Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

292 
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On 𝓟-Stable Functions 

 

M P Jeyaraman1, T G Bhaskar2, H Aaisha Farzana3, M P Rajakumar4 
1Department of Mathematics, Presidency College(Autonomous), Chennai, 600005, Tamil Nadu, India. 

jeyaraman_mp@yahoo.co.in 

2Department of Mathematics, Government Arts College(Autonomous), Nandanam, Chennai, 600035, Tamil Nadu, India. 

tgbhas@yahoo.co.in 

3Department of Mathematics with Computer Applications, A.M. Jain College, Meenambakkam, Chennai, 600061, 

Tamilnadu, India.  h.aaisha@gmail.com 

4Department of Computer Science and Engineering, St.Joseph’s College of Engineering, OMR, Chennai, 600119, 

Tamilnadu, India.  rajranjhu@gmail.com 

 

Article History: 

Received: 14-07-2024 

Revised: 29-08-2024 

Accepted: 10-09-2024 

Abstract  

Let ℎ1, ℎ2 be two analytic functions defined in the open unit disc Δ:= {𝑧 ∈ ℂ: |𝑧| < 1} 

which are normalized by the condition ℎ1(0) = 1 = ℎ2(0). Then ℎ1 is 𝒫-stable with 

respect to ℎ2, whenever  

𝒫𝑛(ℎ1, 𝑧)

ℎ1(𝑧)
≺

1

ℎ2(𝑧)
      (𝑧 ∈ Δ), 

holds for all 𝑛 ∈ ℕ. Here ’≺’ stands for subordination and 𝒫𝑛(ℎ, 𝑧) = 𝒫𝑛(𝑧) ∗ ℎ(𝑧) 

where 𝒫𝑛(𝑧) denote the 𝑛-degree polynomial induced by the (𝑛 + 1)th row entities in 

an admissible lower triangular matrix. The main purpose of this article is to prove that 

the function ((𝐴𝑧 + 1)/(𝐵𝑧 + 1))𝛿  is 𝒫-stable with respect to (𝐵𝑧 + 1)−𝛿 , for 𝛿 ∈

(0,1] and −1 ≤ 𝐵 < 𝐴 ≤ 0 but not 𝒫-stable with respect to itself, when −1 ≤ 𝐵 < 𝐴 <

0 and 𝛿 ∈ (0,1]. As an application, considered different admissible lower triangular 

matrices to derive various results related on stability. 

Keywords: Stable functions, 𝒫-stable functions, Generalized Cesàro stable functions, 

Subordinations, Cesàro mean, Janowski function. 

 

1. Introduction and Preliminaries 

      Let 𝒜 be the class of analytic functions ℎ in the unit disc Δ = {𝑧 ∈ ℂ: |𝑧| < 1}. Let 𝒜0 and 𝒜1 

be the subclass of 𝒜 with the normalization ℎ(0) = 1 and ℎ(0) = ℎ′(0) − 1 = 0, respectively. The 

class 𝒮 ⊂ 𝒜 consists of all univalent functions in Δ. For 0 ≤ 𝛾 < 1, a function ℎ ∈ 𝒜 belongs to the 

class 𝒮∗(𝛾) of starlike of order 𝛾 and 𝒞(𝛾) of convex of order 𝛾, if ℎ maps conformally the unit disc 

Δ onto the domains that are starlike and convex while the analytical characterization of these classes 

are given by 𝑅𝑒 (
𝑧ℎ′(𝑧)

ℎ(𝑧)
) > 𝛾 and 𝑅𝑒 (1 +

𝑧ℎ′′(𝑧)

ℎ′(𝑧)
) > 𝛾 in Δ, respectively. Also, we denote 𝒮∗(0):= 𝒮∗ 

and 𝒞(0):= 𝒞. These subclasses has a proper inclusion as follows: 𝒞 ⊂ 𝒮∗ ⊂ 𝒮 ⊂ 𝒜. By Alexander 

transformation, we have ℎ ∈ 𝒞 if and only if 𝑧ℎ′ ∈ 𝒮∗. A function ℎ ∈ 𝒜1 is pre-starlike of order 𝛾 if 

ℎ ∗ 𝒦𝛾 ∈ 𝒮
∗(𝛾), where 𝒦𝛾(𝑧) =

𝑧

(1−𝑧)2−2𝛾
. If ℎ1, ℎ2 are analytic functions in Δ, we say ℎ1 is 

subordinate to ℎ2, written ℎ1 ≺ ℎ2, if ℎ1 = ℎ2 ∘ 𝜔 for some analytic function 𝜔: Δ → Δ with 𝜔(0) =

0. If ℎ2 is univalent in Δ, then ℎ1 ≺ ℎ2 if and only if ℎ1(Δ) ⊆ ℎ2(Δ) and ℎ1(0) = ℎ2(0). 



Communications on Applied Nonlinear Analysis 

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       For −1 ≤ 𝐵 < 𝐴 ≤ 1 and 𝛿 ∈ (0,1], we define  

                   𝒥𝐴,𝐵
𝛿 (𝑧) = (

𝐴𝑧+1

𝐵𝑧+1
)
𝛿

= 1 + ∑∞𝑗=1 𝑏𝑗
𝐴,𝐵(𝛿)𝑧𝑗      (𝑧 ∈ Δ), (1.1) 

 where 𝑏𝑗: = 𝑏𝑗
𝐴,𝐵(𝛿) = ∑𝑗𝑖=0

[𝛿]𝑖

𝑖!

(𝛿)𝑗−𝑖

(𝑗−𝑖)!
𝐴𝑖(−𝐵)𝑗−𝑖. Also,  

   [𝛿]𝑗 = {
    1,   𝑗 = 0
𝛿[𝛿 − 1]𝑗−1,   𝑗 ≥ 1
  

  and    (𝛿)𝑗 = {
    1,   𝑗 = 0
𝛿(𝛿 + 1)𝑗−1,   𝑗 ≥ 1
  

 

 are the factorial polynomials. Moreover, we have  

 𝒥𝐴,𝐵
𝛿 (𝑧)′ + (

(𝐵−𝐴)𝛿

1+(𝐵+𝐴)𝑧+𝐵𝐴𝑧2
) 𝒥𝐴,𝐵

𝛿 (𝑧) = 0. (1.2) 

Note that, from (1.1) for 𝐴 = 0 and −1 ≤ 𝐵 < 0, we have  

 𝒥0,𝐵
𝛿 (𝑧) = (𝐵𝑧 + 1)−𝛿 = 1 + ∑∞𝑗=1

(𝛿)𝑗

𝑗!
(−𝐵)𝑗𝑧𝑗 = 1 + ∑∞𝑗=1 𝑐𝑗𝑧

𝑗       (𝑧 ∈ Δ). (1.3) 

 

Definition 1.1. [11] For ℎ ∈ 𝒜0 and 0 < 𝑐 < 1 + 𝑏, the 𝑛th Cesàro mean of type (𝑏 − 1, 𝑐) of 

ℎ(𝑧) = ∑𝑛𝑗=0 𝑎𝑘𝑧
𝑘 is defined as  

 𝜎𝑛
𝑏−1,𝑐(ℎ, 𝑧) =

1

𝐵𝑛
∑𝑛𝑗=0   𝐵𝑛−𝑗𝑏𝑗𝑧

𝑗 = 𝜎𝑛
𝑏−1,𝑐(𝑧) ∗ ℎ(𝑧),    𝑛 ∈ ℕ ∪ {0} (1.4) 

where 𝐵𝑗 =
(𝑏)𝑗

(𝑐)𝑗

𝑏−𝑐+1

𝑏
 and 𝐵0 = 1.   

       Note that, for 𝑏 = 1 + 𝛽 and 𝑐 = 1, (1.4) represents 𝑛th Cesàro mean of order 𝛽 ≥ 0 which was 

studied by Mondal and Swaminathan in [6]. Also note that for 𝑏 = 1 and 𝑐 = 1 in (1.4), then we get 

𝑛th partial sum of ℎ ∈ 𝒜0. 

Definition 1.2. [11] Let ℋ𝑛(𝑛 ∈ ℕ) be a non-empty set consisting of lower triangular matrices 𝐻 =

(ℎ𝑖𝑗) of order (𝑛 + 1) with ℎ𝑖𝑗 ≥ 0, for all 𝑖, 𝑗 = 0,1,2, . . . , 𝑛 is called admissible lower triangular 

matrix set if each matrix satisfies the following admissible conditions: 

(𝑖) ℎ𝑖0 = 1, ∀  0 ≤ 𝑖 ≤ 𝑛, 

(𝑖𝑖)  𝑓𝑜𝑟  𝑒𝑎𝑐ℎ  𝑓𝑖𝑥𝑒𝑑  𝑖 ≥ 1, ℎ𝑖𝑗 = ℎ𝑖1ℎ𝑖−1,𝑗−1, ∀  1 ≤ 𝑗 ≤ 𝑛, 

(𝑖𝑖𝑖)  𝑓𝑜𝑟  𝑒𝑎𝑐ℎ  𝑓𝑖𝑥𝑒𝑑  𝑖 ≥ 1, {ℎ𝑖𝑗}  𝑖𝑠  𝑎  𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔  𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒. 

Then (𝑛 + 1)th row of each (ℎ𝑖𝑗) ∈ ℋ𝑛 induces a 𝑛-degree polynomial 𝒫𝑛 defined as  

 𝒫𝑛(𝑧)   =   1 + ∑𝑛𝑗=1 ℎ𝑛𝑗𝑧
𝑗 . (1.5) 

The convolution of  ℎ(𝑧) = ∑𝑛𝑗=0 𝑎𝑘𝑧
𝑘 ∈ 𝒜0 with 𝑛-degree polynomial 𝒫𝑛 is given by  

 𝒫𝑛(ℎ, 𝑧) = 𝒫𝑛(𝑧) ∗ ℎ(𝑧) = 1 + ∑
𝑛
𝑗=1 ℎ𝑛𝑗𝑏𝑗𝑧

𝑗 , 𝑛 ∈ ℕ. (1.6) 

 

 



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Also we have,  

 𝒫𝑛(ℎ, 𝑧) =   ℎ𝑛1𝒫𝑛−1(ℎ, 𝑧) + ∑
𝑛−1
𝑗=0 (ℎ𝑛𝑗 − ℎ𝑛1ℎ𝑛−1,𝑗)𝑏𝑗𝑧

𝑗 + ℎ𝑛𝑛𝑏𝑛𝑧
𝑛. (1.7) 

      Here we provide some examples of admissible lower triangular matrix which will be useful in 

deriving new and existing results from our main results.    

Example 1.1. If we choose the admissible matrix as  

 𝐻 = (ℎ𝑖𝑗) = {
  1   0 ≤ 𝑗 ≤ 𝑖
  0   𝑗 ≥ 𝑖 + 1,
  

 

then (1.5) represents the 𝑛th partial sum of ℎ ∈ 𝒜0.  

Example 1.2. For the choice of the admissible matrix  

   𝐻 = (ℎ𝑖𝑗) =

{
 
 

 
 
    1   𝑗 = 0
(1+𝛽)𝑖−𝑗

(𝑖−𝑗)!

𝑖!

(1+𝛽)𝑖
  1 ≤ 𝑗 ≤ 𝑖

    0   𝑗 ≥ 𝑖 + 1,
  

 

 then (1.5) represents the 𝑛th Cesàro mean of order 𝛽 ≥ 0.  

Example 1.3. If the admissible matrix is  

   𝐻 = (ℎ𝑖𝑗) =

{
 
 

 
     1   𝑗 = 0
𝐵𝑖−𝑗

𝐵𝑖
  1 ≤ 𝑗 ≤ 𝑖

    0   𝑗 ≥ 𝑖 + 1,
  

 

then (1.5) represents 𝑛th Cesàro mean of type (𝑏 − 1, 𝑐).  

Definition 1.3. [5] For fixed 𝑛 ∈ ℕ and ℎ1, ℎ2 ∈ 𝒜0 we say that ℎ1 is 𝒫𝑛-stable with respect to ℎ2, if  

 
𝒫𝑛(ℎ1,𝑧)

ℎ1(𝑧)
≺

1

ℎ2(𝑧)
. (1.8) 

In particular, ℎ1 is 𝒫𝑛-stable with respect to itself then it is just 𝒫𝑛-stable. Suppose the above 

condition holds for every 𝑛, then ℎ1 is called as 𝒫-stable (with respect to ℎ2).  

       Note that, for the choices of matrix 𝐻 = (ℎ𝑖𝑗) given in Example 1.1, Example 1.2 and Example 

1.3, the equation (1.8) represents the definition of stable[9], Cesàro stable[6] and generalized Cesàro 

stable[11], respectively. 

       The concept of stable functions was first introduced by Ruscheweyh and Salinas[9] and they 

proved that 𝒥0,−1
𝛿 (𝑧) is stable with respect to itself, for 0 < 𝛿 ≤ 1. For related works on stable 

functions and its application on various subclasses, we refer ([1], [2], [8], [10], [12]). Sangal and 

Swaminathan[11] developed 𝑛th Cesàro mean of type (𝑏 − 1, 𝑐), for 𝑏 + 1 > 𝑐 > 0 which give rise 

to generalized Cesàro stable function and as an application they proved that 𝒥0,−1
𝛿 (𝑧), (−1 ≤ 𝛿 ≤ 1) 

is generalized Cesàro stable with respect to itself. Recently, Jeyaraman and Bhaskar[3] proved that 

𝒥𝐴,𝐵
𝛿 (𝑧) is generalized Cesàro stable with respect to 𝒥0,−1

𝛿 (𝑧), for 𝛿 ∈ (0,1] and −1 ≤ 𝐵 < 𝐴 ≤ 0. 



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Mondal 𝑒𝑡  𝑎𝑙.[5] introduced the concept of 𝒫-stable function and proved that 𝒥1−2𝛼,−1
𝛿  is 𝒫-stable 

with respect to 𝒥0,−1
𝛿 , for 0 < 𝛿 ≤ 1 and 1/2 ≤ 𝛼 < 1. It is worth mentioning that the authors in [5] 

considered different admissible lower triangular matrices to derive various results on stability. 

        In this paper, motivated by the aforesaid works, our aim is to prove that 𝒥𝐴,𝐵
𝛿 (𝑧) is 𝒫-stable with 

respect to 𝒥0,𝐵
𝛿 (𝑧), for −1 ≤ 𝐵 < 𝐴 ≤ 0 and 𝛿 ∈ (0,1] but not 𝒫-stable with respect to itself, when 

−1 ≤ 𝐵 < 𝐴 < 0 and 𝛿 ∈ (0,1]. As an application, for various choices of admissible lower 

triangular matrices we obtain existing results on stability, Cesàro stability and generalized Cesàro 

stability. 

       We need the following Lemmas, in order to prove our main results.   

Lemma 1.1. [2, p.3] Suppose 𝒥𝐴,𝐵
𝛿 (𝑧) is the function defined in (1.1), then for 𝛿 ∈ (0,1] and −1 ≤

𝐵 < 𝐴 ≤ 0, we have  

 (𝑖)  𝑏𝑚1
≥ 0, 

 (𝑖𝑖)  (𝑚1 + 1)(𝑚2 + 1)𝑏𝑚1+1 +𝑚1𝑚2𝐵𝑏𝑚1
≥ 0, 𝑓𝑜𝑟  𝑎𝑙𝑙  𝑚1, 𝑚2 ∈ ℕ. 

Lemma 1.2. [7, p.54] Let 𝜐 and 𝜈 are prestarlike function and starlike function, respectively of order 

𝛾 ∈ [0,1). Then for any analytic function 𝜔 in Δ, we have  

 
𝜐∗(𝜈𝜔)

𝜐∗𝜈
(Δ) ⊂ 𝑐𝑜(𝜔(Δ)), 

where 𝑐𝑜(⋅) is the closed convex hull of a set.   

Lemma 1.3. [4, p.57] Let −1 ≤ 𝐵 < 0 and 𝛿1, 𝛿2 > 0. If 𝐻1 ≺ [𝒥0,𝐵
𝛿1 (𝑧)]−1 and 𝐻2 ≺ [𝒥0,𝐵

𝛿2 (𝑧)]−1, 

then 𝐻1𝐻2 ≺ [𝒥0,𝐵
𝛿1+𝛿2(𝑧)]−1, for 𝑧 ∈ Δ.  

 

2. 𝓟-stability of 𝓙𝑨,𝑩
𝜹 (𝒛) 

      For a constant 𝑎 ∈ ℝ and −1 ≤ 𝐵 < 𝐴 ≤ 0, using (1.1), (1.6) and (1.7) we have the following 

relations: 

 

𝑎𝒫′𝑛(𝒥𝐴,𝐵
𝛿 (𝑧), 𝑧) = 𝒫𝑛(𝑎𝒥𝐴,𝐵

𝛿 (𝑧)′, 𝑧) −𝑎(𝑛 + 1)ℎ𝑛𝑛𝑏𝑛+1𝑧
𝑛  

 +𝑎 ∑𝑛−1𝑗=0 (ℎ𝑛,𝑗+1 − ℎ𝑛𝑗)(𝑗 + 1)𝑏𝑗+1𝑧
𝑗  

𝑎𝑧𝒫′𝑛(𝒥𝐴,𝐵
𝛿 (𝑧), 𝑧) = 𝒫𝑛(𝑎𝑧𝒥𝐴,𝐵

𝛿 (𝑧)′, 𝑧)   

𝑎𝑧2𝒫′𝑛(𝒥𝐴,𝐵
𝛿 (𝑧), 𝑧) = 𝒫𝑛(𝑎𝑧

2𝒥𝐴,𝐵
𝛿 (𝑧)′, 𝑧) +𝑎𝑛ℎ𝑛𝑛𝑏𝑛𝑧

𝑛+1  

 +𝑎 ∑𝑛𝑗=2 (ℎ𝑛,𝑗−1 − ℎ𝑛𝑗)(𝑗 − 1)𝑏𝑗−1𝑧
𝑗.  

   }
  
 

  
 

 (2.1) 

Similarly, for a constant 𝑎 ∈ ℝ and −1 ≤ 𝐵 < 0, using (1.3) and (1.6) the following relations hold.  



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𝑎𝒫′𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧) = ℎ𝑛1𝒫𝑛−1(𝑎𝒥0,𝐵

𝛿 (𝑧)′, 𝑧)  

𝑎𝑧𝒫′𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧) = 𝒫𝑛(𝑎𝑧𝒥0,𝐵

𝛿 (𝑧)′, 𝑧)  

𝒫𝑛(𝑎𝑧𝒥0,𝐵
𝛿 (𝑧)′, 𝑧) = ℎ𝑛1𝒫𝑛−1(𝑎𝑧𝒥0,𝐵

𝛿 (𝑧)′, 𝑧) + 𝑎∑𝑛−1𝑗=0 (ℎ𝑛𝑗 − ℎ𝑛1ℎ𝑛−1,𝑗)𝑗𝑐𝑗𝑧
𝑗 + 𝑎ℎ𝑛𝑛𝑛𝑐𝑛𝑧

𝑛.  

}

                                                                                                                                                

(2.2) 

Theorem 2.1. Let 𝐻 = (ℎ𝑖𝑗) be admissible lower triangular matrix with ℎ𝑖1 ≤ 1, ∀  𝑖 ≥ 1. Then for 

−1 ≤ 𝐵 < 𝐴 ≤ 0 and 𝛿 ∈ (0,1], the function 𝒥𝐴,𝐵
𝛿 (𝑧) is 𝒫-stable with respect to 𝒥0,𝐵

𝛿 (𝑧).  

Proof. To prove that 𝒥𝐴,𝐵
𝛿 (𝑧) is 𝒫-stable with respect to 𝒥0,𝐵

𝛿 (𝑧), we must show that  

                    
𝒫𝑛(𝒥𝐴,𝐵

𝛿 (𝑧),𝑧)

𝒥𝐴,𝐵
𝛿 (𝑧)

≺
1

𝒥0,𝐵
𝛿 (𝑧)

    (𝑧 ∈ Δ). 

Therefore, it is enough to prove that  

 |
(𝐵𝑧+1)[𝒫𝑛(𝒥𝐴,𝐵

𝛿 (𝑧),𝑧)]

1
𝛿

(𝐴𝑧+1)
− 1| ≤ 1. 

For fixed 𝑛 and 𝛿, let us consider  

 𝑄(𝑧) = 1 −
(𝐵𝑧+1)[𝒫𝑛(𝒥𝐴,𝐵

𝛿 (𝑧),𝑧)]

1
𝛿

(𝐴𝑧+1)
. 

A simple computation yields  

 𝑄′(𝑧) =
(𝐴−𝐵)[𝒫𝑛(𝒥𝐴,𝐵

𝛿 (𝑧),𝑧)]

1
𝛿
−1

(𝐴𝑧+1)2
[𝒫𝑛(𝒥𝐴,𝐵

𝛿 (𝑧), 𝑧) − (
1+(𝐴+𝐵)𝑧+𝐴𝐵𝑧2

𝛿(𝐴−𝐵)
)𝒫′𝑛(𝒥𝐴,𝐵

𝛿 (𝑧), 𝑧)]. (2.3) 

Using (2.1) and (1.2) in (2.3), we have  

𝑄′(𝑧) =
(𝐴 − 𝐵)[𝒫𝑛(𝒥𝐴,𝐵

𝛿 (𝑧), 𝑧)]
1
𝛿
−1

(𝐴𝑧 + 1)2
[𝒫𝑛(𝒥𝐴,𝐵

𝛿 (𝑧) − (
1 + (𝐴 + 𝐵)𝑧 + 𝐴𝐵𝑧2

𝛿(𝐴 − 𝐵)
)𝒥𝐴,𝐵

𝛿 (𝑧)′, 𝑧)

 +
1

𝛿(𝐴 − 𝐵)
((𝑛 + 1)ℎ𝑛𝑛𝑏𝑛+1𝑧

𝑛 −∑

𝑛−1

𝑗=0

(ℎ𝑛,𝑗+1 − ℎ𝑛𝑗)(𝑗 + 1)𝑏𝑗+1𝑧
𝑗)

 −
𝐴𝐵

𝛿(𝐴 − 𝐵)
(𝑛ℎ𝑛𝑛𝑏𝑛  𝑧

𝑛+1 +∑

𝑛

𝑗=2

(ℎ𝑛,𝑗−1 − ℎ𝑛𝑗)(𝑗 − 1)𝑏𝑗−1𝑧
𝑗)].

 

 

 



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=
[𝒫𝑛(𝒥𝐴,𝐵

𝛿 (𝑧),𝑧)]

1
𝛿
−1

𝛿
[((𝑛 + 1)𝑏𝑛+1 + ∑

∞
𝑛1=1

((𝑛1 + 1)(𝑛 + 1)𝑏𝑛+1 + 𝑛1𝑛𝐵𝑏𝑛)(−𝐴)
𝑛1𝑧𝑛1)ℎ𝑛𝑛𝑧

𝑛

 +((ℎ𝑛0 − ℎ𝑛1)𝑏1)(∑
∞
𝑛1=0

(𝑛1 + 1)(−𝐴)
𝑛1𝑧𝑛1)

 +∑𝑛−1𝑗=1 (ℎ𝑛𝑗 − ℎ𝑛,𝑗+1)𝑧
𝑗((𝑗 + 1)𝑏𝑗+1 − 𝐴𝐵𝑗𝑏𝑗𝑧)(∑

∞
𝑛1=0

(𝑛1 + 1)(−𝐴)
𝑛1𝑧𝑛1)]

 
=

[𝒫𝑛(𝒥𝐴,𝐵
𝛿 (𝑧),𝑧)]

1
𝛿
−1

𝛿
[((𝑛 + 1)𝑏𝑛+1 + ∑

∞
𝑛1=1

((𝑛1 + 1)(𝑛 + 1)𝑏𝑛+1 + 𝑛1𝑛𝐵𝑏𝑛)(−𝐴)
𝑛1𝑧𝑛1)ℎ𝑛𝑛𝑧

𝑛

 +((ℎ𝑛0 − ℎ𝑛1)𝑏1)(∑
∞
𝑛1=0

(𝑛1 + 1)(−𝐴)
𝑛1𝑧𝑛1)

 +∑𝑛−1𝑗=1 (ℎ𝑛𝑗 − ℎ𝑛,𝑗+1)𝑧
𝑗((𝑗 + 1)𝑏𝑗+1 + ∑

∞
𝑛1=1

((𝑛1 + 1)(𝑗 + 1)𝑏𝑗+1 + 𝐵𝑛1𝑗𝑏𝑗)(−𝐴)
𝑛1𝑧𝑛1)].

                                                                                                                                                 (2.4) 

Using Lemma 1.1 and Definition 1.2, from (2.4) it follows that the expression  

 [((𝑛 + 1)𝑏𝑛+1 + ∑
∞
𝑛1=1

((𝑛1 + 1)(𝑛 + 1)𝑏𝑛+1 + 𝑛1𝑛𝐵𝑏𝑛)(−𝐴)
𝑛1𝑧𝑛1)ℎ𝑛𝑛𝑧

𝑛 

 +((ℎ𝑛0 − ℎ𝑛1)𝑏1)(∑
∞
𝑛1=0

(𝑛1 + 1)(−𝐴)
𝑛1𝑧𝑛1) + ∑𝑛−1𝑗=1 (ℎ𝑛𝑗 − ℎ𝑛,𝑗+1)𝑧

𝑗 

 ((𝑗 + 1)𝑏𝑗+1 + ∑
∞
𝑛1=1

((𝑛1 + 1)(𝑗 + 1)𝑏𝑗+1 + 𝐵𝑛1𝑗𝑏𝑗)(−𝐴)
𝑛1𝑧𝑛1)], 

 represents a series of positive Taylor’s coefficients about 𝑧 = 0. Again by Lemma 1.1, we have 𝑏𝑗 >

0 for all 𝑗 ∈ ℕ, it follows that 𝒫𝑛(𝒥𝐴,𝐵
𝛿 (𝑧), 𝑧) has a series representation with positive Taylor’s 

coefficients about 𝑧 = 0 which implies that |𝒫𝑛(𝒥𝐴,𝐵
𝛿 (𝑧), 𝑧)| ≤ 𝒫𝑛(𝒥𝐴,𝐵

𝛿 (𝑧), |𝑧|). Applying these 

results in (2.3), we conclude that 𝑄′(𝑧) is a series with positive Taylor’s coefficients about 𝑧 = 0. 

Hence, |𝑄′(𝑧)| ≤ 𝑄′(|𝑧|). Since 𝑄(0) = 0 and 𝑄(−𝐵) = 1, it follows that  

 |𝑄(𝑧)| = |∫
𝑧

0
𝑄′(𝑡)  𝑑𝑡| ≤ ∫

−𝐵

0
|𝑄′ (−

𝑡𝑧

𝐵
)|   𝑑𝑡 ≤ ∫

−𝐵

0
𝑄′(𝑡)  𝑑𝑡 = 1. 

 Therefore, 𝒥𝐴,𝐵
𝛿 (𝑧) is 𝒫𝑛-stable with respect to 𝒥0,𝐵

𝛿 (𝑧) for all 𝑛 ∈ ℕ. Hence the proof.  

Remark 2.1.  (i) For 𝐴 = 1 − 2𝛼 and 𝐵 = −1, Theorem 2.1 reduces to [5, Theorem 2.2].  

(ii) For the choices of matrix 𝐻 = (ℎ𝑖𝑗) as given in Example 1.1, Example 1.2 and Example 1.3 in 

Theorem 2.1, we have the result of stability[2, Theorem 3], Cesàro stability[3, Corollary 2.3] and 

generalized Cesàro stability[3, Theorem 2.1], respectively.  

(iii) If we take 𝐴 = 1 − 2𝛼, 𝐵 = −1 with the choice of matrix 𝐻 = (ℎ𝑖𝑗) as per Example 1.1, 

Example 1.2 and Example 1.3 in Theorem 2.1, then we have the result obtained in [1, Theorem 2.1], 

[5, Theorem 2.3] and [5, Theorem 2.4], respectively.  

       Now, by extending the range of the parameter 𝛿 from (0,1] to [−1,1] and letting 𝐴 = 0 in 

Theorem 2.1, we have the following: 

Theorem 2.2. Let 𝐻 = (ℎ𝑖𝑗) be admissible lower triangular matrix with ℎ𝑖1 ≤ 1, ∀  𝑖 ≥ 1. Then for 

−1 ≤ 𝐵 < 0 and 𝛿 ∈ [−1,1], the function 𝒥0,𝐵
𝛿 (𝑧) is 𝒫-stable.   

Proof. To prove that 𝒥0,𝐵
𝛿 (𝑧) is 𝒫-stable, we must show that  

 
𝒫𝑛(𝒥0,𝐵

𝛿 (𝑧),𝑧)

𝒥0,𝐵
𝛿 (𝑧)

≺
1

𝒥0,𝐵
𝛿 (𝑧)

    (𝑧 ∈ Δ). 



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Therefore, it is enough to prove that  

 |(𝐵𝑧 + 1)[𝒫𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧)]

1

𝛿 − 1| ≤ 1. 

For fixed 𝑛 and 𝛿, let us consider  

 𝑅(𝑧) = 1 − (𝐵𝑧 + 1)[𝒫𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧)]

1

𝛿. (2.5) 

We note that for 𝐴 = 0 in (1.2), we get  

 𝒥0,𝐵
𝛿 (𝑧)′ + (

𝐵𝛿

1+𝐵𝑧
)𝒥0,𝐵

𝛿 (𝑧) =   0. (2.6) 

Clearly for 𝛿 = 0, we have 𝒥0,𝐵
𝛿 (𝑧) = 1 and hence |𝑅(𝑧)| ≤ 1. For 𝛿 ≠ 0, a simple calculation using 

(2.5) yields  

 𝑅′(𝑧) = (−𝐵)[𝒫𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧)]

1

𝛿
−1
[𝒫𝑛(𝒥0,𝐵

𝛿 (𝑧), 𝑧) + (
𝐵𝑧+1

𝐵𝛿
)𝑃′𝑛(𝒥0,𝐵

𝛿 (𝑧), 𝑧)]. (2.7) 

Using (2.2) and (2.6) in (2.7) , we have  

𝑅′(𝑧) = (−𝐵)[𝒫𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧)]

1
𝛿
−1
[ℎ𝑛1𝑃𝑛−1 (𝒥0,𝐵

𝛿 (𝑧) +
𝐵𝑧 + 1

𝐵𝛿
𝒥0,𝐵
𝛿 (𝑧)′, 𝑧) 

 +∑𝑛−1𝑗=0 (ℎ𝑛𝑗 − ℎ𝑛1ℎ𝑛−1,𝑗) (𝑐𝑗 +
𝑗𝑐𝑗

𝛿
) 𝑧𝑗 + ℎ𝑛𝑛 (𝑐𝑛 +

𝑛𝑐𝑛

𝛿
) 𝑧𝑛]. 

Therefore,  

𝑅′(𝑧) = (−𝐵)[𝒫𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧)]

1
𝛿
−1
[ℎ𝑛𝑛 (

(−1)𝑛𝐵𝑛(𝛿 + 1)𝑛
𝑛!

) 𝑧𝑛 

 +∑𝑛−1𝑗=0 ℎ𝑛1(ℎ𝑛−1,𝑗−1 − ℎ𝑛−1,𝑗) (
(−1)𝑗𝐵𝑗(𝛿+1)𝑗

𝑗!
) 𝑧𝑗]. (2.8) 

Using the Definition 1.2 in (2.8), it follows that the expression  

 [ℎ𝑛𝑛 (
(−1)𝑛𝐵𝑛(𝛿+1)𝑛

𝑛!
) 𝑧𝑛 + ∑𝑛−1𝑗=0 ℎ𝑛1(ℎ𝑛−1,𝑗−1 − ℎ𝑛−1,𝑗) (

(−1)𝑗𝐵𝑗(𝛿+1)𝑗

𝑗!
) 𝑧𝑗], 

represents a series of positive Taylor’s coefficients about 𝑧 = 0. 

Case (i): For 𝛿 ∈ (0,1], since 𝑐𝑗 > 0 for all 𝑗 ∈ ℕ, 𝒫𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧) has a series representation with 

positive Taylor’s coefficients about 𝑧 = 0. Hence, |𝒫𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧)| ≤ 𝒫𝑛(𝒥0,𝐵

𝛿 (𝑧), |𝑧|).   

  

Case (ii): For 𝛿 ∈ [−1,0), the series (1.3) has coefficients 𝑐𝑗 < 0 for all 𝑗 ∈ ℕ. Thus, we can write  

 𝒫𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧) = 1 + ∑𝑛𝑗=1 ℎ𝑛𝑗𝑐𝑗𝑧

𝑗 = 1 − 𝜏(𝑧), 

where 𝜏(𝑧) is series with positive Taylor’s coefficients about 𝑧 = 0. Therefore,  

 [𝒫𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧)]

1

𝛿
−1 = [1 − 𝜏(𝑧)]

1

𝛿
−1 = 1 + ∑∞𝑗=1

(1−
1

𝛿
)𝑗

𝑗!
(𝜏(𝑧))𝑗. 



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Thus, [𝒫𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧)]

1

𝛿
−1

 is a series with positive Taylor’s coefficients about 𝑧 = 0. Hence, 

|[𝒫𝑛(𝒥0,𝐵
𝛿 (𝑧), 𝑧)]

1

𝛿
−1| ≤ [𝒫𝑛(𝒥0,𝐵

𝛿 (𝑧), |𝑧|)]
1

𝛿
−1

. 

From the case (i) and case (ii), we observe that 𝑅′(𝑧) is a series with positive Taylor’s coefficients 

about 𝑧 = 0. Hence, |𝑅′(𝑧)| ≤ 𝑅′(|𝑧|). Since 𝑅(0) = 0 and 𝑅(−𝐵) = 1, it follows that  

 |𝑅(𝑧)| = |∫
𝑧

0
𝑅′(𝑡)  𝑑𝑡| ≤ ∫

−𝐵

0
|𝑅′ (−

𝑡𝑧

𝐵
)|   𝑑𝑡 ≤ ∫

−𝐵

0
𝑅′(𝑡)  𝑑𝑡 = 1. 

Therefore, 𝒥0,𝐵
𝛿 (𝑧) is 𝒫𝑛-stable for all 𝑛 ∈ ℕ. Hence the proof.  

Remark 2.2. (i) For the choice of matrix 𝐻 = (ℎ𝑖𝑗) given in Example 1.3, Theorem 2.2 reduces to  

[3, Theorem 2.2]. 

(ii) For the choice 𝐵 = −1 and matrix 𝐻 = (ℎ𝑖𝑗) given in Example 1.1, Example 1.2 and Example 

1.3, we have the results of [10, Theorem 1.1], [6, Theorem 2.2] and [11, Theorem 2.1], respectively. 

 

Theorem 2.3. Let 𝐻 = (ℎ𝑖𝑗) be admissible lower triangular matrix with ℎ𝑖1 ≤ 1, ∀  𝑖 ≥ 1. Then for 

−1 ≤ 𝐵 < 𝐴 < 0 and 𝛿 ∈ (0,1], the function 𝒥𝐴,𝐵
𝛿 (𝑧) is not 𝒫-stable with respect to itself.   

Proof. To prove the result, it is enough to show that 

     
𝒫𝑛(𝒥𝐴,𝐵

𝛿 (𝑧),𝑧)

𝒥𝐴,𝐵
𝛿 (𝑧)

⊀
1

𝒥𝐴,𝐵
𝛿 (𝑧)

    (𝑧 ∈ Δ), 

or equivalently, 𝑆(𝑧) =
(𝐵𝑧+1)[𝒫𝑛(𝒥𝐴,𝐵

𝛿 (𝑧),𝑧)]

1
𝛿

(𝐴𝑧+1)
 is not subordinate to 𝑇(𝑧) =

𝐵𝑧+1

𝐴𝑧+1
. 

If 𝑆(𝑧) ≺ 𝑇(𝑧), then by definition of subordination and Schwarz lemma, we have |𝑆′(0)| ≤ |𝑇′(0)| 

and 𝑆(|𝑧| ≤ 𝑟) ⊆ 𝑇(|𝑧| ≤ 𝑟), where 0 ≤ 𝑟 < 1.  

To prove 𝑆(𝑧) ≺ 𝑇(𝑧), for 𝑧 =
𝐵𝑤+1

𝐴𝑤+1
, we have to show that there exist atleast a point 𝑧0 ∈ Δ with 

|𝑧0| ≤ 𝑟0, for which 𝑆(𝑧0) lies outside the disc, |𝑤 −
𝑟2𝐴−𝐵

𝐵2−𝑟2𝐴2
| ≤

𝑟(𝐴−𝐵)

𝐵2−𝑟2𝐴2
, for −1 ≤ 𝐵 < 𝐴 ≤ 0. 

On choosing 𝑧0 = 0.907512 + 0.395628𝑖, 𝑟0 = 0.99, 𝐴0 = −0.4, 𝐵0 = −1, 𝛿0 = 0.4 and 𝑛 = 1, 

we obtain 
𝑟0
2𝐴0−𝐵0

𝐵0
2−𝑟0

2𝐴0
2 = 0.721028862 and 

𝑟0(𝐴0−𝐵0)

𝐵0
2−𝑟0

2𝐴0
2 = 0.70447257. To complete the proof, we need to 

show that 𝑆(|𝑧| ≤ 𝑟) ⊈ 𝑇(|𝑧| ≤ 𝑟) for the values of ℎ11 ≥ 0, which is possible for various choices 

of admissible lower triangular matrices. Here, we provide the table for some values of ℎ11. 

It is clear from the Table 1 that 𝑆(𝑧0) lies outside the disc |𝑤 − 0.721028862| ≤ 0.70447257 for 

the values of ℎ11 in the interval [0,1]. Also from the Figure 1(A) and Figure 1(B), we observe that 

the 𝑆(𝑧) is not subordinate to 𝑇(𝑧) for the values of ℎ11 = 0.3 and ℎ11 = 0.5 respectively. 

Therefore, 𝒥𝐴,𝐵
𝛿 (𝑧) is not 𝒫1-stable. Hence, 𝒥𝐴,𝐵

𝛿 (𝑧) is not 𝒫-stable, which completes the proof.  

 



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Sl. No. Values of ℎ11 𝑆(𝑧0) |𝑆(𝑧0) −
r0
2𝐴0 − 𝐵0

𝐵0
2 − r0

2A0
2 | 

1 0.1 0.311155-0.5744904i 0.7057165 

2 0.3 0.373144-0.6225204i 0.7131308 

3 0.5 0.4403595-0.6719527i 0.7282141 

4 0.8 0.5512408-0.748699i 0.7677097 

5 1 0.6320364-0.8015772i 0.8065021 

TABLE 1 

 

Figure 1 (A) Boundary curves for h11 = 0.3 

 

Figure 1 (B) Boundary curves for h11 = 0.5 

Remark 2.3. It is clear that for 𝛽 ≥ 0, 𝛿 ∈ (0,1] and −1 ≤ 𝐵 < 𝐴 < 0, 𝒥𝐴,𝐵
𝛿 (𝑧) is not a Cesàro stable 

with respect to itself.  

3.  On 𝓟-stability of 𝒉 ∈ 𝑺∗(𝜹) 

           Also 𝑧ℎ𝛿(𝑧) =
𝑧

(1−𝑧)𝛿
∈ 𝑆∗(1 − 𝛿/2), 𝛿 ∈ (0,1] is an extremal function which plays an 

important role in studying several properties like growth, distortion, etc., it is observed that for 𝛿 ∈

(0,1] and 𝐵 = −1 in Theorem 2.2, we have  



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𝒫𝑛(ℎ𝛿,𝑧)

ℎ𝛿
≺

1

ℎ𝛿
. (3.1) 

Theorem 3.1. Let ℎ ∈ 𝑆∗(1 − 𝛿/2), for 𝛿 ∈ (0,1]. Then  

 
𝒫𝑛(ℎ/𝑧,𝑧)

ℎ(𝑧)/𝑧
≺ (1 − 𝑧)𝛿     (𝑧 ∈ Δ). 

Proof. If ℎ ∈ 𝑆∗(1 − 𝛿/2), then ℎ(𝑧) = 𝑧ℎ𝛿 ∗ 𝐻(𝑧), where 𝐻(𝑧) is an unique prestarlike function of 

order (1 − 𝛿/2). Using ℎ(𝑧) = 𝑧ℎ𝛿 ∗ 𝐻(𝑧), we have  

 
𝒫𝑛(ℎ/𝑧,𝑧)

ℎ(𝑧)/𝑧
=

𝑧𝒫𝑛(𝑧)∗ℎ(𝑧)

ℎ(𝑧)
=

𝐻(𝑧)∗[(𝑧ℎ𝛿)
𝒫𝑛(ℎ𝛿,𝑧)

ℎ𝛿
]

𝐻(𝑧)∗𝑧ℎ𝛿
∈ 𝑐𝑜 (

𝒫𝑛(ℎ𝛿,𝑧)

ℎ𝛿
(Δ)). 

By Lemma 1.2, we see that the range of 
𝒫𝑛(ℎ/𝑧,𝑧)

ℎ(𝑧)/𝑧
 lies in the closed convex hull of image of 

𝒫𝑛(ℎ𝛿,𝑧)

ℎ𝛿
 

under Δ. Now applying (3.1), for 𝛿 ∈ (0,1], we have  

 
𝒫𝑛(ℎ/𝑧,𝑧)

ℎ(𝑧)/𝑧
≺ (1 − 𝑧)𝛿     (𝑧 ∈ Δ). 

Hence, the proof.  

Remark 3.1. For the choice of matrix 𝐻 = (ℎ𝑖𝑗) as given in Example 1.1, Example 1.2 and Example 

1.3, Theorem 3.1 reduces to [9, Theorem 1.1], [6, Theorem 2.3] and [11, Theorem 2.2], respectively.  

Now if we consider 0 < 𝛿 ≤ 𝜇 ≤ 1 in Theorem 2.2, then we have the following result. 

Theorem 3.2. Let −1 ≤ 𝐵 < 0 and 0 < 𝛿 ≤ 𝜇 ≤ 1. Then  

 
𝒫𝑛(𝒥0,𝐵

𝛿 (𝑧),𝑧)

𝒥0,𝐵
𝜇 (𝑧)

≺
1

𝒥0,𝐵
𝜇 (𝑧)

    (𝑧 ∈ Δ). 

Proof.  Using Theorem 2.2 and Lemma 1.3, we have  

𝑙𝑜𝑔 [
𝒫𝑛(𝒥0,𝐵

𝛿 (𝑧), 𝑧)

𝒥0,𝐵
𝜇
(𝑧)

]

1
𝜇

= 𝑙𝑜𝑔 [
𝒫𝑛(𝒥0,𝐵

𝛿 (𝑧), 𝑧)

𝒥0,𝐵
𝜇−𝛿

(𝑧)𝒥0,𝐵
𝛿 (𝑧)

]

1
𝜇

= 𝑙𝑜𝑔 [
1

𝒥0,𝐵
𝜇−𝛿

(𝑧)
]

1
𝜇

+ 𝑙𝑜𝑔 [
𝒫𝑛(𝒥0,𝐵

𝛿 (𝑧), 𝑧)

𝒥0,𝐵
𝛿 (𝑧)

]

1
𝜇

= 𝑙𝑜𝑔[(𝐵𝜔1(𝑧) + 1)
𝜇−𝛿]

1
𝜇 + 𝑙𝑜𝑔[(𝐵𝜔2(𝑧) + 1)

𝛿]
1
𝜇 ≺ 𝑙𝑜𝑔 [

1

𝒥0,𝐵
𝜇
(𝑧)
]

1
𝜇

,

 

 for some analytic functions |𝜔1(𝑧)| ≤ |𝑧| and |𝜔2(𝑧)| ≤ |𝑧| in Δ. Hence the proof.  

 

Acknowledgements. The authors are thankful to the referees for their insightful suggestions. 

References 

[1]  S. Chakraborty and A. Vasudevarao, On stable functions, Comput. Methods Funct. Theory 18 (4), (2018), 677-688. 

[2]  K. Chandrasekaran, D. J. Prabhakaran and P. Sangal, Stable functions of Janowski type, J. Math. Inequal. 15, 

(2021), 969-979. 

[3]  M. P. Jeyaraman and T. G. Bhaskar, Some results on generalized Cesàro stable of Janowski function,  Afr. Mat. 35 

(39), (2024), 1-9. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

302 
https://internationalpubls.com 

[4]  S. Koumandos and S. Ruscheweyh, On a conjecture for trigonometric sums and starlike functions, J. Approx. 

Theory 149 (1), (2007), 42-58. 

[5]  S. R. Mondal, K. S. Nisar and T. Abdeljawad, Some subordination involving polynomials induced by lower 

triangular matrices, Adv. Differ. Equ., Springer, 538, (2020). 

[6]  S. R. Mondal and A. Swaminathan, Stable functions and extension of Vietoris’ theorem, Results Math. 62 (1-2), 

(2012), 33-51. 

[7]  S. Ruscheweyh, Linear operators between classes of prestarlike functions, Comment. Math. Helv. 52 (4), (1977), 

497-509. 

[8]  S. Ruscheweyh, On the kakeya-enestrom theorem and gegenbauer polynomial sums, SIAM, Oest. J. Math. Anal. 9 

(4), (1978), 682-686. 

[9]  S. Ruscheweyh and L. Salinas, On starlike functions of order 𝜆 ∈ [
1

2
, 1), Ann. Univ. Mariae Curie-Sklodowska, 

Sec. A (54), (2000), 117-123. 

[10]  S. Ruscheweyh and L. Salinas, Stable functions and Vietoris’ theorem, J. Math. Anal. Appl. 291 (2), (2004), 596-

604. 

[11] P. Sangal and A. Swaminathan, On generalised Cesàro stable functions, Math. Inequal. Appl. 22 (1), (2019), 227-

247. 

[12] L. Vietoris, Uber das vorzeichen geiwisser trigonometrishcher summen, Sitzungsber, Oest. Akad. Wiss. 167, 

(1958), 125-135.