Volume 10, Number 2, 2022, Pages 341-353 https://doi.org/10.34198/ejms.10222.341353
Received: June 19, 2022; Accepted: July 20, 2022 2020 Mathematics Subject Classification: 30C45.
Keywords and phrases: analytic function, Borel distribution, strongly starlike, strongly convex, coefficient estimates, integral representation.
*
A Certain Subclass of Multivalent Functions Associated with Borel Distribution Series
Abbas Kareem Wanas1,* and Hussein Kadhim Radhi2
1Department of Mathematics, College of Science, University of Al-Qadisiyah, Iraq e-mail: [email protected]
2Department of Mathematics, College of Science, University of Al-Qadisiyah, Iraq e-mail: [email protected]
Abstract
In this paper, we determine the necessary and sufficient conditions for the power series whose coefficients are probabilities of the Borel distribution to be in the family , , , , of analytic functions which defined in the open unit disk. We derive a number of important geometric properties, such as, coefficient estimates, integral representation, radii of starlikeness and convexity. Also we discuss the extreme points and neighborhood property for functions belongs to this family.
1. Introduction
Indicate by the family of all functions of the form
= + , 1.1
which are analytic and multivalent in the open unit disk = ∈ ℂ: | | < 1!. Also, let " denote the subfamily of consisting of functions of the form:
= − ≥ 0, ∈ & . 1.2
The function ∈ " is said to be starlike of order ( 0 ≤ ( < if it satisfies the
condition:
*+ , - . > ( ∈ ,
A function ∈ " is said to be convex of order ( 0 ≤ ( < if it satisfies the condition
*+ ,1 + --- . > ( ∈ .
A function ∈ " is said to be close-to-convex of order ( 0 ≤ ( < if it satisfies the condition
*+ , -0 . > ( 1 .
Let the function and 5 be analytic in . We say that the function is subordinate to 5, if there exists a Schwarz function C analytic in with C 0 = 0 and |C | <
1 ∈ such that = 5DC E. This subordinate is denoted by ≺ 5 or ≺ 5 ∈ . It is well known that (see [6]), if the function 5 is univalent in , then
≺ 5 if and only if 0 = 5 0 and ⊂ 5 .
Denote by I∗ and K the families of starlike and convex functions of order(, respectively. These families were introduced and studied by Silverman [9].
The elementary distributions such as the Poisson, the Pascal, the Logarithmic, the Binomial have been partially studied in the Geometric Function Theory from a theoretical point of view (see [1,3,4,7,8,10,11]).
A discrete random variable L is said to have a Borel distribution if it takes the values 1,2,3, … with the probabilities OPQ! ,STOS!PUQ,VTUX!OPWQ, … respectively, where is called the parameter.
Very recently, Wanas and Khuttar [12] introduced the Borel distribution (BD) whose probability mass function is
Y L = Z = Z [0 +0T[
Z! , Z = 1,2,3, … .
Wanas and Khuttar [12] introduced a series whose coefficients are probabilities of the Borel distribution (BD)
\ ; = − D ^ − E 0 0 +0T 0
^ − ! = − Φ , ,
where 0 < ≤ 1 and
Φ , =D ^ − E 0 0 +0T 0
^ − ! .
We consider a linear operator ` , ∶ " ⟶ " defined by the convolution or Hadamard product
` , = \ ; ∗ = − D ^ − E 0 0 +0T 0
^ − ! ,
where ≥ 0,0 < ≤ 1 and ∈ .
We now recall the following Lemmas that will be used to prove our main results.
Lemma 1.1 [5]. If and 5 are analytic in with ≺ 5, then c d Z+ef dg
Sh i
jk ≤ c d5 Z+ef dgjk,
Sh
i
where l > 0, = Z+ef ^j 0 < Z < 1 .
Lemma 1.2 [2]. Let ≥ 0. Then *+ C > if and only if |C − + | <
|C + − |, where Cbe any complex number.
2. Main Results
We begin this section by defining the family , , , , as follows:
Definition 2.1. A function of the form (1.2) is said to be in the class , , , , if satisfies the following condition:
*+
⎩⎪⎪
⎨
⎪⎪⎧
` , -+ SD` , E--
qD` , E-+ D` , , E--r + 1 − q D` , E-+ 1 − ` , r⎭⎪⎪⎬
⎪⎪⎫
> , 2.1
where 0 ≤ < 1, 0 ≤ < 1, 0 ≤ < 1 and 0 < ≤ 1.
Theorem 2.1. Let ∈ " , then ∈ , , , , if and only if
^ − 1 + 1 ^ − − ^ − 1 v,
≤ − 1 + 1 − − − 1 , 2.2 where 0 < ≤ 1, 0 ≤ < 1, 0 < ≤ 1, 0 ≤ ≤ 1 and ∈ &.
The result is sharp for the function
= − − 1 + 1 − − − 1
^ − 1 + 1 ^ − − ^ − 1 Φ , ,
^ ≥ + 1, ∈ & . 2.3 Proof. Assume that ∈ , , , , , so we have
*+
⎩⎪⎪
⎨
⎪⎪⎧
` , ′ + S ` , ′′
qD` , E-+ D` , , E--r + 1 − q D` , E-+ 1 − ` , r⎭⎪⎪⎬
⎪⎪⎫
> .
Then
*+
⎩⎪
⎨
⎪⎧
− ∑ v, ^ + − 1 − ∑ v, ^ ^ − 1
− ∑ v, ^ + − 1 − ∑ v, ^ ^ − 1
+ 1 − − ∑ v, 1 − ^ + 1 − 1 − D − ∑ v, E⎭⎪⎬
⎪⎫
> .
Or equivalently
*+ , − 1 + 1 − − − 1 − ∑ ^ − 1 + 1 ^ − − ^ − 1 Φ ,
D1 + − 1 ED1 + − 1 E − ∑ Φ , D1 + ^ − 1 ED1 + ^ − 1 E . > 0.
This inequality is correct for a ∈ . Letting → 10 yields
*+ , − 1 + 1 − − − 1
− ^ − 1 + 1 ^ − − ^ − 1 Φ , z > 0.
Therefore
^ − 1 + 1 ^ − − ^ − 1 Φ ,
≤ − 1 + 1 − − − 1 .
Conversely, let (2.2) hold. We will prove that (2.1) is correct and then ∈ , , , , . By Lemma 1.2, we put
C = ` , ′ + S ` , ′′
qD` , E-+ D` , , E--r
+ 1 − q D` , E-+ 1 − ` , r
.
Or show that { = 1
|& | | D` , E-+ SD` , E--− + D` , E-− + ` , ′′d = },
where
& = qD` , E-+ D` , , E--r
+ 1 − q D` , E-+ 1 − D` , Er
and it is easy to verify that } − { > 0.
And so the proof is complete.
Corollary 2.1. If ∈ , , , , , then
≤ − 1 + 1 − − − 1
^ − 1 + 1 ^ − − ^ − 1 Φ , , ^ ≥ + 1, ∈ & . Theorem2.2. Let a function ∈ , , , , . Then is -valently close-to-convex of order ( 0 ≤ ( < in the disk | | < * , where
* = inf , − ( ^ − 1 + 1 ^ − − ^ − 1 Φ ,
− 1 + 1 − − − 1 .
•P€~
,
^ ≥ + 1; ∈ & . The result is sharp, with the extermal function given by (2.3).
Proof. It is sufficient to show that
• -0 − • ≤ − ( 0 ≤ ( < , for | | < * , we have that
• -0 − • ≤ Φ , ^ | | 0 .
Thus
• -0 − • ≤ − (, if
Φ , | | 0
− ( ≤ 1. 2.4 Hence, by Theorem 2.1, (2.4) will be true if
1
− ( | | 0 ≤ ^ − 1 + 1 ^ − − ^ − 1 Φ ,
− 1 + 1 − − − 1 ,
and hence
| | ≤ , − ( ^ − 1 + 1 ^ − − ^ − 1 Φ ,
− 1 + 1 − − − 1 .
•P€~
,
^ ≥ + 1; ∈ & . The result is sharp for the function given by (2.3).
Theorem 2.3. Let ∈ , , , , . Then is -valently starlike of order ( 0 ≤ ( < in the disk | | < *S, where
*S= inf , − ( ^ − 1 + 1 ^ − − ^ − 1 Φ ,
^ − ( − 1 + 1 − − − 1 .
•P€~
,
^ ≥ + 1; ∈ & . The result is sharp for the function given by (2.3).
Proof. It is sufficient to show that
• - − • ≤ − ( 0 ≤ ( < , for | | < *S, we have
• - − • ≤∑ Φ , ^ − | | 0
1 − ∑ ƒ| | 0 .
Thus
• - − • ≤ − (, if
^ − (
− ( Φ , | | 0 ≤ 1. 2.5 Hence, by Theorem 2.1, (2.5) will be true if
^ − (
− ( | | 0 ≤ ^ − 1 + 1 ^ − − ^ − 1
− 1 + 1 − − − 1 ,
and hence
| | ≤ , − ( ^ − 1 + 1 ^ − − ^ − 1
^ − ( − 1 + 1 − − − 1 .
•P€~
, ^ ≥ + 1; ∈ & . Setting | | = *S, we get the desired result.
Theorem 2.4. Let ∈ , , , , . Then is -valently convex of order ( 0 ≤ ( < in the disk | | < *X, where
*X= inf , − ( ^ − 1 + 1 ^ − − ^ − 1
^ − ( − 1 + 1 − − − 1 .
•P€~
, ^ ≥ + 1; ∈ & . The result is sharp with the extermal function given by (2.3).
Proof. It is sufficient to show that
•1 + --- − • ≤ − ( 0 ≤ ( < ,
for | | < *X, we have
•1 + --- − • ≤∑ Φ , ^ ^ − | | 0
− ∑ Φ , | | 0 .
Thus
•1 + --- − • ≤ − (, if
^ ^ − (
− ( Φ , | | 0 ≤ 1. 2.6 Hence, by Theorem 2.1, (2.6) will be true if
^ ^ − (
− ( | | 0 ≤ ^ − 1 + 1 ^ − − ^ − 1
− 1 + 1 − − − 1 ,
and hence
| | ≤ , − ( ^ − 1 + 1 ^ − − ^ − 1
^ ^ − ( − 1 + 1 − − − 1 .
•P€~
, ^ ≥ + 1; ∈ & . Setting | | = *X, we get the desired result.
Theorem 2.5. Let the functions [ defined by
[ = − ,[ , D ,[ ≥ 0, ^ ≥ + 1, ∈ &, Z = 1,2, … , †E, 2.7 be in the class , , , , for every Z = 1,2, … , †.
Then the function ℎ defined by
ℎ = − + , + ≥ 0, ^ ≥ + 1, ∈ & ,
also belongs to the class , , , , where + =1
† ,[
‰ [
, ^ ≥ + 1, ∈ & .
Proof. Since [ ∈ , , , , it follows from Theorem 2.1 that
^ − 1 + 1 ^ − − ^ − 1 Φ , ,[
≤ − 1 + 1 − − − 1 ,
for every Z = 1,2, … , †. Hence
^ − 1 + 1 ^ − − ^ − 1 Φ , +
= ^ − 1 + 1 ^ − − ^ − 1 Φ , Š1
† ,[
‰ [
‹
=1
† Š ^ − 1 + 1 ^ − − ^ − 1 Φ , ,[‹
‰
[
≤ − 1 + 1 − − − 1 .
By Theorem 2.1, it follows that ℎ ∈ , , , , .
Theorem 2.6. Let the functions [ defined by (2.7) be in the class ∈ , , , , for every Z = 1,2, … , †. Then the function ℎS defined by
ℎS = Œ[ [
‰
[
is also in the class ∈ , , , , where
Œ[ = 1, Œ[≥ 0 .
‰
[
Proof. By Theorem 2.1 for every Z = 1,2, … , †, we have
^ − 1 + 1 ^ − − ^ − 1 Φ , ,[
≤ − 1 + 1 − − − 1 .
But
ℎS = Œ[ [
‰ [
= Œ[Š − ,[ ‹ =
‰ [
− Š Œ[ ,[
‰ [
‹ .
Therefore
^ − 1 + 1 ^ − − ^ − 1 Φ , Š Œ[ ,[
‰ [
‹
= Œ[
‰ [
Š ^ − 1 + 1 ^ − − ^ − 1 Φ , ,[‹
≤ Œ[
‰ [
− 1 + 1 − − − 1
= − 1 + 1 − − − 1
and the proof is complete.
Theorem 2.7. Let • > 0. If ∈ , , , , and suppose that Ž is defined by
Ž = − − 1 + 1 − − − 1
• − 1 + 1 • − − • − 1 ΦŽ, Ž,
• ≥ + 1; ∈ & . If there exists an analytic function C defined by
DC EŽ0 = • − 1 + 1 • − − • − 1 ΦŽ,
− 1 + 1 − − − 1 0 .
Then, for = Z+ef and 0 < Z < 1 , c | |•jk
Sh i
≤ c | Ž |•jk, • > 0 . 2.8
Sh
i
Proof. We show that
c ’1 − 0 ’
•
jk
Sh i
≤ c •1 − − 1 + 1 − − − 1
• − 1 + 1 • − − • − 1 ΦŽ,
Ž0 ••jk.
Sh
i
By applying Lemma 1.1, it suffices to show that
1 − 0 ≺ 1 − − 1 + 1 − − − 1
• − 1 + 1 • − − • − 1 ΦŽ, Ž0 . Set
1 − 0 = 1 − − 1 + 1 − − − 1
• − 1 + 1 • − − • − 1 ΦŽ, DC EŽ0 . We find that
DC EŽ0 = • − 1 + 1 • − − • − 1 ΦŽ,
− 1 + 1 − − − 1 0 ,
which readily yield C 0 = 0.
Furthermore using (2.2), we obtain
|C |Ž0 = ’ • − 1 + 1 • − − • − 1 ΦŽ,
− 1 + 1 − − − 1 0 ’
≤ | | ’ ^ − 1 + 1 ^ − − ^ − 1 Φ ,
− 1 + 1 − − − 1 ’
≤ | | < 1.
Next, the proof for the first derivative
Theorem 2.8. Let • > 0. If ∈ , , , , and
Ž = − − 1 + 1 − − − 1
• − 1 + 1 • − − • − 1 vŽ, Ž,
• ≥ + 1; ∈ & . Then for = Z+ef and 0 < Z < 1 ,
c | - |•jk
Sh i
≤ c | Ž- |•jk, • > 0 . 2.9
Sh
i
Proof. It is sufficient to show that
1 − ^ 0 ≺ 1 − • − 1 + 1 − − − 1
• − 1 + 1 • − − • − 1 ΦŽ, Ž0 . This follows because
|C |Ž0 = ’ • − 1 + 1 • − − • − 1 ΦŽ,
• − 1 + 1 − − − 1
^ 0
’
≤ | | ’ ^ − 1 + 1 ^ − − ^ − 1 Φ ,
− 1 + 1 − − − 1 ’
≤ | | < 1.
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