Using a general Hurwitz-Lerch Zeta for BI-Univalent Analytic Functions to Estimate a Second Hankel Determinant

Using a General Hurwitz-Lerch Zeta for BI-Univalent Analytic Functions to Estimate a Second Hankel Determinant

Alaa Ali Aljamie¹, Nagat Muftah Alabbar^2*

¹Mathematics Department, Faculty of Science, University of Derna, Derna, Libya.

²Mathematics Department, Faculty of Education of Benghazi, University of Benghazi.Libya.

^*Corresponding author

DOI: https://doi.org/10.51244/IJRSI.2025.120500143

Received: 07 May 2025; Accepted: 13 May 2025; Published: 17 June 2025

ABSTRACT

In this paper, we introduce and investigate a new class of bi- univalent functions defined in the open unit disk  involving a general integral operator associated with the general Hurwitz- Lerch Zeta function denoted by  . The main result of the investigation is to estimate the upper bounds for the initial Taylor–Maclaurin coefficients of functions  and   for this class. Following, we find the second Hankel determinant. Several new results are shown after specializing the parameters employed in our main results.

Keywords- Hankel determinant, Bi-univalent functions, coefficient bounds, Hurwitz -Lerch zeta function.  

INTRODUCTION

Let  denote the class of all analytic functions in the open unit disk and normalized by the conditions  and given by the power series 

 (1.1)

Let  denote the subclass of A consisting of univalent functions. The well-known Koebe one-quarter theorem (see [1]) every univalent function   contains a disk of radius , the inverse of   is a univalent analytic function on the disk  Therefore, for each function , there is an inverse function  of   defined by

  

where

If and  are univalent function in U, then we say the function  is bi-univalent function in U.

The class of bi-univalent function in  given by (1.1) denoted by Σ. The following are some important examples of bi-univalent functions in U

Σ

Lewin [8] investigated in 1967 and showed a bound of the coefficient on the class  of bi-univalent functions and estimated   Following, Brannan and Clunie [9] showed the result of Lewin and established that  Afterwards, Netanyahu [10] showed that  Then many authors approximate the Taylor-Maclaurin coefficient  and defined new subclasses of bi-univalent analytic functions unit disk see (e.g. [11]- [13]). However, the problem to estimate the coefficients of  still an open.  In current paper, employing the techniques of Srivastava et al [13] which have brought back interest in the study of analytic and bi-univalent functions, we introduce a new class and estimates of the coefficients  and , although we estimate a second Hankel determinant for a class  .                                                                                             

In 1976 Noonan and Thomas [2] defined the Hankel determinant of a function   for  and  

 

 The determinant has been extensively studied with  referring to the second Hankel determinant which is defined by  It has also been investigated by several authors (e.g. [3]-[7]). 

Definition.1.1 [14] A general Hurwitz–Lerch Zeta function defined by

where  when  ,  and when () 

 Nagat and Darus [17], introduced the generalized integral operator associated with the general Hurwitz- Lerch Zeta function, denoted by  for 𝑓 ∈ A as follows:

For    the generalized integral operator  is defined by

 

 

where 

.

Many other works on analytic and univalent functions related to this operator can be see (e.g. [15],[17], [18]).

By using a generalized integral operator, a new class of bi-univalent functions are considered as the following. 

Definition.1.1: For   and  a function   and of the form (1.1) is said to be in the class  if the following conditions are satisfied:

 

and

 

where  .

It is of interest to note that by taking  and  in Definition 1.1, we state the following class  due to Frasin et al. [19] in the next remark.

Remark 1: A function   and of the form (1.1) is said to be in the class   if the following conditions are satisfied:

 

   and                     

 

where  . 

It is of interest to note that by taking  , and  in Definition1.1, we state the following class  due to Srivastava et al. [20] in the next remark.

Remark 2: A function   and of the form (1.1) is said to be in the class  if the following conditions are satisfied:

and 

    

where . 

PRELIMINARY RESULTS

In order to derive our main results, we have to recall here the following lemmas

Lemma 2.1.[1] Let  be the class of all analytic functions  of the form 

  

  satisfying and Then 

Lemma 2.2. [21] if the function  is given by the series 

                                                                                             
 

for some with  and 

Lemma 2.3. [22] The power series for  given in (2.1) converges in U to a function in  if and only if the Toeplitz determinants

   

And  and all non-negative. They are strictly positive except for

 and  for  in this case  for  and  for 

MAIN RESULTS

Theorem 3.1 Let   be an analytic and bi-univalent function given by (1.1) be in the class , Then          

  

 

Proof: Since  there exists two analytic functions  and :  with  satisfying the following conditions.

and 

 

Define the functions  and  by

  (3.3)

And

 

Applying (3.3) and (3.4) in (3.1) and (3.2), respectively

 

 

and 

 

 

 Now, by comparing the coefficients in (3.5), we see that:

 

 

 

And by comparing the coefficients in (3.6), we see that:

 

 

 

From (3.7) and (3.10), gives 

and

 

Also, from (3.8) and (3.11), we get:

 

Thus, we have

 

 

Subtracting (3.8), (3.11), we get

 

Upon substituting the value of  from (3.14), we obtain 

 

Applying lemma 2.1

As applications of Theorem3.1 about upper bounds for coefficients and for the analytic and bi-univalent functions in the new class , we obtain and improve the known results by [19] in the following corollaries by setting the particular values of the parameters  ,  

 Corollary 1. ([19]) Let   be an analytic and bi-univalent function given by (1.1) be in the class  , , by taking  ,   and  Then the upper bounds for two initial coefficients are

 

 

Corollary 2. [20] Let   be an analytic and bi-univalent function given by (1.1) be in the class  , by taking  ,    and  Then the upper bounds for two initial coefficients are

 

 

Theorem 3.2 Let   be an analytic and bi-univalent function given by (1.1) be in the class, ,  Then the upper bound for the Second Hankel is

 

 

Proof: since  and from (3.13) in Theorem3.1, we get: 

From (3.8) and (3.11) and using (3.13), we get:

Then 

 

  

 

Also, from (3.9) and (3.12), and using (3.13), we get:

 

 

 

 

 

From (3.13), (3.15) and (3.16), we stabilize that

  

According to lemma2.2, we have

and 

then, 

and further 

 

 

 

Now letting where  with  we obtain:

 

Differentiating ,we get:

 

By using elementary calculus, one can show that   for  hence  is an increasing function and thus, the upper bound for  corresponds to  ,in which case 

 

Assuming  has a maximum value in an interior of by elementary calculations, we find 

Then implies that real’s critical point   . 

Now we will find the value of ,

 

We came to the following conclusions after some calculations:

Case (1): when  ,we observe that, is  is out of the interval , therefore the maximum value of  occurs at  or   which contradiction our assumption of having the maximum value at the interior point of  . Since  is an increasing function in the interval , maximum point of  must be on the boundary of  that is.

Thus, we have:

 

Case (2): when  ,we observe that that is is an interior of the interval  so the maximum value  occurs at  .Thus we have 

         

 

 

This completes the proof of Theorem 3.2. 

In particular, Theorem 3.2 gives the following corollaries

By preferring ,   and  in Theorem3.2 

Corollary 1. ([23]) Let   be an analytic and bi-univalent function given by (1.1) be in the class  , , by taking  ,   and  Then the upper bound for the Second Hankel is

 

Corollary 2. ([24]) Let   be an analytic and bi-univalent function given by (1.1) be in the class  , , by taking  ,,   and  Then the upper bound for the Second Hankel is

 

CONCLUSION 

In this study, A new class of bi-univalent functions in the open unit disk has been introduced and defined via a general integral operator. Derived estimates for the initial coefficients of functions and further obtained an upper bound for the second Hankel determinant for this class. Several existing and new results may be identified as special cases of our main result. By varying the parameters involved, these results contribute to the further development of the theory of bi-univalent functions and open new avenues for future studies of other special functions operators. Many research papers have been utilized to investigate various problems related to this area, can be see (e.g.  [25-26])., Journal of Humanitarian and Applied Sciences, 8,496-504.

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