Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1872 
https://internationalpubls.com 

A New Subclass of Meromorphic Functions Associated with Sălăgean 

Operator    

 

Debasmita Dash 1, Saumya Singh 2* 

1, 2 Department of Mathematics, O. P. Jindal University, Raigarh, 496109 India; 

* Correspondence should be addressed to Saumya Singh; email- saumya.singh@opju.ac.in 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  In this paper, we propose a novel subclass of meromorphic functions within the 

class 𝒜𝑝
∗  which includes functions of the form 𝑓(𝑧) = 𝑧−𝑝 + ∑ 𝑎𝑗𝑧𝑗∞

𝑗=𝑝 , where 𝑧 ∈ △∗ and 

𝑝 ∈ ℕ. The study specifically focuses on using the Sălăgean operator 𝐷𝑝,𝜆
𝑛 𝑓(𝑧) to established 

this new subclass denoted as ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝) which is defined by the parameters 0 ≤

𝛼 < 1, 0 ≤ 𝛽 < 1 , 0 ≤ 𝛾 < 1,  𝑘 ≥ 0,  0 < 𝜆 ≤ 1  and 𝑛 ∈ ℕ0. 

𝑅𝑒 (
𝑧(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))′−(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))

𝛼𝑧(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′+(1−𝑟)(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))
) > 𝑘 |

𝑧(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′−(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))

𝛼𝑧(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′+(1−𝑟)(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))
− 1| + 𝛽 by the given 

condition for this class we will next to prove this class of univalent function then we derive 

the related results. The mathematical formulation of  𝐷𝑝,𝜆
𝑛 𝑓(𝑧) , where the operator influences 

the series expansion of the meromorphic functions, is also elaborated in the study.  

Keywords: Meromorphic functions, Function class 𝒜𝑝
∗ , Sălăgean operator, Starlikeness, 

Convexity, Subclass of meromorphic functions, Series expansion, Analytic function 

properties. 

1. Introduction 

The study of meromorphic functions has been a significant area of interest in complex analysis, 

particularly in the context of geometric function theory. These functions, which are analytic except for 

isolated poles, play a crucial role in various applied & mathematical fields. Within the class of 

meromorphic functions, researchers have explored different subclasses defined by specific function 

properties and operator-induced transformations. One such transformation involves differential and 

integral operators that influence the function's behaviour in the punctured unit disk. The role of 

multiplier transformations in defining new function classes has been studied extensively. For some 

basic concepts like Univalent functions, which are holomorphic and injective, are fundamental in 

geometric function theory for a comprehensive study, (see [7], [8]). Alharayzeh and Ghanim (2022) 

recently introduced a k-uniformly univalent subclass with negative coefficients (see also [1]), proving 

key coefficient estimates and geometric properties. Building upon their findings, this paper introduces 

a new subclass incorporating additional parameters, leading to broader applications in function theory. 

In (2016), Amourah and Darus defined a new class of univalent functions by applying a generalized 

differential operator that includes negative coefficients, highlighting its geometric and analytic 

properties in paper [3]. Recent advancements in geometric function theory have led to the development 

of new subclasses of analytic functions. In (2024), Carroll discusses key techniques in paper [4], for 

analyzing these classes, which are the foundation of our investigation. In (2010), Ali and Ravichandran 

defined a subclass of meromorphic functions based on 𝛼- convexity and examined its analytic and 

mailto:saumya.singh@opju.ac.in


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1873 
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geometric properties in paper [2]. Darus et al. established univalence criteria in paper [5], for analytic 

functions using a generalized differential operator. Building on their findings, this paper explores new 

subclasses by extending the operator’s properties of generalized Sălăgean operator and derived new 

results. For computational purposes we refer to paper [6], which is based on the convexity criterion 

established by Deniz and Erhan in (2013). For further study on univalence criteria, we refer also papers 

[11] and [12]. In (2009), Lee et al. established coefficient bounds in paper [9] for such functions, 

providing key insights into their structure and meromorphic properties.  

Definition: Let the class 𝒜p
∗  consists of all analytic functions in the punctured unit disk △∗ , of the 

form 𝑓(𝑧) = 𝑧−𝑝 + ∑ 𝑎𝑗𝑧𝑗∞
𝑗=𝑝 , 𝑧 ∈△∗  and 𝑝 ∈ ℕ . For the function 𝑓 ∈ 𝒜p

∗  , here we introduced a 

new subclass for  

0 ≤ 𝛼 < 1, 0 ≤ 𝛽 < 1 , 0 ≤ 𝛾 < 1,  𝑘 ≥ 0,  0 < 𝜆 ≤ 1, 𝑝 ∈ ℕ , 𝑛 ∈ ℕ0 = {0,1,2, … },   

By using the operator 𝐷𝑝,𝜆
𝑛   by 𝐷𝑝,𝜆

0 = 𝑓(𝑧) which is introduced in paper [1] 

Let ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝)  

𝑅𝑒 (
𝑧(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))′ − (𝐷𝑝,𝜆
𝑛 𝑓(𝑧))

𝛼𝑧(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′ + (1 − 𝑟)(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))
) > 𝑘 |

𝑧(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′ − (𝐷𝑝,𝜆

𝑛 𝑓(𝑧))

𝛼𝑧(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′ + (1 − 𝑟)(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))
− 1| + 𝛽     (1)  

Where,  

𝐷𝑝,𝜆
𝑛 𝑓(𝑧) = 𝑧−𝑝 + ∑ {(1 − 𝜆) (1 +

𝑗

𝑝
) 𝑎𝑗𝑧𝑗}

𝑛

, 𝑧 ∈△∗  𝑎𝑛𝑑 𝑎𝑗 > 0  

∞

𝑗=𝑝

 

Here 𝐷𝑝,𝜆
𝑛 𝑓(𝑧) be the Sălăgean operator. 

2. Objectives 

The primary objective of this study is to investigate and analyze various newly defined subclasses of 

analytic functions within the unit disc. In this study we introduced a new subclass of meromorphic 

functions ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝), which lies within the class 𝒜𝑝
∗  and consists of functions of the form 

𝑓(𝑧) = 𝑧−𝑝 + ∑ 𝑎𝑗𝑧𝑗∞
𝑗=𝑝 , where 𝑧 ∈  △∗ and 𝑝 ∈ ℕ. This study specifically focuses on using the 

Sălăgean operator 𝐷𝑝,𝜆
𝑛 𝑓(𝑧) to established the foundational properties of this class, proving its 

univalence, and analyzing the influence of the operator on the function’s geometric behavior. The 

mathematical formulation of  𝐷𝑝,𝜆
𝑛 𝑓(𝑧) , where the operator influences on the series expansion of the 

meromorphic functions, is also elaborated in the study.  

3. Methods  

In this paper, we introduced the subclass ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝) of meromorphic functions defined in 

the punctured unit disk △∗ , associated with the generalized Sălăgean operator 𝐷𝑝,𝜆
𝑛  which is defined 

as 𝐷𝑝,𝜆
𝑛 𝑓(𝑧) = 𝑧−𝑝 + ∑ {(1 − 𝜆) (1 +

𝑗

𝑝
) 𝑎𝑗𝑧𝑗}

𝑛

 ∞
𝑗=𝑝 where 𝑎𝑗 > 0. Coefficient estimates are obtained 

by applying a differential subordination condition for function belongs to the subclass 

ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝). Convexity of 𝑓(𝑧) is established under the condition 𝑅𝑒 (1 +
𝑧(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))′′

(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′

) > 0, 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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and Univalence criteria is ensured by the condition |
𝑧(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))′′

(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′ | < 1 . These methods provide 

conditions on the coefficients and geometric criteria for functions defined in the above subclass.  

4. Results 

We begin by proving several theorems based on the operator classification. 

Theorem: 1 

Let 0 ≤ 𝛼 < 1, 0 ≤ 𝛽 < 1 , 0 ≤ 𝛾 < 1,  𝑘 ≥ 0,  0 < 𝜆 ≤ 1, 𝑝 ∈ ℕ , 𝑛 ∈ ℕ0 = {0,1,2, … }. A function 𝑓 

given by 𝑓(𝑧) = 𝑧−𝑝 + ∑ 𝑎𝑗𝑧𝑗∞
𝑗=𝑝  be an analytic function belonging to the class ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝), 

then 𝑓 satisfies the condition: 

𝑅𝑒 (
𝑧 (𝐷𝑝,𝜆

𝑛 𝑓(𝑧))
′

− (𝐷𝑝,𝜆
𝑛 𝑓(𝑧))

𝛼𝑧 (𝐷𝑝,𝜆
𝑛 𝑓(𝑧))

′

+ (1 − 𝑟) (𝐷𝑝,𝜆
𝑛 𝑓(𝑧))

) > 𝑘 |
𝑧 (𝐷𝑝,𝜆

𝑛 𝑓(𝑧))
′

− (𝐷𝑝,𝜆
𝑛 𝑓(𝑧))

𝛼𝑧 (𝐷𝑝,𝜆
𝑛 𝑓(𝑧))

′

+ (1 − 𝑟) (𝐷𝑝,𝜆
𝑛 𝑓(𝑧))

− 1| + 𝛽 

where, 𝐷𝑝,𝜆
𝑛 𝑓(𝑧) = 𝑧−𝑝 + ∑ {(1 − 𝜆) (1 +

𝑗

𝑝
) 𝑎𝑗𝑧𝑗}

𝑛

, 𝑧 ∈△∗  𝑎𝑛𝑑 𝑎𝑗 > 0  ∞
𝑗=𝑝  

Then the coefficients 𝑎𝑗  of the Taylor series expansion satisfy the inequality  

𝑎𝑗 ≤
(1 − 𝛽)(𝛼𝑗 + 1 − 𝑟)

(𝑘 + 1)|2 − 𝛾 − 𝑛(1 − 𝛼)|(1 + 𝑗)
   , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 ≥ 2 

This result establishes an upper bound on the growth of the coefficients of functions in this subclass 

ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝). 

Proof: Given that ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝) if and only if the condition (1) is satisfied,  

Let 

𝜔 =  
𝑧(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))′ − (𝐷𝑝,𝜆
𝑛 𝑓(𝑧))

𝛼𝑧(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′ + (1 − 𝑟)(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))
 

Subject to the condition that, 𝑅𝑒(𝜔) ≥ 𝑘|𝜔 − 1| + 𝛽 if and only if  (𝑘 + 1)|𝜔 − 1| ≤ 1 − 𝛽 

Now    

(𝑘 + 1)|𝜔 − 1| = (𝑘 + 1) |
(1 − 𝑝)𝑧−𝑝 + ∑ {(1 − 𝜆) (1 +

𝑗
𝑝)}

𝑛

(1 + 𝑗)𝑎𝑗𝑧𝑗∞
𝑗=2   

(1 − 𝑟 − 𝛼𝑝)𝑧−𝑝 + ∑ (𝛼𝑗 + 1 − 𝑟) {(1 − 𝜆) (1 +
𝑗
𝑝)}

𝑛

𝑎𝑗𝑧𝑗∞
𝑗=2

− 1|   

≤ 1 − 𝛽 

is equivalent to   

(𝑘 + 1) |
(1 − 𝑝)𝑧−𝑝 + ∑ {(1 − 𝜆) (1 +

𝑗
𝑝)}

𝑛

(1 + 𝑗)𝑎𝑗𝑧𝑗∞
𝑗=2   

(1 − 𝛼𝑝 − 𝑟)𝑧−𝑝 − ∑ (𝛼𝑗 + 1 − 𝑟) {(1 − 𝜆) (1 +
𝑗
𝑝)}

𝑛

𝑎𝑗𝑧𝑗∞
𝑗=2

− 1|   ≤ 1 − 𝛽 



Communications on Applied Nonlinear Analysis 

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So  

(𝑘 + 1) |
(1 − 𝑝)𝑧−𝑝 + ∑ |2 − 𝛾 − 𝑛(1 − 𝛼)| {(1 − 𝜆) (1 +

𝑗
𝑝

)}
𝑛

(1 + 𝑗)𝑎𝑗𝑧𝑗 − (1 − 𝛼𝑝 − 𝑟)𝑧−𝑝∞
𝑗=2   

(1 − 𝛼𝑝 − 𝑟)𝑧−𝑝 − ∑ (𝛼𝑗 + 1 − 𝑟) {(1 − 𝜆) (1 +
𝑗
𝑝

)}
𝑛

𝑎𝑗𝑧𝑗∞
𝑗=2

|   

≤ 1 − 𝛽 

Further the above inequality  

(𝑘 + 1) |
(𝛼𝑝 + 𝑟 − 𝑝)𝑧−𝑝 + ∑ |2 − 𝛾 − 𝑛(1 − 𝛼)| {(1 − 𝜆) (1 +

𝑗
𝑝)}

𝑛

(1 + 𝑗)𝑎𝑗𝑧𝑗∞
𝑗=2   

(1 − 𝛼𝑝 − 𝑟)𝑧−𝑝 − ∑ (𝛼𝑗 + 1 − 𝑟) {(1 − 𝜆) (1 +
𝑗
𝑝)}

𝑛

𝑎𝑗𝑧𝑗∞
𝑗=2

|  ≤ 1 − 𝛽 

By factoring out the term 𝑧−𝑝 

(𝑘 + 1) |
|
𝑧−𝑝 ((𝛼𝑝 + 𝑟 − 𝑝) + ∑ |2 − 𝛾 − 𝑛(1 − 𝛼)| {(1 − 𝜆) (1 +

𝑗
𝑝)}

𝑛

(1 + 𝑗)𝑎𝑗𝑧𝑗+𝑝∞
𝑗=2 )  

𝑧−𝑝 ((1 − 𝛼𝑝 − 𝑟) − ∑ (𝛼𝑗 + 1 − 𝑟) {(1 − 𝜆) (1 +
𝑗
𝑝)}

𝑛

𝑎𝑗𝑧𝑗+𝑝∞
𝑗=2 )

|
|

≤ 1 − 𝛽 

cancel 𝑧−𝑝 from both numerator and denominator  

(𝑘 + 1) |
|
((𝛼𝑝 + 𝑟 − 𝑝) + ∑ |2 − 𝛾 − 𝑛(1 − 𝛼)| {(1 − 𝜆) (1 +

𝑗
𝑝)}

𝑛

(1 + 𝑗)𝑎𝑗𝑧𝑗+𝑝∞
𝑗=2 )  

((1 − 𝛼𝑝 − 𝑟) − ∑ (𝛼𝑗 + 1 − 𝑟) {(1 − 𝜆) (1 +
𝑗
𝑝)}

𝑛

𝑎𝑗𝑧𝑗+𝑝∞
𝑗=2 )

|
| ≤ 1 − 𝛽 

Then, (𝑘 + 1) [(𝛼𝑝 + 𝑟 − 𝑝) + ∑ |2 − 𝛾 − 𝑛(1 − 𝛼)| {(1 − 𝜆) (1 +
𝑗

𝑝
)}

𝑛
(1 + 𝑗)𝑎𝑗𝑧𝑗+𝑝∞

𝑗=2 ]  

≤ (1 − 𝛼𝑝 − 𝑟)(1 − 𝛽) − (1 − 𝛽) (∑(𝛼𝑗 + 1 − 𝑟) {(1 − 𝜆) (1 +
𝑗

𝑝
)}

𝑛

𝑎𝑗𝑧𝑗+𝑝

∞

𝑗=2

) 

Now expand the terms of left-hand side (LHS) 

(𝑘 + 1) [(𝛼𝑝 + 𝑟 − 𝑝) + ∑|2 − 𝛾 − 𝑛(1 − 𝛼)| {(1 − 𝜆) (1 +
𝑗

𝑝
)}

𝑛

(1 + 𝑗)𝑎𝑗𝑧𝑗+𝑝

∞

𝑗=2

] 

The LHS consists of a constant term (𝑘 + 1)(𝛼𝑝 + 𝑟 − 𝑝) and a series denoted by 𝑆1, where 

𝑆1 = ∑|2 − 𝛾 − 𝑛(1 − 𝛼)| {(1 − 𝜆) (1 +
𝑗

𝑝
)}

𝑛

(1 + 𝑗)𝑎𝑗𝑧𝑗+𝑝

∞

𝑗=2

 

So, the LHS becomes  

(𝑘 + 1)(𝛼𝑝 + 𝑟 − 𝑝) + (𝑘 + 1)𝑆1 



Communications on Applied Nonlinear Analysis 

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Right-hand side (RHS) 

(1 − 𝛼𝑝 − 𝑟)(1 − 𝛽) − (1 − 𝛽) (∑(𝛼𝑗 + 1 − 𝑟) {(1 − 𝜆) (1 +
𝑗

𝑝
)}

𝑛

𝑎𝑗𝑧𝑗+𝑝

∞

𝑗=2

) 

The RHS also has a constant term (1 − 𝛼𝑝 − 𝑟)(1 − 𝛽) and a series denoted by 𝑆2, where  

𝑆2 = ∑(𝛼𝑗 + 1 − 𝑟) {(1 − 𝜆) (1 +
𝑗

𝑝
)}

𝑛

𝑎𝑗𝑧𝑗+𝑝

∞

𝑗=2

 

So, the RHS becomes  

(1 − 𝛼𝑝 − 𝑟)(1 − 𝛽) − (1 − 𝛽)𝑆2 

Then using the expressions for LHS & RHS  

(𝑘 + 1)(𝛼𝑝 + 𝑟 − 𝑝) + (𝑘 + 1)𝑆1 ≤  (1 − 𝛼𝑝 − 𝑟)(1 − 𝛽) − (1 − 𝛽)𝑆2 

Rearranging terms  

(𝑘 + 1)(𝛼𝑝 + 𝑟 − 𝑝) − (1 − 𝛼𝑝 − 𝑟)(1 − 𝛽) ≤  −(𝑘 + 1)𝑆1 − (1 − 𝛽)𝑆2 

Then expand and simplify the constants  

(𝑘 + 1)(𝛼𝑝 + 𝑟 − 𝑝) − (1 − 𝛼𝑝 − 𝑟)(1 − 𝛽) 

Distribute (1 − 𝛽) in the second term 

(𝑘 + 1)(𝛼𝑝 + 𝑟 − 𝑝) − (1 − 𝛽)(1) + (1 − 𝛽)(𝛼𝑝 + 𝑟) 

Combine like terms: 

𝑘(𝛼𝑝 + 𝑟 − 𝑝) + (𝛼𝑝 + 𝑟 − 𝑝) − (1 − 𝛽) + (1 − 𝛽)(𝛼𝑝 + 𝑟) 

Simplify further, then we get 

(𝛼𝑝 + 𝑟 − 𝑝) + 𝑘(𝛼𝑝 + 𝑟 − 𝑝) + (1 − 𝛽)(𝛼𝑝 + 𝑟 − 1) 

For the series 𝑆1 and 𝑆2,the terms of 𝑧𝑗+𝑝. Since 𝑧𝑗+𝑝 terms are independent of the constant parts, we 

can combine the inequalities to solve for the coefficients 𝑎𝑗 , given the following condition must hold: 

(𝑘 + 1)|2 − 𝛾 − 𝑛(1 − 𝛼)| {(1 − 𝜆) (1 +
𝑗

𝑝
)}

𝑛

(1 + 𝑗)𝑎𝑗

≤ (1 − 𝛽)(𝛼𝑗 + 1 − 𝑟) {(1 − 𝜆) (1 +
𝑗

𝑝
)}

𝑛

𝑎𝑗 

Then dividing both sides by {(1 − 𝜆) (1 +
𝑗

𝑝
)}

𝑛

, we get 

(𝑘 + 1)|2 − 𝛾 − 𝑛(1 − 𝛼)|(1 + 𝑗)𝑎𝑗 ≤ (1 − 𝛽)(𝛼𝑗 + 1 − 𝑟)𝑎𝑗 

For non- zero 𝑎𝑗 , divide through by 𝑎𝑗 

(𝑘 + 1)|2 − 𝛾 − 𝑛(1 − 𝛼)|(1 + 𝑗) ≤ (1 − 𝛽)(𝛼𝑗 + 1 − 𝑟) 



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Then expand and simplify  

(𝑘 + 1)|2 − 𝛾 − 𝑛(1 − 𝛼)| ≤ (1 − 𝛽)
(𝛼𝑗 + 1 − 𝑟)

(1 + 𝑗)
 

This gives a bound for the coefficients 𝑎𝑗 

Then the solution depends on analysing the coefficients 𝑎𝑗 satisfying: 

𝑎𝑗 ≤
(1 − 𝛽)(𝛼𝑗 + 1 − 𝑟)

(𝑘 + 1)|2 − 𝛾 − 𝑛(1 − 𝛼)|(1 + 𝑗)
 

The result is bounded by the choice of parameters 𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑝 and 𝑛. 

Corollary: (Boundedness of Coefficients for Special Cases) 

If we take 𝛼 = 0, 𝛽 = 0, 𝛾 = 0 and 𝑘 = 0 then the bound on the coefficients simplifies to: 

𝑎𝑗 ≤
1

|2 − 𝑛|(1 + 𝑗)
 

For all 𝑗 ≥ 2. This shows that in this special case, the coefficients decay inversely with respect to (1 +

𝑗), ensuring the function remains in a bounded subclass of analytic functions. 

Theorem: 2 (Convexity Condition) 

Let 𝑓(𝑧) ∈ ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝). If  𝐷𝑝,𝜆
𝑛 𝑓(𝑧) satisfies the following condition: 

𝑅𝑒 (1 +
𝑧(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))′′

(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′

) > 0, 

Then 𝑓(𝑧) is convex in △∗ .   

Proof: Here the operator  𝐷𝑝,𝜆
𝑛 𝑓(𝑧) is defined as  

𝐷𝑝,𝜆
𝑛 𝑓(𝑧) = 𝑧−𝑝 + ∑ {(1 − 𝜆) (1 +

𝑗

𝑝
) 𝑎𝑗𝑧𝑗}

𝑛

,   𝑧 ∈ △∗ 𝑎𝑛𝑑 𝑎𝑗 > 0

∞

𝑗=𝑝

 

For simplicity, Let 𝑔(𝑧) = 𝐷𝑝,𝜆
𝑛 𝑓(𝑧) 

Compute the 1st and 2nd derivative of 𝑔(𝑧), as follows then we get 

𝑔′(𝑧) = −𝑝𝑧−𝑝−1 + ∑ 𝑛

∞

𝑗=𝑝

[(1 − 𝜆) (1 +
𝑗

𝑝
) 𝑎𝑗𝑧𝑗−1] (1 +

𝑗

𝑝
) 

𝑔′′(𝑧) = 𝑝(𝑝 + 1)𝑧−𝑝−2 + ∑ 𝑛 (𝑛 − 1)

∞

𝑗=𝑝

[(1 − 𝜆) (1 +
𝑗

𝑝
) 𝑎𝑗𝑧𝑗−2] (1 +

𝑗

𝑝
)

2

 

Substitute these derivatives into the condition from 

𝑅𝑒 (1 +
𝑧𝑔′′(𝑧)

𝑔′(𝑧)
) > 0 



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⇒ 𝑅𝑒 (1 +
𝑧(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))′′

(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′

) > 0 

⇒ 𝑅𝑒 (1 +
𝑧 {𝑝(𝑝 + 1)𝑧−𝑝−2 + ∑ 𝑛 (𝑛 − 1)∞

𝑗=𝑝 [(1 − 𝜆) (1 +
𝑗
𝑝) 𝑎𝑗𝑧𝑗−2] (1 +

𝑗
𝑝)

2

}

−𝑝𝑧−𝑝−1 + ∑ 𝑛∞
𝑗=𝑝 [(1 − 𝜆) (1 +

𝑗
𝑝) 𝑎𝑗𝑧𝑗−1] (1 +

𝑗
𝑝)  

) > 0 

In the punctured unit disk △∗ , |𝑧| < 1, so higher-order terms 𝑧𝑗 ,becomes negligible as |𝑧| → 0 for 

leading order terms, then the condition will be    

𝑅𝑒 (1 +
𝑧[𝑝(𝑝 + 1)𝑧−𝑝−2]

−𝑝𝑧−𝑝−1
) > 0 

⇒ 𝑅𝑒 (1 +
𝑝(𝑝 + 1)𝑧−𝑝−1

−𝑝𝑧−𝑝−1
) > 0 

⇒ 𝑅𝑒 (1 + (
𝑝(𝑝 + 1)

−𝑝
)) > 0 

⇒ 𝑅𝑒(1 − (𝑝 + 1)) > 0 

⇒ 𝑅𝑒(1 − 𝑝 − 1) > 0 

⇒ 𝑅𝑒(−𝑝) > 0 

This means the real part of −𝑝, must be greater than 0. Let 𝑝 = 𝑎 + 𝑏𝑖, where 𝑎 and 𝑏 be the real part 

and 𝑖 be the imaginary part of this complex number, then 

−𝑝 = −𝑎 − 𝑏𝑖 

Thus, the real part of  −𝑝 is −𝑎, the inequality becomes   

−𝑎 > 0 

    𝑎 < 0 

Therefore,  

𝑅𝑒 (1 +
𝑧𝑔′′(𝑧)

𝑔′(𝑧)
) > 0 

This inequality holds when the real part of 𝑝 is negative. 

Theorem: 3 (Univalence Criterion) 

Let 𝑓(𝑧) ∈ ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝), If the following inequality holds: 

|
𝑧(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))′′

(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′

| < 1, 

Then 𝑓(𝑧) is univalent in △∗ . 

Proof: Given the form of 𝐷𝑝,𝜆
𝑛 𝑓(𝑧) , here the 1st derivative of this operator will be  



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(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′ = −𝑝𝑧−𝑝−1 + ∑ 𝑛

∞

𝑗=𝑝

[(1 − 𝜆) (1 +
𝑗

𝑝
) 𝑎𝑗𝑧𝑗−1] (1 +

𝑗

𝑝
) 

2nd derivative will be  

(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′′ = 𝑝(𝑝 + 1)𝑧−𝑝−2 + ∑ 𝑛 (𝑛 − 1)

∞

𝑗=𝑝

[(1 − 𝜆) (1 +
𝑗

𝑝
) 𝑎𝑗𝑧𝑗−2] (1 +

𝑗

𝑝
)

2

 

Substitute these derivatives in to the conditions we get, 

|
𝑧(𝐷𝑝,𝜆

𝑛 𝑓(𝑧))′′

(𝐷𝑝,𝜆
𝑛 𝑓(𝑧))′

| < 1, 

= ||
𝑧 𝑝(𝑝 + 1)𝑧−𝑝−2 + ∑ 𝑛 (𝑛 − 1)∞

𝑗=𝑝 [(1 − 𝜆) (1 +
𝑗
𝑝) 𝑎𝑗𝑧𝑗−1] (1 +

𝑗
𝑝)

2

−𝑝𝑧−𝑝−1 + ∑ 𝑛∞
𝑗=𝑝 [(1 − 𝜆) (1 +

𝑗
𝑝) 𝑎𝑗𝑧𝑗−1] (1 +

𝑗
𝑝)

|| < 1, 

= ||
 𝑝(𝑝 + 1)𝑧−𝑝−1 + ∑ 𝑛 (𝑛 − 1)∞

𝑗=𝑝 [(1 − 𝜆) (1 +
𝑗
𝑝) 𝑎𝑗𝑧𝑗−2] (1 +

𝑗
𝑝)

2

−𝑝𝑧−𝑝−1 + ∑ 𝑛∞
𝑗=𝑝 [(1 − 𝜆) (1 +

𝑗
𝑝

) 𝑎𝑗𝑧𝑗−1] (1 +
𝑗
𝑝

)
|| < 1, 

= ||
𝑧−𝑝−1 {𝑝(𝑝 + 1) + ∑ 𝑛 (𝑛 − 1)∞

𝑗=𝑝 [(1 − 𝜆) (1 +
𝑗
𝑝) 𝑎𝑗𝑧𝑗+𝑝+1] (1 +

𝑗
𝑝)

2

}

𝑧−𝑝−1 {−𝑝 + ∑ 𝑛∞
𝑗=𝑝 [(1 − 𝜆) (1 +

𝑗
𝑝) 𝑎𝑗𝑧𝑗+𝑝+1] (1 +

𝑗
𝑝)}

|| < 1, 

After factoring 𝑧−𝑝−1 out the fraction simplifies to,  

= ||
{𝑝(𝑝 + 1) + ∑ 𝑛 (𝑛 − 1)∞

𝑗=𝑝 [(1 − 𝜆) (1 +
𝑗
𝑝) 𝑎𝑗𝑧𝑗+𝑝+1] (1 +

𝑗
𝑝)

2

}

{−𝑝 + ∑ 𝑛∞
𝑗=𝑝 [(1 − 𝜆) (1 +

𝑗
𝑝) 𝑎𝑗𝑧𝑗+𝑝+1] (1 +

𝑗
𝑝)}

|| < 1, 

𝑝(𝑝 + 1) + ∑ 𝑛 (𝑛 − 1)

∞

𝑗=𝑝

[(1 − 𝜆) (1 +
𝑗

𝑝
) 𝑎𝑗𝑧𝑗+𝑝+1] (1 +

𝑗

𝑝
)

2

 

Here the leading term is 𝑝(𝑝 + 1) and the remaining part is a series that depends on 𝑧. 

−𝑝 + ∑ 𝑛

∞

𝑗=𝑝

[(1 − 𝜆) (1 +
𝑗

𝑝
) 𝑎𝑗𝑧𝑗+𝑝+1] (1 +

𝑗

𝑝
) 

Here the leading term is −𝑝, with the remaining part as a series. 

To evaluate the modulus 

At 𝒛 → 𝟎, 



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The series terms involving 𝑧𝑗+𝑝+1 will reduce, making the leading terms 𝑝(𝑝 + 1) in the numerator 

and −𝑝 in the denominator dominant.   

The expression becomes approximately 

|
𝑝(𝑝 + 1)

−𝑝
| = |−(𝑝 + 1)| = (𝑝 + 1) < 1 

This needs to be less than 1, so for small 𝑧 this inequality holds only if  (𝑝 + 1) < 1, which suggests 

𝑝 < 0. 

At 𝒛 → ∞, 

The series terms involving higher powers of 𝑧 will dominate, and we need to analyse the growth rate 

of the series terms relative to each other. For large 𝑧, we compare the asymptotic behaviour of the 

series in both the numerator and denominator.   

5. Discussion 

In this work, a new subclass ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝) of meromorphic functions defined in the punctured 

unit disk △∗ = {𝑧 ∈ ℂ ∶ 0 < |𝑧| < 1} is introduced using a generalized Sălăgean-type differential 

operator 𝐷𝑝,𝜆
𝑛 . The functions in this class are characterized by satisfying a specific real-part inequality 

involving the parameters 𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝 which controls their geometric behaviour. A sharp upper 

bound is derived for the coefficients 𝑎𝑗 of functions in this class. This bound depends on the involved 

parameters and effectively restricts the growth of the series coefficients. Such bounds are crucial for 

understanding the geometric structure and distortion properties of the functions. If the Sălăgean 

operator 𝐷𝑝,𝜆
𝑛 𝑓(𝑧) satisfies a specific positivity condition involving its logarithmic derivative, then the 

function 𝑓 is convex in △∗ . This result connects analytic properties of the differential operator to the 

geometric shape (convexity) of the function image. Theorem 3 provides a sufficient condition for 

univalence: if the modulus of the logarithmic derivative of 𝐷𝑝,𝜆
𝑛 𝑓(𝑧) remains less than one, then the 

function 𝑓  is univalent (injective) in △∗ .This ensures that the function preserves distinctness of points 

under mapping, a foundational property in geometric function theory.  

Overall, these theorems collectively establish important structural properties coefficient bounds, 

convexity, and univalence of the functions in the class ℳ(𝛼, 𝛽, 𝛾, 𝑘, 𝜆, 𝑛, 𝑝), highlighting the impact 

of the Sălăgean operator on the analytic and geometric behaviour of meromorphic functions. 

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