Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

  

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Fekete-Szegö and Second Hankel Determinant for a Class of 𝖕-Valent 

Functions Related to Modified Sigmoid Functions 

 

Dr. Saleem Ahmed1,   Dr. M. Musthafa Ibrahim2,*, Dr. Badriya Nasser Mohammed Al Hashmi3 

1College of Engineering, University of Buraimi, saleem.a@uob.edu.om 

2,*College of Engineering, University of Buraimi, musthafa.i@uob.edu.om 

3College of Engineering, University of Buraimi, badriya.n@uob.edu.om 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: In this paper, we study the initial coefficient bounds for a noval class 𝑀𝜆(∗)(𝜂, 𝜑𝑛,𝑚) 

of 𝒫-valently analytic functions related to Sigmoid functions. Furthermore, the famous 

classical Fekete-Szegö inequality for this class are discussed.  

Conclusions: In this paper, We introduced and investigated the 𝔭-univalent function for the class 

𝑀𝜆,(∗)(𝜂, 𝜑𝑛,𝑚) related to the to modified Sigmoid functions. Thus, we obtained second, third 

and fourth Taylor–Maclaurin coefficients of functions in this class. These results were an 

improvement on the estimates obtained in the recent studies. 

2020 Mathematics Subject Classification. Primary 30C45; 30C50; 30C80; Secondary 11B65, 

47B38.  

Keywords: Analytic functions, modified Hadamard product, Sigmoid function, Fekete-Szegö 

inequality  

1. INTRODUCTION AND MOTIVATION 

Let 𝒜𝔭 denote the class of functions of the form 

 ℑ(𝑧) = 𝑧𝔭 + ∑∞
𝑘=1 𝔞𝑘+𝔭 𝑧𝑘+𝔭, (1.1) 

 which are 𝔭-valently analytic in the open unit disk: 

 𝕌 = {𝑧 ∈ ℂ: 0 ≤ |𝑧| < 1} 

The investigation of 𝔭-valently analytic functions regarding many aspects like starlikeness, 

subordination, the introduction of new subclasses are still inspiring with interesting outcomes. 

Special functions are composed of large number of highly interconnected processing elements 

(neurons) working together to solve a specific task. They play a vital role in univalent function theory. 

These functions have been overshadowed by other fields like algebra, differential equations, topology, 

functional analysis and real analysis, among others, because they work in the same way the brain does. 

An example of such functions is the activation function. Activation function increases the size of 

hypothesis space that a network can represent. The most popular activation function is the sigmoid 

function. The Sigmoid function of the form  

 ℵ(𝑧) =
1

1+𝑒−𝑧 (1.2) 

 is differentiable and has the following properties.  

mailto:saleem.a@uob.edu.om
mailto:musthafa.i@uob.edu.om


Communications on Applied Nonlinear Analysis 

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    • It outputs real numbers between 0 and 1.  

    • It maps from a very large input domain to a small range of outputs.  

    • never loses information because it is a one-to-one function.  

    • increases monotonically.  

 These properties enable us to use Sigmoid function in univalent function theory. 

We briefly recall the following definitions needed our investigation. 

Definition 1.1  ([14]) Let ℑ(𝑧) = 𝑧𝔭 + ∑∞
𝑘=1 𝔞𝑘+𝔭 𝑧𝑘+𝔭, and 𝔤(𝑧) = 𝑧𝔭 + ∑∞

𝑘=1 𝔟𝑘+𝔭 𝑧𝑘+𝔭. The 

modified Hadamard product of two functions ℑ and 𝔤 which belong to 𝒜𝔭 is defined by  

 ℑ(𝑧) = (ℑ ∗ 𝔤)(𝑧) = 𝑧𝔭 + ∑∞
𝑘=1 𝔞𝑘+𝔭𝔟𝑘+𝔭 𝑧𝑘+𝔭 (1.3) 

 Definition 1.2  ([15]) Let ℑ ∈ 𝐴. Then the 𝔮𝑡ℎ Hankel determinant of ℑ is defined for 𝔮 ≥ 1 and 𝑛 ≥

1 by  

 𝐻𝔮(𝑛) = |
|

𝔞𝑛 𝔞𝑛+1 ⋯ 𝔞𝑛+𝑞−1

𝔞𝑛+1 𝔞𝑛+2 ⋯ 𝔞𝑛+𝑞

⋮ ⋮ ⋮ ⋮
𝔞𝑛+𝑞−1 𝔞𝑛+𝑞 ⋯ 𝔞𝑛+2𝑞−2

|
| (1.4) 

Thus, the second Hankel determinant 

 𝐻2(2) = |

𝔞2 𝔞3

𝔞3 𝔞4| = 𝔞2𝔞4 − 𝔞3
2 (1.5) 

 For two analytic functions ℑ and 𝔤, the function ℑ is subordinate to 𝔤, written as follows:  

 ℑ(𝑧) ≺ 𝔤(𝑧) 

if there exists an analytic function 𝑤, with 𝑤(0) = 0 and |𝑤(𝑧)| < 1 such that ℑ(𝑧) = 𝔤(𝑤(𝑧)). In 

particular, if the function 𝔤 is univalent in 𝕌, then ℑ(𝑧) ≺ 𝔤(𝑧) is equivalent to ℑ(0) = 𝔤(0) and 

ℑ(𝑈) ⊂ 𝔤(𝑈). 

Definition 1.3  ([9]) Let 𝜂 ∈ ℂ/{0} and the class 𝑀𝜆(𝜂, 𝜑𝑛,𝑚) denote the subclass of 𝒜𝔭 consisting of 

functions ℑ of the form (1.1), and satisfying the following subordination condition  

 1 +
1

𝜂
[

𝑧ℑ′(𝑧)

ℑ(𝑧)
+ 𝜆

𝑧2ℑ′′(𝑧)

ℑ(𝑧)
− 1] ≺ 𝜑𝑛,𝑚 (1.6) 

 for 0 ≤ 𝜆 ≤ 1 and 𝜑𝑛,𝑚 is a simple logistic Sigmoid activation function.  

In this study, we solve the Fekete-Szegö problem for functions in the class 𝑀𝜆(∗)(𝜂, 𝜑𝑛,𝑚) and 

in the special instances, as well as provide bound estimates for the coefficients and an upper bound 

estimate for the second Hankel determinant. 

 



Communications on Applied Nonlinear Analysis 

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Definition 1.4  Let 𝜂 ∈ ℂ/{0} and the class 𝑀𝜆(∗)(𝜂, 𝜑𝑛,𝑚) denote the subclass of 𝒜𝔭 consisting of 

functions ℑ of the form (1.1), and satisfying the following subordination condition  

 1 +
1

𝜂
[

𝑧(ℑ∗𝔤)′(𝑧)

(ℑ∗𝔤)(𝑧)
+ 𝜆

𝑧2(ℑ∗𝔤)′′(𝑧)

(ℑ∗𝔤)(𝑧)
− 1] ≺ 𝜑𝑛,𝑚 = 1 + ∑∞

𝑚=1
(−1)𝑚

2𝑚 (∑∞
𝑛=1

(−1)𝑚

𝑛!
𝑧𝑛)

𝑚

 (1.7) 

 for 0 ≤ 𝜆 ≤ 1 and 𝜑𝑛,𝑚 is a simple logistic Sigmoid activation function.  

2.  Preliminary results 

The following results are needed for our investigation 

Let 𝑃 be the family of all functions 𝑝 analytic in 𝕌 for which ℜ{𝛼(𝑧)} > 0 and  

 𝑝(𝑧) = 1 + 𝑃1𝑧 + 𝑃2𝑧2 + ⋯,   (𝑓𝑜𝑟𝑧 ∈ 𝕌) 

Lemma 2.1 ([8]) If 𝑝 ∈ 𝑃, then |𝑃𝑘| ≤ 2 (2,3,4, ⋯ )  

Lemma 2.2 ([6]) Let 𝑔 be a Sigmoid function defined in (1.2) and  

 𝜑(𝑧) = 2𝑔(𝑧) = 1 + ∑∞
𝑚=1

(−1)𝑚

2𝑚 (∑∞
𝑛=1

(−1)𝑚

𝑛!
𝑧𝑛)

𝑚
 (2.1) 

 then 𝜑(𝑧) ∈ 𝑃, |𝑧| < 1 where 𝜑(𝑧) is a modified Sigmoid function.  

Lemma 2.3 ([6]) Let 𝑔 be a Sigmoid function defined in (1.1) and  

 𝜑𝑛,𝑚(𝑧) = 1 + ∑∞
𝑚=1

(−1)𝑚

2𝑚 (∑∞
𝑛=1

(−1)𝑚

𝑛!
𝑧𝑛)

𝑚
 (2.2) 

 then |𝜑𝑛,𝑚(𝑧)| < 2 .  

Lemma 2.4 ([6]) Let 𝜑(𝑧) ∈ 𝑃 and be starlike, then ℑ is a normalized univalent function of 

the form (1.1). Setting 𝑚 = 1, Fadipe et al. [6] remarked that  

 𝜑(𝑧) = 1 + ∑∞
𝑛=1 𝑐𝑛𝑧𝑛 (2.3) 

 where 𝑐𝑛 =
(−1)𝑛+1

2𝑛!
, then |𝑐𝑛| ≤ 2 for 𝑛 = 2,3,4, ⋯ and the result is sharp for each 𝑛.  

3. Some coefficient estimates for the class of 𝑴𝝀,(∗)(𝜼, 𝝋𝒏,𝒎) 

 In this section, we will find the estimates on the coefficients 𝔞𝔭+1𝔟𝔭+1, 𝔞𝔭+2𝔟𝔭+2 and 𝔞𝔭+3𝔟𝔭+3 

for functions in the class 𝑀𝜆,(∗)(𝜂, 𝜑𝑛,𝑚).  

Theorem 3.1  Let 

 𝜑𝑛,𝑚(𝑧) = 1 + ∑∞
𝑚=1

(−1)𝑚

2𝑚 (∑∞
𝑛=1

(−1)𝑚

𝑛!
𝑧𝑛)

𝑚

 

where 𝜑𝑛,𝑚(𝑧) ∈ 𝐴 is a modified logistic Sigmoid activation function and 𝜑𝑛,𝑚
′ (0) > 0. If 𝐹(𝑧) =

(ℑ ∗ 𝔤)(𝑧) given by (1.1) belongs to the class 𝑀𝜆,(∗)(𝜂, 𝜑𝑛,𝑚) then,  

 𝔞𝔭+1𝔟𝔭+1 =
𝜂

2𝔭(1+𝜆(𝔭+1)) (3.1) 

  



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 𝔞𝔭+2𝔟𝔭+2 =
𝜂2

4𝔭(𝔭+1)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))
 (3.2) 

  

 𝔞𝔭+3𝔟𝔭+3 =
𝜂(3𝜂2−𝔭(𝔭+1)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2)))

24𝔭(𝔭+1)(𝔭+2)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))(1+𝜆(𝔭+3))
 (3.3) 

 Proof. Let ℑ(𝑧) = 𝑧𝔭 + ∑∞
𝑘=1 𝔞𝑘+𝔭 𝑧𝑘+𝔭, and 𝔤(𝑧) = 𝑧𝔭 + ∑∞

𝑘=1 𝔟𝑘+𝔭 𝑧𝑘+𝔭. Then we can write the 

following equalities:  

ℑ(𝑧) = (ℑ ∗ 𝔤)(𝑧) = 𝑧𝔭 + ∑∞
𝑘=1 𝔞𝑘+𝔭𝔟𝑘+𝔭 𝑧𝑘+𝔭 ⇒ (ℑ ∗ 𝔤)′(𝑧) = 𝔭𝑧𝑝−1 + ∑∞

𝑘=1 (𝑘 +

𝔭)𝔞𝑘+𝔭𝔟𝑘+𝔭 𝑧𝑘+𝔭−1 

 ⇒ (ℑ ∗ 𝔤)′′(𝑧) = 𝔭(𝔭 − 1)𝑧𝑝−2 + ∑∞
𝑘=1 (𝑘 + 𝔭)(𝑘 + 𝔭 − 1)𝔞𝑘+𝔭𝔟𝑘+𝔭 𝑧𝑘+𝔭−2 

Thus, we obtain 

 𝑧(ℑ ∗ 𝔤)′(𝑧) + 𝜆𝑧2(ℑ ∗ 𝔤)′′(𝑧) = 𝔭(1 − 𝜆 + 𝜆𝔭)𝑧𝔭 + ∑∞
𝑘=1 (𝑘 + 𝔭)(1 + (𝑘 + 𝔭 −

1)𝜆)𝔞𝑘+𝔭𝔟𝑘+𝔭 𝑧𝑘+𝔭 

and  

𝑧(ℑ ∗ 𝔤)′(𝑧) + 𝜆𝑧2(ℑ ∗ 𝔤)′′(𝑧) − (ℑ ∗ 𝔤)(𝑧) = (𝔭 − 1)(1 + 𝜆𝔭)𝑧𝔭 + ∑∞
𝑘=1 (𝑘 + 𝔭 − 1)(1 +

(𝑘 + 𝔭)𝜆)𝔞𝑘+𝔭𝔟𝑘+𝔭 𝑧𝑘+𝔭 

If 𝐹 ∈ 𝑀𝜆(∗)(𝜂, 𝜑𝑛,𝑚), then we have  

 
1

𝜂
[

𝑧(ℑ∗𝔤)′(𝑧)+𝜆𝑧2(ℑ∗𝔤)′′(𝑧)−(ℑ∗𝔤)(𝑧)

(ℑ∗𝔤)(𝑧)
] = 𝜑𝑛,𝑚 − 1 (3.4) 

 where 𝜑𝑛,𝑚 is a modified Sigmoid function given by  

 𝜑𝑛,𝑚 = 1 +
1

2
𝑧 −

1

24
𝑧3 +

1

240
𝑧5 −

17

40320
𝑧7 + ⋯ (3.5) 

 In view of (3.4) and (3.5), expanding in series forms we have  

 

1

𝜂
[(𝔭 − 1)(1 + 𝜆𝔭)𝑧𝔭 + ∑∞

𝑘=1 (𝑘 + 𝔭 − 1)(1 + (𝑘 + 𝔭)𝜆)𝔞𝑘+𝔭𝔟𝑘+𝔭 𝑧𝑘+𝔭] =

[𝑧𝔭 + ∑∞
𝑘=1 𝔞𝑘+𝔭𝔟𝑘+𝔭 𝑧𝑘+𝔭] [

1

2
𝑧 −

1

24
𝑧3 +

1

240
𝑧5 −

17

40320
𝑧7 + ⋯ ]

 (3.6) 

 Comparing the coefficients of 𝑧𝔭+1, 𝑧𝔭+2 and 𝑧𝔭+3 in(3.6), we obtain  

 𝔞𝔭+1𝔟𝔭+1 =
𝜂

2𝔭(1+𝜆(𝔭+1)) (3.7)  

 𝔞𝔭+2𝔟𝔭+2 =
𝜂2

4𝔭(𝔭+1)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))
 (3.8)  

 𝔞𝔭+3𝔟𝔭+3 =
𝜂(3𝜂2−𝑝(𝔭+1)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2)))

24𝑝(𝔭+1)(𝔭+2)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))(1+𝜆(𝔭+3))
 (3.9) 

  

 



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Corollary 3.2  For coefficient 𝔞𝔭+1𝔟𝔭+1,  

 |𝔞𝔭+1𝔟𝔭+1| =
|𝜂|

2𝔭(1+𝜆(𝔭+1))
 

is written and since 𝜑(𝜆) =
1

(1+𝜆(𝔭+1))
, 𝜑′(𝜆) < 0 in the interval 0 ≤ 𝜆 ≤ 1 and 𝜑(𝜆) is decreasing, it 

will be  

 
|𝜂|

2𝔭(𝔭+2)
≤ |𝔞𝔭+1𝔟𝔭+1| ≤

|𝜂|

2𝔭
 (3.10) 

 for 
1

2
≤

1

(1+𝜆(𝔭+1))
≤ 1. 

Similarly, since the coefficients 𝔞𝔭+1𝔟𝔭+1, 𝔞𝔭+2𝔟𝔭+2 and 𝔞𝔭+3𝔟𝔭+3 depend on 𝜆 and are 

decreasing with respect to 𝜆, the following inequalities can be written easily: 

 
|𝜂2|

4(𝔭+1)(𝔭+2)(𝔭+3)
≤ |𝔞𝔭+2𝔟𝔭+2| ≤

|𝜂|2

4𝑝(𝔭+1)
 (3.11)  

 
|(3𝜂3−𝑝(𝔭+1)(𝔭+2)(𝔭+3)𝜂)|

24(𝔭+1)(𝔭+2)2(𝔭+3)(𝔭+4)
≤ |𝔞𝔭+3𝔟𝔭+3| ≤

|(𝜂3−𝑝(𝔭+1)𝜂)|

24𝑝(𝔭+1)(𝔭+2)
 (3.12)  

  

4.  Some results connected with the Fekete-Szegö inequality and Hankel coefficient for the class 

of 𝑴𝝀,(∗)(𝜼, 𝝋𝒏,𝒎) 

The Fekete-Szegö problem may be considered one of the most important results about 

univalent functions, which is related to coefficients an of a function’s Taylor series and was introduced 

by Fekete-Szegö [1]. The problem of maximizing the absolute value of functional 𝑎3 − 𝜇𝑎2
2 is called 

the Fekete-Szegö problem. This result is sharp and is studied thoroughly by many researchers. The 

equality holds true for the Koebe function. In 1969, Keogh and Merkes [2] obtained the sharp upper 

bound of the Fekete-Szegö functional |𝑎3 − 𝜇𝑎2
2| for some subclasses of univalent function.  

Recently, Murugusundarmoorthy and Janani [3], Olantunji et al. [5], Olantunji [4], and Orhan 

and Çağlar [7]have studied Sigmoid function for various classes of analytic and univalent functions. 

In this section, we first prove the following Fekete-Szegö result for the function in the classes 

𝑀𝜆,(∗)(𝜂, 𝜑𝑛,𝑚) with the values of 𝔞𝔭+1𝔟𝔭+1 and 𝔞𝔭+2𝔟𝔭+2. 

Theorem 4.1  If 𝐹(𝑧) ∈ 𝒜𝒫 given by (1.1) belongs to the class 𝑀𝜆,(∗)(𝜂, 𝜑𝑛,𝑚) then,  

 |𝔞𝔭+2𝔟𝔭+2 − 𝜇(𝔞𝔭+1𝔟𝔭+1)
2

| =
|𝜂|2

4𝔭(𝔭+1)
(1 + |𝜇|

(𝔭+1)

𝔭
) (4.1) 

Proof. If the values of 𝔞𝔭+1𝔟𝔭+1 and 𝔞𝔭+2𝔟𝔭+2 determined by (3.7) and (3.8) are written instead of 

𝔞𝔭+2𝔟𝔭+2 − 𝜇(𝔞𝔭+1𝔟𝔭+1)
2
, we get  

 𝔞𝔭+2𝔟𝔭+2 − 𝜇(𝔞𝔭+1𝔟𝔭+1)
2

=
𝜂2

4𝔭(𝔭+1)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))
− 𝜇 (

𝜂

2𝔭(1+𝜆(𝔭+1))
)

2

 

                                     =
𝜂2

4𝔭(𝔭+1)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))
− 𝜇

(𝜂)2

4𝑝2(1+𝜆(𝔭+1))2. 



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Taking absolute value on both sides of the above equation and applying triangle inequality, we get  

 |𝔞𝔭+2𝔟𝔭+2 − 𝜇(𝔞𝔭+1𝔟𝔭+1)
2

| ≤
|𝜂|2

4𝑝(𝔭+1)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))
+ |𝜇|

|(𝜂)|2

4𝔭2(1+𝜆(𝔭+1))2. 

Here 𝜁1 =
1

(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))
 and 𝜁2 =

1

(1+𝜆(𝔭+1))2 are taken and these functions depending on 𝜆 are 

considered to be decreasing in the interval 0 ≤ 𝜆 ≤ 1, since  

 max
0≤𝜆≤1

1

(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))
  = 1 

and  

 max
0≤𝜆≤1

1

(1+𝜆(𝔭+1))2   = 1 

we get  

 |𝔞𝔭+2𝔟𝔭+2 − 𝜇(𝔞𝔭+1𝔟𝔭+1)
2

| ≤
|𝜂|2

4𝔭(𝔭+1)
+ |𝜇|

|𝜂|2

4𝑝2. 

thus we obtain  

 |𝔞𝔭+2𝔟𝔭+2 − 𝜇(𝔞𝔭+1𝔟𝔭+1)
2

| ≤
|𝜂|2

4𝔭(𝔭+1)
(1 + |𝜇|

(𝔭+1)

𝑝
) 

Hence, we have reached the desired assertion of the Theorem(4.1),  

 |𝔞𝔭+2𝔟𝔭+2 − 𝜇(𝔞𝔭+1𝔟𝔭+1)
2

| ≤ {

|𝜂|2

4𝔭(𝔭+1)
(1 + |𝜇|

(𝔭+1)

𝑝
) ,        𝜇 ≥ 0

|𝜂|2

4𝔭(𝔭+1)
(1 − |𝜇|

(𝔭+1)

𝑝
) ,            𝜇 ≤ 0

   

This completes the proof of the Theorem.  

In the theory of singularities [10] and the investigation of power series with integral 

coefficients, the Hankel determinant is very important. The reader is encouraged to read [15] for more 

information. For several subfamilies of univalent functions, the growth of 𝐻𝑞(𝑛) has been explored. 

We know that the function 𝐻2(1) = 𝑎3 − 𝑎2
2 for 𝑞 = 2 and 𝑛 = 1 is a well recognized Fekete-Szegö 

functional. For the bi-convex and bi-starlike classes, the second Hankel determinant 𝐻2(2) is given by 

𝐻2(2) = 𝑎2𝑎4 − 𝑎3
2 [12].  

The following theorem will give some results related to Hankel determinant for the functions 

belonging to classes 𝑀𝜆,(∗)(𝜂, 𝜑𝑛,𝑚).  

Theorem 4.2  If 𝐹(𝑧) ∈ 𝒜𝒫 given by (1.1) belongs to the class 𝑀𝜆,(∗)(𝜂, 𝜑𝑛,𝑚) then,  

|(𝔞𝔭+1𝔟𝔭+1)(𝔞𝔭+3𝔟𝔭+3) − (𝔞𝔭+2𝔟𝔭+2)
2

| ≤
|𝜂|2

48𝔭2(𝔭+1)2(𝔭+2)
((𝔭 + 1)|3𝜂2 − 𝑝(𝔭 + 1)| + 3(𝔭 +

2)|𝜂|2) (4.2) 

 Proof. From (3.7), (3.8)and (3.9) , we get 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

  

2656 
https://internationalpubls.com 

 𝐻2(2) = |

𝔞2 𝔞3

𝔞3 𝔞4| = 𝔞2𝔞4 − 𝔞3
2 (4.3) 

 

(𝔞𝔭+1𝔟𝔭+1)(𝔞𝔭+3𝔟𝔭+3) − (𝔞𝔭+2𝔟𝔭+2)
2

=                                        

(
𝜂

2𝔭(1+𝜆(𝔭+1))
) (

𝜂(3𝜂2−𝔭(𝔭+1)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2)))

24𝔭(𝔭+1)(𝔭+2)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))(1+𝜆(𝔭+3))
)

− (
𝜂2

4𝔭(𝔭+1)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))
)

2

 (4.4) 

 
=

(3𝜂4−𝔭(𝔭+1)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))𝜂2)

48𝔭2(𝔭+1)(𝔭+2)(1+𝜆(𝔭+1))2(1+𝜆(𝔭+2))(1+𝜆(𝔭+3))
−

𝜂4

16𝔭2(𝔭+1)2(1+𝜆(𝔭+1))2(1+𝜆(𝔭+2))2

 (4.5) 

and thus  

 

|(𝔞𝔭+1𝔟𝔭+1)(𝔞𝔭+3𝔟𝔭+3) − (𝔞𝔭+2𝔟𝔭+2)
2

| ≤                                

|(3𝜂4−𝑝(𝔭+1)(1+𝜆(𝔭+1))(1+𝜆(𝔭+2))𝜂2)|

48𝔭2(𝔭+1)(𝔭+2)(1+𝜆(𝔭+1))2(1+𝜆(𝔭+2))(1+𝜆(𝔭+3))

+
|𝜂|4

16𝔭2(𝔭+1)2(1+𝜆(𝔭+1))2(1+𝜆(𝔭+2))2

 (4.6) 

Here 𝜁3 =
1

(1+𝜆(𝔭+1))2(1+𝜆(𝔭+2))(1+𝜆(𝔭+3))
 , 𝜁4 =

1

(1+𝜆(𝔭+1))(1+𝜆(𝔭+3))
 and 𝜁5 =

1

(1+𝜆(𝔭+1))2(1+𝜆(𝔭+2))2 are taken and these functions depending on 𝜆 are considered to be decreasing in 

the interval 0 ≤ 𝜆 ≤ 1, since  

 max
0≤𝜆≤1

1

(1+𝜆(𝔭+1))2(1+𝜆(𝔭+2))(1+𝜆(𝔭+3))
  = 1,  

 max
0≤𝜆≤1

1

(1+𝜆(𝔭+1))(1+𝜆(𝔭+3))
  = 1 

and  max
0≤𝜆≤1

1

(1+𝜆(𝔭+1))2(1+𝜆(𝔭+2))2   = 1 

thus we obtain  

|(𝔞𝔭+1𝔟𝔭+1)(𝔞𝔭+3𝔟𝔭+3) − (𝔞𝔭+2𝔟𝔭+2)
2

| ≤
|𝜂|2

48𝔭2(𝔭+1)(𝔭+2)
((𝔭 + 1)|3𝜂2 − 𝔭(𝔭 + 1)| + 3(𝔭 +

2)|𝜂|2)  (4.7) 

 

This completes the proof of the Theorem.  

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https://internationalpubls.com 

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