1) − ( + )(1 + − )] ,r
∞
e
≤ vr
u r
< @( + 1) − ( + )(1 + − )=
= @( + 1) − ( + )(1 + − ) and the proof is complete.
Corollary 4.1. The class ( , , , , ) is close under convex linear combination.
5. Extreme Points
We get here an extreme point of the class ( , , , , ).
Theorem 5.1. Let ( ) = and
( ) = − @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ) , (5.1) where ∈ ∆, 0 ≤ < , ≥ 0, 0 ≤ ≤ 1, ≥ + 1 and ∈ℕ= 1,2, … .
Then ( ) ∈ ( , , , , ) if and only if it can be expressed as:
( ) = * + * ( ),
∞
(5.2)
where <* ≥ 0, * ≥ 0, ≥ + 1= and * + ∑∞ * = 1.
Proof. Suppose that ( ) is represented in the form (5.2). Then
( ) = * + * i − @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ) j
∞
= − @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ) *
∞
.
Hence
@( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − )
∞
× @( + 1) − ( + )(1 + − )*
@( + 1) − ( + )(1 + − )
= *
∞
= 1 − * ≤ 1.
Then ( ) ∈ ( , , , , ).
Conversely, suppose that ( ) ∈ ( , , , , ). We may set
* = @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ) , where is given by (2.5). Then
( ) = −
∞
= − @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ) *
∞
= − [
∞
− ( )]* = d1 − *
∞
e + * ( )
∞
= * + * ( ).
∞
This completes the proof of Theorem 5.1.
6. Integral Means Inequalities
By using Theorem 2.1 and Lemma 1.3, we prove the following theorems.
Theorem 6.1. Let [ > 0. If ( ) ∈ ( , , , , ) and suppose that x( ) is defined by
x( ) = − @( + 1) − ( + )(1 + − )
v@( + 1) − ( + )(1 + v − ) x, (v ≥ + 1; ∈ℕ).
If there is a holomorphic function ;( ) defined by
<;( )=x7 = v@( + 1) − ( + )(1 + v − )
@( + 1) − ( + )(1 + − ) 7 .
∞
Then, for = V-PQ and (0 < V < 1), T | ( )|W X ≤
@Y Z
T | x( )|W X, ([ > 0).
@Y Z
(6.1)
Proof. We must show that
T y1 − 7
∞
y
W
X ≤
@Y Z
T B1 − @( + 1) − ( + )(1 + − )
v@( + 1) − ( + )(1 + v − ) x7 BW X
@Y Z
.
By using Lemma 1.3, it suffices to show that
1 − 7
∞
≺ 1 − @( + 1) − ( + )(1 + − ) v@( + 1) − ( + )(1 + v − ) x7 . Put
1 − 7
∞
= 1 − @( + 1) − ( + )(1 + − )
v@( + 1) − ( + )(1 + v − ) <;( )=x7 . We find that
<;( )=x7 = v@( + 1) − ( + )(1 + v − )
@( + 1) − ( + )(1 + − ) 7 ,
∞
that yield easily ;(0) = 0.
In addition by using (2.1), we get
|;( )|x7 = yv@( + 1) − ( + )(1 + v − )
@( + 1) − ( + )(1 + − ) 7
∞
y
≤ | | y
∞ @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ) y
≤ | | < 1.
Next, the proof for the first derivative.
Theorem 6.2. Let [ > 0. If ( ) ∈ ( , , , , ) and
x( ) = − @( + 1) − ( + )(1 + − )
v@( + 1) − ( + )(1 + v − ) x, (v ≥ + 1; ∈ℕ).
Then, for = V-PQ and (0 < V < 1), T | ′( )|W X ≤
@Y Z
T U x′( )UW X, ([ > 0). (6.
@Y Z
2)
Proof. It is sufficient to show that
1 − 7
∞
≺ 1 −v< @( + 1) − ( + )(1 + − )=
<v@( + 1) − ( + )(1 + v − )= x7 . This follows because
|;( )|x7 = y <v@( + 1) − ( + )(1 + v − )=
v< @( + 1) − ( + )(1 + − )= 7
∞
y
≤ | |y
∞ @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ) y
≤ | | < 1.
Theorem 6.3. Let ℎ( ) = − ∑z & ( ∈ ∆; & ≥ 0; ≥ + 1; ∈ ℕ = 1,2, … ) and ( ) ∈ ( , , , , ) be of the form (1.2) and let for some v ∈ ℕ,
|x
&x = min~ |
& , where
| = @( + 1) − ( + )(1 + − )
@( + 1) − ( + )(1 + − ).
Also, let for such v ∈ ℕ, the functions x and ℎx be defined by
x( ) = − @( + 1) − ( + )(1 + − ) v@( + 1) − ( + )(1 + v − ) x,
ℎx( ) = − &x x. (6.3)
If there is a holomorphic function ;( ) defined by
<;( )=x7 = v@( + 1) − ( + )(1 + v − )
< @( + 1) − ( + )(1 + − )=&x & 7 ,
z
then, for [ > 0, = V-PQ and (0 < V < 1), T |( ∗ ℎ)( )|W X ≤
@Y Z
T |( x∗ ℎx)( )|W X, ([ > 0).
@Y
Z
Proof. Convolution of k(s) and h(s) is defined by ( ∗ ℎ)( ) = − &
z
.
Similarly, from (6.3), we get
( x∗ ℎx)( ) = − < @( + 1) − ( + )(1 + − )=&x v@( + 1) − ( + )(1 + v − ) x. To prove the theorem, we must show that for [ > 0, = V-PQ and (0 < V < 1),
T y1 − & 7
z
y
W
X
@Y Z
≤ T B1 −< @( + 1) − ( + )(1 + − )=&x v@( + 1) − ( + )(1 + v − ) x7 B
W
X
@Y Z
.
Therefore, using Lemma 1.3, it is sufficient to show that
1 − & 7
z
≺ 1 −< @( + 1) − ( + )(1 + − )=&x
v@( + 1) − ( + )(1 + v − ) x7 . (6.4) If the subordination (6.4) is correct, then there is a holomorphic function ;( ) with
|;( )| < 1 and ;(0) = 0 such that
1 − & 7
z
= 1 −< @( + 1) − ( + )(1 + − )=&x
v@( + 1) − ( + )(1 + v − ) <;( )=x7 .
According to the assumption of the theorem, there is a holomorphic function ;( ) given by
<;( )=x7 = v@( + 1) − ( + )(1 + v − )
< @( + 1) − ( + )(1 + − )=&x & 7 ,
z
which readily yield ;(0) = 0. So for such function ;( ), using the assumption in the coefficient inequality for the class ( , , , , ), we have
|;( )|x7 = y v@( + 1) − ( + )(1 + v − )
< @( + 1) − ( + )(1 + − )=&x & 7
z
y
≤ | | y v@( + 1) − ( + )(1 + v − )
< @( + 1) − ( + )(1 + − )=&x &
z
y
≤ | | < 1.
Therefore, the subordination (6.4) holds true.
7. Integral Operators
In this segment, we consider integral transforms of functions in the class ( , , , , ).
Theorem 7.1. Let ( ) ∈ ( , , , , ) be defined by (1.2) and & be any real number such that & > − . Then the integral operator
‚( ) =& +
ƒ T vƒ7 (v) v( & > − ),
5 Z
(7.1)
also in the class ( , , , , ).
Proof. By virtue of (7.1) it follows from (1.2) that
‚( ) =& +
ƒ T vƒ7 dv −
z
v e v
5 Z
=& +
ƒ T dv ƒ7 −
z
v ƒ7 e v
5 Z
= − „& +
& + …
z
= − ℎ
z
,
whereℎ = †ƒƒ ‡ . But
[ @( + 1) − ( + )(1 + − )]ℎ
z
= [ @( + 1) − ( + )(1 + − )] „& +
& + …
z
.
Since †ƒƒ ‡ ≤ 1 and by (2.1), the last expression is less than or equal to @( + 1) − ( + )(1 + − ). This ends the proof.
References
[1] R. Agarwal, J. Gupta and G. S. Paliwal, Geometric properties and neighborhood results for a subclass of analytic functions defined by convolution, International Bulletin of Mathematical Research 2(4) (2015), 5-15.
[2] M. K. Aouf, A. O. Mostafa and A. A. Hussain, Certain subclass of p-valent starlike and convex uniformaly functions defined by convolution, Int. J. Open Problems Compt.
Math. 9(1) (2016), 36-60. https://doi.org/10.12816/0026353
[3] E. S. Aqlan, Some problems connected with geometric function theory, Ph.D. Thesis, Pune University, Pune, 2004.
[4] W. G. Atshan and N. A. J. Al-Ziadi, A new subclass of harmonic univalent functions, J.
Al-Qadisiyah Comput. Sci. Math. 9(2) (2017), 26-32.
https://doi.org/10.29304/jqcm.2017.9.2.140
[5] W. G. Atshan, On a certain class of multivalent functions defined by Hadamard product, Journal of Asian Scientific Research 3(9) (2013), 891-902.
[6] R. Bharati, R. Parvatham and A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math. 28(1) (1997), 17-32. https://doi.org/10.5556/j.tkjm.28.1997.4330
[7] S. H. Hadi, M. Darus, C. Part and J. R. Lee, Some geometric properties of multivalent functions associated with a new generalized q-Mittag-Leffler function, AIMS Mathematics 7(7) (2022), 11772–11783. https://doi.org/10.3934/math.2022656
[8] S. M. Khaimar and M. More, On a subclass of multivalent β-uniformly starlike and convex functions defined by a linear operator, LAENG Internat. J. Appl. Math. 39(3) (2009), Article ID IJAM_39_06.
[9] A. Y. Lashin, On a certain subclass of starlike functions with negative coefficients, J.
Ineq. Pure Appl. Math. 10(2) (2009), 1-8.
[10] J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc.
23(2) (1925), 481-519. https://doi.org/10.1112/plms/s2-23.1.481
[11] S. Owa, The quasi-Hadamard products of certain analytic functions, in Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa, (Editors), World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992.
https://doi.org/10.1142/9789814355896_0019
[12] G. S. Sălăgean, H. M. Hossen and M. K. Aouf, On certain class of p-valent functions with negative coefficients. II, Studia Univ. Babes-Bolyai 69(1) (2004), 77-85.
[13] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109-116. https://doi.org/10.1090/S0002-9939-1975-0369678-0
[14] A. K. Wanas and H. M. Ahsoni, Some geometric properties for a class of analytic functions defined by beta negative binomial distribution series, Earthline Journal of Mathematical Sciences 9(1) (2022), 105-116. https://doi.org/10.34198/ejms.9122.105116
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