COEFFICIENT ESTIMATES OF A NEW SPECIAL SUBCATEGORY OF BI-UNIVALENT FUNCTIONS

 

Kosrat O. Mohammed¹ , Khalid I. Abdullah^1,*, Nafya H. Mohammed¹, Abubakr M. Adarbar¹, Hedayat M. Sharifi¹

 

¹College of Basic Education, University of Raparin, Rania, Kurdistan Region-Iraq

Corresponding author Email: kosrat.husman@uor.edu.krd,

 

ABSTRACT:                                                                                                                                

In this article, based on the concepts of subordination and q-derivative operator, we introduce and study a new subcategory  of analytic and bi-univalent functions in the open unit disk D. Upper bounds for the second and third coefficients of the functions belonging to the subcategory  are found and several particular outcomes of the main finding are also presented.

1.     introduction

        Suppose that  is the set of all complex numbers in the plane and represents the open unit disk in , we signify the class of all functions that are analytic in  and normalized by . Then every function  in  has power series representation:

An analytic function  is called univalent in  if it is injective in . The Koebe function is an example of univalent                   functions (see (Duren, 1983)). For more than a century, the theory of univalent functions is a very active field of study. A significant portion of its history is connected to the well-known Bieberbach conjecture that  for . This well-known conjecture from 1916 rose to prominence as one of mathematics' most well-known problems.

        Now, we denote the class of all functions in  that they are univalent in  by  (Srivastava & Owa, 1992; Ma & Minda, 1992).  Let the functions and  be analytic in , then we say that the function  is subordinate to the function  and we write  if there is an analytic function in  such that where   and  ( is a Schwarz function). Specifically, if  is univalent in  then the following equivalence relationship is valid

        Using the subordination concept, Ma and Minda (1992) introduced the subcategories of starlike and convex functions. Here, we assume a function  has a positive real part in ,  is symmetric about the real axis, ,  and using the power series representation

…; ).         (1.2)

The Ma-and-Minda subcategories of functions are introduced as follows:

and

        Here, we need to recall the Koebe one-quarter theorem that stated by Duren (2001):

The range of every function of the class  contains the disk .

        We note that the Koebe one-quarter theorem sensures that the image of  under any function  contains a disk with the center at the origin and the radius  . Thus, every univalent function  has an inverse , such that

and

        where the radius  depends on the function . Note that the inverse function  is defined by

                              (1.3)

        We say that the function  is bi-univalent in  if  and  are univalent functions in  and we denote by   the class of all bi-univalent functions in  given by (1.1). Some examples of bi-univalent functions in   are:

 , ,   ,

(see Alrefai & Ali, 2020). However, ∑ does not contain the renowned Koebe function. Additionally, several functions that belong to the class S, like  and   are not in the class ∑.

        A long time ago, scientists were tried to estimate the coefficients of the power series of some classes of bi-univalent functions. For example, it has been proved by Lewin (1967) that  and then conjectured by Brannan and Clunie (1980) that . Furthermore, Netanyahu (1969) proved that . All that we have mentioned above is related to the geometric properties of analytic functions. In this last decade, some efforts have been made in this field, some of which are mentioned here.

        Subclasses of analytic functions have been investigated from different perspectives. For example, Alimohammadi et al. (2020) investigated the strong starlikeness properties as well as the close-to-convexity properties for a new subcategory  which is a subclass of analytic functions. Also, we mention a work of Mohammed et al. (2022) who introduced a subcategory of normalized analytic functions that is defined using a differential inequality and they studied several geometric properties of it.

        Lately, the q-calculus (quantum calculus) has become crucial in univalent functions theory, especially for estimating sharp inequality bounds for different subcategory of univalent and bi-univalent functions. The idea of ​​this merger is inspired by Jackson's works. Jackson (1909, 1910) introduced and studied q-derivative operator  of a function  as follows:

Assume that 

and  if  exists.

For any function , the simple computation implies

where, 

We observe that

 

                                               (1.4)

        The first one that used this idea in relation with univalent functions was Srivastava (1989). (see also Seoudy, 2014; Toklu, 2019).

        However, the problem of estimating the coefficients for every Taylor-Maclaurin series coefficients   is still an open problem.

In this paper, the authors presented a new subclass of the function class Σ based on q-derivative operator (Jackson-derivative operator)  for functions   in this new subclass and estimated the upper bounds for the coefficients and , using the techniques previously used by Frasin and Aouf (2011) and Saravanan and Muthunagai (2019) (see also Mohammed, 2021; Abdullah, 2022).

        Now, we introduce the category as follows:

Definition 1.1 A function , as defined by (1.1), belongs to the class  if it satisfies the following two conditions:  

   

and

   

where  and   is the function given by (1.2). The main result can be demonstrated using the following lemma.

Lemma 1.2. (Duren, 2001) If  , where  represents the class of all functions that are analytic in , with

 

(1.5)

then  for every .  

2.     main Results

        This section presents some interesting estimate coefficients for the functions in the mentioned subcategory of . Now let us to explain our main result.

Theorem 2.1. Let  be a function in the class , then

also,

Proof. Suppose that the function  and  in this case there are two Schwarz functions , such that  

	,			(2.1)
and
			(2.2)

 

Now, we define two auxiliary functions  and  by

and

In other words, we have

(2.3)

and

(2.4)

Then the two functions  and  are analytic in ,  Given that , the real parts of  and  are nonnegative in  and by using lemma (1.2),  and  for every . Recall that

so, we have

    

However,

The equations (2.1), (2.3), (2.5), and (2.6) can be equated to obtain:

By comparing the coefficients in the above equation, we get: 

,

(2.7)

and 

(2.8)

Once more, given

 

we have

Once more, since 

The equations (2.2), (2.4), (2.9) and (2.10) can be equated to obtain:

                          

Hence, by comparing the coefficients in the above equation, we get: 

and, also

By comparing the equations (2.7) and (2.11) we have

also

Now, apply the equations (2.8) and (2.13) in (2.12) to obtain 

Given that ,  (2.14) yields

and we obtain

Alternatively, obtain that

It is easy to conclude that

Using Lemma (1.2), we have   and  and after some e computation this implies that

Finally, we get

The next step is to find an upper bound for and for this purpose we must subtract the equation (2.12) from the equation (2.8) to obtain

Alternatively, we conclude that

(2.16)

Substituting the equation (2.15) in (2.16), we obtain

It is easy to conclude that

Again, by using Lemma (1.2)   and  and this implies that

Finally, we have

        Now, we shall give upper bounds concerning the initial two coefficients of the function . Since  (by (1.3)), the upper bound that is obtained for  also holds for . Additionally, in order to obtain the upper bound for  we must perform some calculations based on the equation  that we will explain it in the following corollary.

Corollary 2.2. If the function  is in the category , then 

Proof. Using the equation (2.15) and the equation (2.16) in the proof of the Theorem 2.1, we have

Then, by using Lemma (1.2) and triangle inequality, we conclude that

At this step, we want to highlight some interesting findings for some special cases of  in Theorem 2.1. We consider the Taylor-Maclaurin expansion of

and then we obtain a special case of Theorem 2.1 by assuming . In this case ,    and we acquire the following result.

Corollary 2.3 If the function  is in the category , then simple computations yield

Also,

 Now, by taking

we have  and we obtain the next result.

Corollary 2.4. If the function  is in the category  such that  then simple computations yield

 and

Finally, if we take

        

we get   and we acquire the following result.

Corollary 2.5. If the function  is in the category  such that , then simple computations yield

 and

 

 

Acknowledgements

        The efforts and acknowledgements of the reviewers for improving this article are entirely appreciated by the authors.

References

Alimohammadi, D., Cho, N. E., Adegani, E. A., & Motamednezhad, A. (2020). Argument and coefficient estimates for certain analytic functions. Mathematics, 8(1), 88. https://doi.org/10.3390/math8010088.

Alrefai, O., & Ali, M. (2020). General coefficient estimates for bi-univalent functions: a new approach. Turkish Journal of Mathematics, 44(1), 240-251. 10.3906/mat-1910-100.

Brannan, D. A., & Clunie, J. (1980). Aspects of contemporary complex analysis. (No Title).

Duren, P. L. (1983). Grundlehren der Mathematischen Wissenchaffen. Univalent Functions; Springer: New York, NY, USA; Berlin/Heidelberg, Germany, 259.

Duren, P. L. (2001). Univalent functions (Vol. 259). Springer Science & Business Media.

Frasin, B. A., & Aouf, M. K. (2011). New subclasses of bi-univalent functions. Applied Mathematics Letters, 24(9), 1569-1573. https://doi.org/10.1016/j.aml.2011.03.048.

Jackson, F. H. (1909). XI.—On q-functions and a certain difference operator. Earth and Environmental Science Transactions of the Royal Society of Edinburgh, 46(2), 253-281. https://doi.org/10.1017/S0080456800002751.

Jackson, F. H. (1910). On q-definite integrals. Quart. J. Pure Appl. Math, 41(15), 193-203.

Lewin, M. (1967). On a coefficient problem for bi-univalent functions. Proceedings of the American mathematical society, 18(1), 63-68. https://doi.org/10.1090/S0002-9939-1967-0206255-1.

Ma, W. (1992). A unified treatment of some special classes of univalent functions. In Proceedings of the Conference on Complex Analysis, 1992. International Press Inc.

Hameed Mohammed, N. H. (2021). Coefficient Bounds for a New Class of Bi-Univalent Functions Associated with Subordination. Mathematical Analysis and Convex Optimization, 2(2), 73-82. 10.52547/maco.2.2.8.

Mohammed, N. H. (2022). Bounds for the coefficients of two new subclasses of bi-univalent functions. Science Journal of University of Zakho, 10(2), 66-69. https://doi.org/10.25271/sjuoz.2022.10.2.922.

Mohammed, N. H., Adegani, E. A., Bulboacă, T. E. O. D. O. R., & Cho, N. E. (2022). A family of holomorphic functions defined by differential inequality. Math. Inequal. Appl, 25, 27-39. http://dx.doi.org/10.7153/mia-2022-25-03.

Netanyahu, E. (1969). The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in. Archive for rational mechanics and analysis, 32(2), 100-112. 10.1007/BF00247676.

Saravanan, G., & Muthunagai, K. (2019). Coefficient bounds for a new subclass of bi-univalent functions defined by q-fractional derivative operator. Recent Developments in Mathematical Analysis and Computing, 2095(1), 030023. 10.1063/1.5097534.

Seoudy, T. M., & Aouf, M. K. (2014). Convolution Properties for Certain Classes of Analytic Functions Defined by q‐Derivative Operator. In Abstract and Applied Analysis (Vol. 2014, No. 1, p. 846719). Hindawi Publishing Corporation. https://doi.org/10.1155/2014/846719.

Srivastava, H. M. (1989). Univalent functions, fractional calculus, and associated generalized hypergeometric functions. Univalent Functions, Fractional Calculus, and Their Applications; Srivastava, HM, Owa, S., Eds, 329-354.

Srivastava, H. M., & Owa, S. (1992). Current topics in analytic function theory. World Scientific.

Toklu, E. (2019). A new subclass of bi-univalent functions defined by q-derivative. TWMS Journal of Applied and  Engineering Mathematics, 9(1), 84-90.