Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

254 
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Subclass of Analytic Functions Associated with Differential Operator 

 

Katterapalle Sridevi𝟏,  N. Sri Lakshmi Sudha Rani𝟐      
1 Department of Mathematics, 

Dr.B.R.Ambedkar Open University, Hyderabad - 500 033, T.S, India. 

sridevidrk18@gmail.com 
2 Department of Mathematics, 

Dr.B.R.Ambedkar Open University, Hyderabad - 500 033, T.S, India. 

lakshmisudha1848@gmail.com  

 

Article History: 

Received: 16-04-2024 

Revised: 28-05-2024 

Accepted: 13-06-2024 

Abstract:  

Introduction: : In this work, we introduce and investigate a new class �̃�𝑆𝑠
𝑚(ς, ℏ, ℘, ϱ, 𝑡) of 

analytic functions in the open unit disc 𝑈 with negative coefficients. The object of the 

present paper is to determine coefficient estimates, neighborhoods and partial sums for 

functions 𝑓 belonging to this class. 

Keywords: : analytic function, uniformly starlike function, coefficient estimate, 

neighborhood, partial sums. 

AMS Subject Classification: 30C45. 

 

1. Introduction 

Let 𝐴 denote the class of analytic functions 𝑓 defined on the unit disk 𝑈 = {𝑧: |𝑧| < 1} with 

normalization 𝑓(0) = 0 and 𝑓′(0) = 1. Such a function has the Taylor series expansion about the 

origin in the form 

𝑓(𝑧) = 𝑧 + ∑ 𝑎𝑛

∞

𝑛=2

𝑧𝑛                                                                            (1.1), 

denoted by 𝑆, the subclass of 𝐴 consisting of functions that are univalent in 𝑈. 

For 𝑓 ∈ 𝐴 given by (1.1) and 𝑔(𝑧) given by 

𝑔(𝑧) = 𝑧 + ∑ 𝑏𝑛

∞

𝑛=2

𝑧𝑛                                                                          (1.2) 

their convolution (or Hadamard product), denoted by (𝑓 ∗ 𝑔), is defined as 

(𝑓 ∗ 𝑔)(𝑧) = 𝑧 + ∑ 𝑎𝑛

∞

𝑛=2

𝑏𝑛𝑧𝑛 = (𝑔 ∗ 𝑓)(𝑧)  (𝑧 ∈ 𝑈).                     (1.3) 

Note that 𝑓 ∗ 𝑔 ∈ 𝐴. 

A function 𝑓 ∈ 𝐴 is said to be in 𝑈𝑆(ϱ), the class of uniformly starlike functions of order ϱ, 0 ≤ ϱ <

1, if satisfies the condition 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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ℜ {
𝑧𝑓′(𝑧)

𝑓(𝑧)
} > |

𝑧𝑓′(𝑧)

𝑓(𝑧)
− 1| + ϱ,                                 (1.4) 

and a function 𝑓 ∈ 𝐴 is said to be in 𝑈𝐶(ϱ), the class of uniformly convex functions of order ϱ, 0 ≤

ϱ < 1, if satisfies the condition 

ℜ {1 +
𝑧𝑓″(𝑧)

𝑓′(𝑧)
} > |

𝑧𝑓″(𝑧)

𝑓′(𝑧)
| + ϱ.                              (1.5) 

Uniformly starlike and uniformly convex functions were first introduced by Goodman  [8] and then 

studied by various authors. 

In , Sakaguchi [11] defined the class 𝑆𝑠 of starlike functions with respect to symmetric points as 

follows: 

Let 𝑓 ∈ 𝐴. Then 𝑓 is said to be starlike with respect to symmetric points in 𝑈 if and only if 

ℜ {
2𝑧𝑓′(𝑧)

𝑓(𝑧) − 𝑓(−𝑧)
} > 0,  (𝑧 ∈ 𝑈). 

Recently, Owa et al. [10]  defined the class 𝑆𝑠(ς, 𝑡) as follows: 

ℜ {
(1 − 𝑡)𝑧𝑓′(𝑧)

𝑓(𝑧) − 𝑓(𝑡𝑧)
} > ς,  (𝑧 ∈ 𝑈), 

where 0 ≤ ς < 1, |𝑡| ≤ 1, 𝑡 ≠ 1. Note that 𝑆𝑠(0, −1) = 𝑆𝑠 and 𝑆𝑠(ς, −1) = 𝑆𝑠(ς) is called Sakaguchi 

function of order ς. 

In , Darus and Faisal [5] introduced the following differential operator. 

For a function 𝑓 ∈ 𝐴, 

𝒟℘
0 (ς, ℏ)𝑓(𝑧) = 𝑓(𝑧)

𝒟℘
1 (ς, ℏ)𝑓(𝑧) = (

ς − ℏ − ℘

ς
) 𝑓(𝑧) + (

ℏ + ℘

ς
) 𝑧𝑓′(𝑧)

𝒟℘
2 (ς, ℏ)𝑓(𝑧) = 𝒟 (𝒟℘

1 (ς, ℏ)𝑓(𝑧))

                       ⋮

𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) = 𝒟℘ (𝒟℘

𝑚−1(ς, ℏ)𝑓(𝑧))

 

where ς, ℏ, ℘ ≥ 0, ς ≠ 0 and 𝑚 ∈ ℕ0 = ℕ ∪ {0}. 

If 𝑓 is given by (1.1)  then from the definition of the operator 𝒟℘
𝑚(ς, ℏ)𝑓 it is easy to see that 

𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) = 𝑧 + ∑ 𝜙𝑛

∞

𝑛=2

(ς, ℏ, ℘, 𝑚)𝑎𝑛𝑧𝑛                       (1.6) 

where 

                                                         𝜙𝑛(ς, ℏ, ℘, 𝑚) = (
ς+(ℏ+℘)(𝑛−1)

ς
)

𝑚

                       (1.7) 



Communications on Applied Nonlinear Analysis 

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By specializing the parameters of 𝒟℘
𝑚(ς, ℏ)𝑓(𝑧), we get the following differential operators. If we 

substitute 

(i) ς = 1and ℏ = 0, we get 𝒟𝑚𝑓(𝑧) = 𝑧 + ∑ (1 + ℘(𝑛 − 1))
𝑚∞

𝑛=2 𝑎𝑛𝑧𝑛 oof differential operator 

given by Al-Oboudi [1]. 

(ii) ς = 1, ℏ = 0 and ℘ = 1, we get 𝒟𝑚𝑓(𝑧) = 𝑧 + ∑ (𝑛)𝑚∞
𝑛=2 𝑎𝑛𝑧𝑛 of Salagean differential 

operator[12]. 

Now, by making use of the differential operator 𝒟℘
𝑚(ς, ℏ)𝑓, we define a new subclass of functions 

belonging to the class 𝐴. 

Definition 1.  A function 𝑓 ∈ 𝐴 is said to be in the class 𝛩𝑆𝑠
𝑚(ς, ℏ, ℘, ϱ, 𝑡) if for all 𝑧 ∈ 𝑈 

ℜ {
(1 − 𝑡)𝑧 (𝒟℘

𝑚(ς, ℏ)𝑓(𝑧)) ′

𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧)
} ≥ |

(1 − 𝑡)𝑧 (𝒟℘
𝑚(ς, ℏ)𝑓(𝑧)) ′

𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧)
− 1| + ϱ, 

for ℘ ≥ 0,  𝑚, |𝑡| ≤ 1, 𝑡 ≠ 1,0 ≤ ϱ < 1. 

Furthermore, we say that a function 𝑓 ∈ 𝑈𝑆𝑠
𝑚(ς, ℏ, ℘, ϱ, 𝑡) is in the subclass �̃�𝑆𝑠

𝑚(ς, ℏ, ℘, ϱ, 𝑡) if 

𝑓(𝑧) is of the following form 

𝑓(𝑧) = 𝑧 − ∑ 𝑎𝑛

∞

𝑛=2

𝑧𝑛,  𝑎𝑛 ≥ 0, 𝑛 ∈ ℕ,  𝑧 ∈ 𝑈.                            (1.8) 

The aim of the present paper is to study the coefficient bounds, partial sums and certain 

neighborhood results of the class �̃�𝑆𝑠
𝑚(ς, ℏ, ℘, ϱ, 𝑡). 

Firstly, we shall need the following lemmas. 

Lemma 2.  Let 𝑤 = 𝑢 + 𝑖𝑣. Then ℜ (𝑤) ≥ 𝛽  if and only if  |𝑤 − (1 + 𝛽)| ≤ |𝑤 + (1 − ς)|. 

Lemma 3.  Let 𝑤 = 𝑢 + 𝑖𝑣 and ς, ϱ be real numbers. Then ℜ (𝑤) > 𝛽|𝑤 − 1| +

ϱ  if and only if  ℜ{𝑤(1 + 𝛽𝑒𝑖𝜃) − 𝛽𝑒𝑖𝜃} > ϱ 

2  Coefficient bounds  

Theorem 4.  The function 𝑓 defined by (1.8)  is in the class �̃�𝑆𝑠
𝑚(ς, ℏ, ℘, ϱ, 𝑡) if and only if 

∑ 𝜙𝑛
∞
𝑛=2 (ς, ℏ, ℘, 𝑚)|2𝑛 − 𝑢𝑛(1 + ϱ)|𝑎𝑛 ≤ 1 − ϱ,                                                                             (2.1)  

where ℘ ≥ 0,  𝑚, 𝑘 ≥ 0, |𝑡| ≤ 1, 𝑡 ≠ 1,0 ≤ ϱ < 1 and 𝑢𝑛 = 1 + 𝑡 + ⋯ + 𝑡𝑛−1. 

The result is sharp for the function 𝑓(𝑧) given by 𝑓(𝑧) = 𝑧 −
1−ϱ

𝜙𝑛(ς,ℏ,℘,𝑚)|2𝑛−𝑢𝑛(1+ϱ)|
𝑧𝑛. 

Proof. By Definition 1, we get 

ℜ {
(1 − 𝑡)𝑧 (𝒟℘

𝑚(ς, ℏ)𝑓(𝑧)) ′

𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧)
} ≥ |

(1 − 𝑡)𝑧 (𝒟℘
𝑚(ς, ℏ)𝑓(𝑧)) ′

𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧)
− 1| + ϱ. 

 



Communications on Applied Nonlinear Analysis 

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Then by Lemma 3, we have 

ℜ {
(1 − 𝑡)𝑧 (𝒟℘

𝑚(ς, ℏ)𝑓(𝑧)) ′

𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧)
(1 + 𝑒𝑖𝜃) − 𝑒𝑖𝜃} ≥ ϱ,  −𝜋 < 𝜃 ≤ 𝜋 

or equivalently 

ℜ {
(1 − 𝑡)𝑧 (𝒟℘

𝑚(ς, ℏ)𝑓(𝑧)) ′(1 + 𝑒𝑖𝜃)

𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧)
−

𝑒𝑖𝜃[𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧)]

𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧)
} ≥ ϱ.        (2.2) 

Let 𝐹(𝑧) = (1 − 𝑡)𝑧 (𝒟℘
𝑚(ς, ℏ)𝑓(𝑧)) ′(1 + 𝑒𝑖𝜃) − 𝑒𝑖𝜃[𝒟℘

𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘
𝑚(ς, ℏ)𝑓(𝑡𝑧)] 

and 𝐸(𝑧) = 𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧). 

By Lemma 2 ,  (2.2) is equivalent to 

|𝐹(𝑧) + (1 − ϱ)𝐸(𝑧)| ≥ |𝐹(𝑧) − (1 + ϱ)𝐸(𝑧)|,   for 0 ≤ ϱ < 1. 

But 

|𝐹(𝑧) + (1 − ϱ)𝐸(𝑧)| =  |(1 − 𝑡){(2 − ϱ)𝑧 − ∑ 𝜙𝑛

∞

𝑛=2

(ς, ℏ, ℘, 𝑚)(𝑛 + 𝑢𝑛(1 − ϱ))𝑎𝑛𝑧𝑛

  −𝑒𝑖𝜃 ∑ 𝜙𝑛

∞

𝑛=2

(ς, ℏ, ℘, 𝑚)(𝑛 − 𝑢𝑛)𝑎𝑛𝑧𝑛}|

≥ |1 − 𝑡|{(2 − ϱ)|𝑧| − ∑ 𝜙𝑛

∞

𝑛=2

(ς, ℏ, ℘, 𝑚)|𝑛 + 𝑢𝑛(1 − ϱ)|𝑎𝑛|𝑧𝑛|

  − ∑ 𝜙𝑛

∞

𝑛=2

(ς, ℏ, ℘, 𝑚)|𝑛 − 𝑢𝑛|𝑎𝑛|𝑧𝑛|}.

 

Also 

|𝐹(𝑧) − (1 + ϱ)𝐸(𝑧)| =  |(1 − 𝑡){−ϱ𝑧 − ∑ 𝜙𝑛

∞

𝑛=2

(ς, ℏ, ℘, 𝑚)(𝑛 − 𝑢𝑛(1 + ϱ))𝑎𝑛𝑧𝑛

  −𝑒𝑖𝜃 ∑ 𝜙𝑛

∞

𝑛=2

(ς, ℏ, ℘, 𝑚)(𝑛 − 𝑢𝑛)𝑎𝑛𝑧𝑛}|

≤ |1 − 𝑡|{ϱ|𝑧| + ∑ 𝜙𝑛

∞

𝑛=2

(ς, ℏ, ℘, 𝑚)|𝑛 − 𝑢𝑛(1 + ϱ)|𝑎𝑛|𝑧𝑛|

  + ∑ 𝜙𝑛

∞

𝑛=2

(ς, ℏ, ℘, 𝑚)|𝑛 − 𝑢𝑛|𝑎𝑛|𝑧𝑛|}.

 

 

 



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So 

|𝐹(𝑧) + (1 − ϱ)𝐸(𝑧)| − |𝐹(𝑧) − (1 + ϱ)𝐸(𝑧)|

≥  |1 − 𝑡|{2(1 − ϱ)|𝑧| − ∑ 𝜙𝑛

∞

𝑛=2

(ς, ℏ, ℘, 𝑚)[|𝑛 + 𝑢𝑛(1 − ϱ)| + |𝑛 − 𝑢𝑛(1 + ϱ)| + 2|𝑛 − 𝑢𝑛| ]𝑎𝑛|𝑧𝑛|}

≥  2(1 − ϱ)|𝑧| − ∑ 2

∞

𝑛=2

𝜙𝑛(ς, ℏ, ℘, 𝑚)|2𝑛 − 𝑢𝑛(1 + ϱ)|𝑎𝑛|𝑧𝑛| ≥ 0

 

or 

∑ 𝜙𝑛

∞

𝑛=2

(ς, ℏ, ℘, 𝑚)|2𝑛 − 𝑢𝑛(1 + ϱ)|𝑎𝑛 ≤ 1 − ϱ. 

Conversely, suppose that  (2.1) holds. Then we must show 

ℜ {
(1 − 𝑡)𝑧 (𝒟℘

𝑚(ς, ℏ)𝑓(𝑧)) ′(1 + 𝑒𝑖𝜃) − 𝑒𝑖𝜃[𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧)]

𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧)
} ≥ ϱ. 

Upon choosing the values of 𝑧 on the positive real axis where 0 ≤ |𝑧| = 𝑟 < 1, the above inequality 

reduces to 

ℜ {
(1 − ϱ) − ∑ 𝜙𝑛

∞
𝑛=2 (ς, ℏ, ℘, 𝑚)[𝑛(1 + 𝑒𝑖𝜃) − 𝑢𝑛(ϱ + 𝑒𝑖𝜃)]𝑎𝑛𝑧𝑛−1

1 − ∑ 𝜙𝑛
∞
𝑛=2 (ς, ℏ, ℘, 𝑚)𝑢𝑛𝑎𝑛𝑧𝑛−1

} ≥ 0. 

Since ℜ(−𝑒𝑖𝜃) ≥ −|𝑒𝑖𝜃| = −1, the above inequality reduces to 

ℜ {
(1 − ϱ) − ∑ 𝜙𝑛

∞
𝑛=2 (ς, ℏ, ℘, 𝑚)[2𝑛 − 𝑢𝑛(1 + ϱ]𝑎𝑛𝑟𝑛−1

1 − ∑ 𝜙𝑛
∞
𝑛=2 (ς, ℏ, ℘, 𝑚)𝑢𝑛𝑎𝑛𝑟𝑛−1

} ≥ 0. 

Letting 𝑟 → 1−, we have desired conclusion.  

Corollary 5.  If 𝑓(𝑧) ∈ �̃�𝑆𝑠
𝑚(ς, ℏ, ℘, ϱ, 𝑡) then 𝑎𝑛 ≤

1−ϱ

𝜙𝑛(ς,ℏ,℘,𝑚)|2𝑛−𝑢𝑛(1+ϱ)|
 where ℘ ≥ 0,  𝑚, |𝑡| ≤

1, 𝑡 ≠ 1,0 ≤ ϱ < 1 and 𝑢𝑛 = 1 + 𝑡 + ⋯ + 𝑡𝑛−1. 

3  Neighborhood property 

Following the earlier investigations (based upon the familiar concept of neighborhoods of analytic 

functions) by Goodman [7], Srinivas et al [16] , Altintas et al [2, 3].  and others including Srivastava 

et al.[15] , Orhan [9], Deniz et al. [6], Catas [4]. 

Definition 6.  Let ℘ ≥ 0,  𝑚, |𝑡| ≤ 1, 𝑡 ≠ 1,0 ≤ ϱ < 1, ς ≥ 0 and 𝑢𝑛 = 1 + 𝑡 + ⋯ + 𝑡𝑛−1. We 

define the ς −neighborhood of a function 𝑓 ∈ 𝐴 and denote by 𝑁ς(𝑓) consisting of all functions 

𝑔(𝑧) = 𝑧 − ∑ 𝑏𝑛
∞
𝑛=2 𝑧𝑛 ∈ 𝑆(𝑏𝑛 ≥ 0, 𝑛 ∈ ℕ) satisfying 

∑
𝜙𝑛(ς, ℏ, ℘, 𝑚)|2𝑛 − 𝑢𝑛(1 + ϱ)|

1 − ϱ

∞

𝑛=2

|𝑎𝑛 − 𝑏𝑛| ≤ 1 − ς. 



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Theorem 7.  Let 𝑓(𝑧) ∈ �̃�𝑆𝑠
𝑚(ς, ℏ, ℘, ϱ, 𝑡) and for all real 𝜃 we have ϱ(𝑒𝑖𝜃 − 1) − 2𝑒𝑖𝜃 ≠ 0. For 

any complex number 𝜖 with |𝜖| < ς(ς ≥ 0), if f satisfies the following condition: 
𝑓(𝑧)+𝜖𝑧

1+𝜖
∈

�̃�𝑆𝑠
𝑚(ς, ℏ, ℘, ϱ, 𝑡) then 𝑁ς(𝑓) ⊂ �̃�𝑆𝑠

𝑚(ς, ℏ, ℘, ϱ, 𝑡). 

Proof. It is obvious that 𝑓 ∈ �̃�𝑆𝑠
𝑚(ς, ℏ, ℘, ϱ, 𝑡) if and only if 

|
(1 − 𝑡)𝑧 (𝒟℘

𝑚(ς, ℏ)𝑓(𝑧)) ′(1 + 𝑒𝑖𝜃) − (𝑒𝑖𝜃 + 1 + ϱ) (𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧))

(1 − 𝑡)𝑧 (𝒟℘
𝑚(ς, ℏ)𝑓(𝑧)) ′(1 + 𝑒𝑖𝜃) + (1 − 𝑒𝑖𝜃 − ϱ) (𝒟℘

𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘
𝑚(ς, ℏ)𝑓(𝑡𝑧))

| < 1, 

 (−𝜋 < 𝜃 ≤ 𝜋), 

for any complex number 𝑠 with |𝑠| = 1, we have 

(1 − 𝑡)𝑧 (𝒟℘
𝑚(ς, ℏ)𝑓(𝑧)) ′(1 + 𝑒𝑖𝜃) − (𝑒𝑖𝜃 + 1 + ϱ) (𝒟℘

𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘
𝑚(ς, ℏ)𝑓(𝑡𝑧))

(1 − 𝑡)𝑧 (𝒟℘
𝑚(ς, ℏ)𝑓(𝑧)) ′(1 + 𝑒𝑖𝜃) + (1 − 𝑒𝑖𝜃 − ϱ) (𝒟℘

𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘
𝑚(ς, ℏ)𝑓(𝑡𝑧))

≠ 𝑠. 

In other words, we must have 

(1 − 𝑠)(1 − 𝑡)𝑧 (𝒟℘
𝑚(ς, ℏ)𝑓(𝑧)) ′(1 + 𝑒𝑖𝜃) − (𝑒𝑖𝜃 + 1 + ϱ + 𝑠(−1 + 𝑒𝑖𝜃 + ϱ)) 

× (𝒟℘
𝑚(ς, ℏ)𝑓(𝑧) − 𝒟℘

𝑚(ς, ℏ)𝑓(𝑡𝑧)) ≠ 0. 

which is equivalent to 

𝑧 − ∑
𝜙𝑛(ς, ℏ, ℘, 𝑚) ((𝑛 − 𝑢𝑛)(1 + 𝑒𝑖𝜃 − 𝑠𝑘𝑒𝑖𝜃) − 𝑠(𝑛 + 𝑢𝑛) − 𝑢𝑛ϱ(1 − 𝑠))

ϱ(𝑠 − 1) − 2𝑠

∞

𝑛=2

𝑧𝑛 ≠ 0. 

However, 𝑓 ∈ �̃�𝑆𝑠
𝑚(ς, ℏ, ℘, ϱ, 𝑡) if and only 

(𝑓∗ℎ)

𝑧
≠ 0, 𝑧 ∈ 𝑈 − {0}, where ℎ(𝑧) = 𝑧 − ∑ 𝑐𝑛

∞
𝑛=2 𝑧𝑛 

and 

𝑐𝑛 =
𝜙𝑛(ς, ℏ, ℘, 𝑚) ((𝑛 − 𝑢𝑛)(1 + 𝑒𝑖𝜃 − 𝑠𝑒𝑖𝜃) − 𝑠(𝑛 + 𝑢𝑛) − 𝑢𝑛ϱ(1 − 𝑠))

ϱ(𝑠 − 1) − 2𝑠
 

we note that 

|𝑐𝑛| ≤
𝜙𝑛(ς, ℏ, ℘, 𝑚)|2𝑛 − 𝑢𝑛(1 + ϱ)|

1 − ϱ
 

since 
𝑓(𝑧)+𝜖𝑧

1+𝜖
∈ �̃�𝑆𝑠

𝑚(ς, ℏ, ℘, ϱ, 𝑡), therefore 𝑧−1 (
𝑓(𝑧)+𝜖𝑧

1+𝜖
∗ ℎ(𝑧)) ≠ 0, which is equivalent to 

(𝑓 ∗ ℎ)(𝑧)

(1 + 𝜖)𝑧
+

𝜖

1 + 𝜖
≠ 0.                                                         (3.1) 

Now suppose that |
(𝑓∗ℎ)(𝑧)

𝑧
| < ς. Then by (3.1), we must have 



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|
(𝑓 ∗ ℎ)(𝑧)

(1 + 𝜖)𝑧
+

𝜖

1 + 𝜖
| ≥

|𝜖|

|1 + 𝜖|
−

1

|1 + 𝜖|
|
(𝑓 ∗ ℎ)(𝑧)

𝑧
|

>
|𝜖| − ς

|1 + 𝜖|
≥ 0,

 

this is a contradiction by |𝜖| < ς and however, we have |
(𝑓∗ℎ)(𝑧)

𝑧
| ≥ ς. If 𝑔(𝑧) = 𝑧 − ∑ 𝑏𝑛

∞
𝑛=2 𝑧𝑛 ∈

𝑁ς(𝑓), then 

ς − |
(𝑔 ∗ ℎ)(𝑧)

𝑧
| ≤ |

((𝑓 − 𝑔) ∗ ℎ)(𝑧)

𝑧
| ≤ ∑|𝑎𝑛 − 𝑏𝑛|

∞

𝑛=2

|𝑐𝑛||𝑧𝑛|

< ∑
𝜙𝑛(ς, ℏ, ℘, 𝑚)|2𝑛 − 𝑢𝑛(1 + ϱ)|

1 − ϱ

∞

𝑛=2

|𝑎𝑛 − 𝑏𝑛| ≤ ς.

 

4     Partial sums  

In this section, applying methods used by Silverman [13]  and Silvia [14] , we investigate the ratio of 

a function of the form  (1.8) to its sequence of partial sums 𝑓𝑚(𝑧) = 𝑧 + ∑ 𝑎𝑛
𝑚
𝑛=2 𝑧𝑛. 

Theorem 8.  If 𝑓 of the form  (1.1) satisfies the condition (2.1)  then  

ℜ {
𝑓(𝑧)

𝑓𝑚(𝑧)
} ≥ 1 −

1

𝛿𝑚+1
                                                           (4.1) 

 and  

                                  𝛿𝑛 = {
1,  𝑖𝑓 𝑛 = 2,3 ⋯ 𝑚

𝛿𝑚+1,  𝑖𝑓 𝑛 = 𝑚 + 1, 𝑚 ∗ 2, ⋯
                                        (4.2) 

 

 Where                 𝛿𝑛 =
𝜙𝑛(ς,ℏ,℘,𝑚)|2𝑛−𝑢𝑛(1+ϱ)|

1−ϱ
.                                                                 (4.3)  

The result in (4.1) is sharp for every m, with the extremal function 

                                                   𝑓(𝑧) = 𝑧 +
𝑧𝑚+1

𝛿𝑚+1
.                                                           (4.4) 

Proof. Define the function 𝑤, we may write 

1 + 𝑤(𝑧)

1 − 𝑤(𝑧)
= 𝛿𝑚+1 {

𝑓(𝑧)

𝑓𝑚(𝑧)
− (1 −

1

𝛿𝑚+1
)}                                    (4.5)            

= {
1 + ∑ 𝑎𝑛

𝑚
𝑛=2 𝑧𝑛−1 + 𝛿𝑚+1 ∑ 𝑎𝑛

∞
𝑛=𝑚+1 𝑧𝑛−1

1 + ∑ 𝑎𝑛
𝑚
𝑛=2 𝑧𝑛−1

} .

 

Then, from (4.5), we can obtain 

𝑤(𝑧) =
𝛿𝑚+1 ∑ 𝑎𝑛

∞
𝑛=𝑚+1 𝑧𝑛−1

2 + 2 ∑ 𝑎𝑛
𝑚
𝑛=2 𝑧𝑛−1 + 𝛿𝑚+1 ∑ 𝑎𝑛

∞
𝑛=𝑚+1 𝑧𝑛−1

 

 



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and 

|𝑤(𝑧)| ≤
𝛿𝑚+1 ∑ 𝑎𝑛

∞
𝑛=𝑚+1

2 − 2 ∑ 𝑎𝑛
𝑚
𝑛=2 − 𝛿𝑚+1 ∑ 𝑎𝑛

∞
𝑛=𝑚+1

. 

Now |𝑤(𝑧)| ≤ 1 if 

2𝛿𝑚+1 ∑ 𝑎𝑛

∞

𝑛=𝑚+1

≤ 2 − 2 ∑ 𝑎𝑛

𝑚

𝑛=2

, 

which is equivalent to 

∑ 𝑎𝑛

𝑚

𝑛=2

+ 𝛿𝑚+1 ∑ 𝑎𝑛

∞

𝑛=𝑚+1

≤ 1.                                         (4.6) 

It is suffices to show that the left hand side of (4.6)  is bounded above by ∑ 𝛿𝑛
∞
𝑛=2 𝑎𝑛, which is 

equivalent to 

∑(𝛿𝑛 − 1)

𝑚

𝑛=2

𝑎𝑛 + ∑ (𝛿𝑛 − 𝛿𝑚+1)

∞

𝑛=𝑚+1

𝑎𝑛 ≥ 0. 

To see that the function given by (4.4) gives the sharp result, we observe that for 𝑧 = 𝑟𝑒𝑖𝜋/𝑛, 

𝑓(𝑧)

𝑓𝑚(𝑧)
= 1 +

𝑧𝑚

𝛿𝑚+1
                                (4.7). 

Taking 𝑧 → 1−, we have 

𝑓(𝑧)

𝑓𝑚(𝑧)
= 1 −

1

𝛿𝑚+1
. 

This completes the proof of Theorem 8. 

We next determine bounds for 
𝑓𝑚(𝑧)

𝑓(𝑧)
. 

Theorem 9.  If 𝑓 of the form (1.1) satisfies the condition (2.1) then  

                                           ℜ {
𝑓𝑚(𝑧)

𝑓(𝑧)
} ≥

𝛿𝑚+1

1+𝛿𝑚+1
.                                              (4.8)  

The result is sharp with the function given by (4.4). 

Proof. We may write 

1 + 𝑤(𝑧)

1 − 𝑤(𝑧)
= (1 + 𝛿𝑚+1) {

𝑓𝑚(𝑧)

𝑓(𝑧)
−

𝛿𝑚+1

1 + 𝛿𝑚+1
}

= {
1 + ∑ 𝑎𝑛

𝑚
𝑛=2 𝑧𝑛−1 − 𝛿𝑚+1 ∑ 𝑎𝑛

∞
𝑛=𝑚+1 𝑧𝑛−1

1 + ∑ 𝑎𝑛
∞
𝑛=2 𝑧𝑛−1

} ,

 

 

 



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where 

𝑤(𝑧) =
(1 + 𝛿𝑚+1) ∑ 𝑎𝑛

∞
𝑛=𝑚+1 𝑧𝑛−1

−(2 + 2 ∑ 𝑎𝑛
𝑚
𝑛=2 𝑧𝑛−1 − (1 − 𝛿𝑚+1) ∑ 𝑎𝑛

∞
𝑛=𝑚+1 𝑧𝑛−1)

 

and 

|𝑤(𝑧)| ≤
(1 + 𝛿𝑚+1) ∑ 𝑎𝑛

∞
𝑛=𝑚+1

2 − 2 ∑ 𝑎𝑛
𝑚
𝑛=2 + (1 − 𝛿𝑚+1) ∑ 𝑎𝑛

∞
𝑛=𝑚+1

≤ 1. 

This last inequality is equivalent to 

∑ 𝑎𝑛

𝑚

𝑛=2

+ 𝛿𝑚+1 ∑ 𝑎𝑛

∞

𝑛=𝑚+1

≤ 1.                          (4.9) 

It is suffices to show that the left hand side of (4.9) is bounded above by ∑ 𝛿𝑛
∞
𝑛=2 𝑎𝑛, which is 

equivalent to 

∑(𝛿𝑛 − 1)

𝑚

𝑛=2

𝑎𝑛 + ∑ (𝛿𝑛 − 𝛿𝑚+1)

∞

𝑛=𝑚+1

𝑎𝑛 ≥ 0. 

This completes the proof of Theorem . 

We next turn to ratios involving derivatives. 

Theorem 10.  If 𝑓 of the form (1.1) satisfies the condition  (2.1) then 

ℜ {
𝑓′(𝑧)

𝑓𝑚′(𝑧)
} ≥ 1 −

𝑚 + 1

𝛿𝑚+1
                                             (4.10) 

ℜ {
𝑓𝑚′(𝑧)

𝑓′(𝑧)
} ≥

𝛿𝑚+1

1 + 𝑚 + 𝛿𝑚+1
                                     (4.11) 

 

 where  

𝛿𝑛 ≥ {

1,  𝑖𝑓 𝑛 = 2,3 ⋯ 𝑚

𝑛 
𝛿𝑚+1

𝑚 + 1
,  𝑖𝑓𝑛 = 𝑚 + 1, 𝑚 ∗ 2, ⋯

 

and 𝛿𝑛 is defined by [4.3]. The estimates in (4.10) and (4.11) are sharp with the extremal function 

given by(4.4). 

Proof. Firstly, we will give proof of (4.10). We write 

1 + 𝑤(𝑧)

1 − 𝑤(𝑧)
= 𝛿𝑚+1 {

𝑓′(𝑧)

𝑓𝑚′(𝑧)
− (1 −

𝑚 + 1

𝛿𝑚+1
)}

= {
1 + ∑ 𝑛𝑚

𝑛=2 𝑎𝑛𝑧𝑛−1 +
𝛿𝑚+1

𝑚 + 1
∑ 𝑛∞

𝑛=𝑚+1 𝑎𝑛𝑧𝑛−1

1 + ∑ 𝑎𝑛
𝑚
𝑛=2 𝑧𝑛−1

} ,

 



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where 

𝑤(𝑧) =

𝛿𝑚+1

𝑚 + 1
∑ 𝑛∞

𝑛=𝑚+1 𝑎𝑛𝑧𝑛−1

2 + 2 ∑ 𝑛𝑚
𝑛=2 𝑎𝑛𝑧𝑛−1 +

𝛿𝑚+1

𝑚 + 1
∑ 𝑛∞

𝑛=𝑚+1 𝑎𝑛𝑧𝑛−1
 

and 

|𝑤(𝑧)| ≤

𝛿𝑚+1

𝑚 + 1
∑ 𝑛∞

𝑛=𝑚+1 𝑎𝑛

2 − 2 ∑ 𝑛𝑚
𝑛=2 𝑎𝑛 +

𝛿𝑚+1

𝑚 + 1
∑ 𝑛∞

𝑛=𝑚+1 𝑎𝑛

. 

Now |𝑤(𝑧)| ≤ 1 if and only if 

∑ 𝑛

𝑚

𝑛=2

𝑎𝑛 +
𝛿𝑚+1

𝑚 + 1
∑ 𝑛

∞

𝑛=𝑚+1

𝑎𝑛 ≤ 1,                                           (4.12) 

since the left hand side of  (4.12)is bounded above by ∑ 𝛿𝑛
∞
𝑛=2 𝑎𝑛. 

The proof of (4.11) follows the pattern of that in Theorem 9. 

This completes the proof of Theorem . 

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