Kyungpook Mathematical Journal 2022; 62(2): 257-269
Published online June 30, 2022
Copyright © Kyungpook Mathematical Journal.
Coefficient Estimates for a Subclass of Bi-univalent Functions Associated with Symmetric q-derivative Operator by Means of the Gegenbauer Polynomials
Ala Amourah, Basem Aref Frasin*, Tariq Al-Hawary
Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid, Jordan
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Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan
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Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan
e-mail : email@example.com
Received: July 3, 2021; Revised: November 5, 2021; Accepted: November 16, 2021
In the present paper, a subclass of analytic and bi-univalent functions is defined using a symmetric
Keywords: Gegenbauer polynomials, bi-univalent functions, analytic functions, Fekete-Szegö, problem
1. Definitions and Preliminaries
Let the functions
If the function
It is well known that every function
A function is said to be bi-univalent in
However, the familiar Koebe function is not a member of
are also not members of
The coefficient estimate problem for each of the Taylor-Maclaurin coefficients
Similar to the familiar subclasses
Orthogonal polynomials have been studied extensively as early as they were discovered by Legendre in 1784 . In mathematical treatment of model problems, orthogonal polynomials arise often to find solutions of ordinary differential equations under certain conditions imposed by the model.
The importance of the orthogonal polynomials for the contemporary mathematics, as well as for wide range of their applications in the physics and engineering, is beyond any doubt. It is well-known that these polynomials play an essential role in problems of the approximation theory. They occur in the theory of differential and integral equations as well as in the mathematical statistics. Their applications in the quantum mechanics, scattering theory, automatic control, signal analysis and axially symmetric potential theory are also known [7, 11].
A special case of orthogonal polynomials are Gegenbauer polynomials. They are representatively related with typically real functions
Typically real functions play an important role in the geometric function theory because of the relation
Very recently, Amourah et al.  considered the Gegenbauer polynomials
with the initial values
Special cases of Gegenbauerpolynomials
The theory of
For the convenience, we provide some basic definitions and concept details of
Definition 1.1. () For
From (1.8), we have
is sometimes called
For a function
From (1.11), we deduce that
Clearly, we have the following relations
Recently, many researchers have been exploring bi-univalent functions associated with orthogonal polynomials, few to mention (see,, ). For Gegenbauer polynomial, as far as we know, there is little work associated with bi-univalent functions in the literatures. Inspired by the works of Amourah et al. , we introduce the following new subclasses of bi-univalent functions, as follows:
Definition 1.3. Let α is a nonzero real constant. A function
We note that
Definition 1.4. Let α is a nonzero real constant. A function
Remark 1.5. We note that the subclasses
The following result will be required for proving our results.
Lemma 1.6. () Let
In this paper, we use the Gegenbauer polynomial expansions to provide estimates for the initial coefficients of the subclass of bi-univalent functions
Unless otherwise mentioned, we assume in the reminder of this paper that,
2. Coefficient Bounds of the Class
This section isdevoted to find initial coefficient bounds of the class
Theorem 2.1. Let
Next, define the functions
In the following, one can derive
3. Fekete-Szegö Problem for the Function Class
Fekete-Szegö inequality is one of the famous problem related to coefficients of univalent analytic functions. It was first given by , who stated that, if
This bound is sharp when η is real.
In this section, we aim to provide Fekete-Szegö inequalities for functions in the class
Theorem 3.1. Let
Which completes the proof of Theorem 3.1.
4. Corollaries and Consequences
The authors would like to thank the referees for their helpful comments and suggestions.
- I. Aldawish, T. Al-Hawary and B. A. Frasin,
Subclasses of bi-univalent functions defined by Frasin differential operator, Afr. Mat., 30(3-4)(2019), 495-503.
- H. Aldweby and M. Darus,
Some subordination results on q-analogue of Ruscheweyh differential operator, Abstr. Appl. Anal., 2014(2014).
- S. Altinkaya and S. Yal¸cin,
Estimates on coefficients of a general subclass of biunivalent functions associated with symmetric q-derivative operator by means of the Chebyshev polynomials, Asia Pac. J. Math., 4(2)(2017), 90-99.
- A. Amourah, B. A. Frasin and T. Abdeljawad,
Fekete-Szeg¨o inequality for analytic and bi-univalent functions subordinate to Gegenbauer polynomials, J. Funct. Spaces, (2021). 5, Bi-Bazilevič functions of order ϑ+iδ associated with (p,q)-Lucas polynomials, AIMS Mathematics, 6(5)(2021), 4296-4305.
- A. Amourah, T. Al-Hawary and B. Frasin,
Application of Chebyshev polynomials to certain class of bi-Bazilevič functions of order α+iβ, Afr. Mat., 32(2021), 1059-1066.
- H. Bateman. Higher Transcendental Function. McGraw-Hil; 1953.
- K. L. Brahim and Y. Sidomou,
On some symmetric q-special functions, Matematiche(Catania), 68(2)(2013), 107-122.
- D. A. Brannan and J. G. Clunie. Aspects of contemporary complex analysis. New York and London: Academic Press; 1980.
- D. A. Brannan and T. S. Taha. On some classes of bi-univalent functions. KFAS Proc. Ser. Oxford: Pergamon; 1988.
- B. Doman,
The classical orthogonal polynomials, World Scientific, (2015).
- M. Fekete and G. Szegö,
Eine Bemerkung Ãber ungerade schlichte Funktionen, J. London Math. Soc., 1(2)(1933), 85-89.
- B. A. Frasin and M. K. Aouf,
New subclasses of bi-univalent functions, Appl. Math. Lett., 24(2011), 1569-1573.
- G. Gasper and M. Rahman. Basic Hypergeometric Series. Cambridge, MA: Cambridge Univ. Press; 1990.
- F. H. Jackson,
On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46(1908), 253-281.
- S. Kanas and D. Răducanu,
Some subclass of analytic functions related to conic domains, Math. Slovaca, 64(5)(2014), 1183-1196.
- K. Kiepiela, I. Naraniecka and J. Szynal,
The Gegenbauer polynomials and typically real functions, J. Comput. Appl. Math., 153(1-2)(2003), 273-282.
- A. Legendre,
Recherches sur laattraction des sph´eroides homog´enes, Universittsbibliothek Johann Christian Senckenberg, 10(1785), 411-434.
- M. Lewin,
On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18(1967), 63-68.
- A. Mohammed and M. Darus,
A generalized operator involving the q-hypergeometric function, Mat. Vesnik, 65(4)(2013), 454-465.
- G. Murugusundaramoorthy, N. Magesh and V. Prameela,
Coefficient bounds for certain subclasses of bi-univalent function, Abstr. Appl. Anal., (2013).
- E. Netanyahu,
The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |ξ|<1, Arch. Rational Mech. Anal., 32(1969), 100-112.
- C. Pommerenke. Univalent Functions. Gottingen: Vandenhoeck and Ruprecht; 1975.
- C. Ramachandran, T. Soupramanien and B. A. Frasin,
New subclasses of analytic function associated with q-difference operator, Eur. J. Pure Appl. Math., 10(2)(2017), 348-362.
- M. Reimer,
Multivariate polynomial approximation, Birkh Auser, (2012).
- T. M. Seoudy and M. K. Aouf,
Coefficient estimates of new classes of q-starlike and q-convex functions of complex order, J. Math. Inequal., 10(1)(2016), 135-145.
- H. M. Srivastava,
Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett.,, 23(2010), 1188-1192.
- H. M. Srivastava,
Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Trans. A., 44(3)(2020), 327-344.
- T. S. Taha.
- F. Yousef, S. Alroud and M. Illafe,
A comprehensive subclass of bi-univalent functions associated with Chebyshev polynomials of the second kind, Bol. Soc. Mat. Mex.(3), 26(2)(2020), 329-339.
- F. Yousef, B. A. Frasin and T. Al-Hawary,
Fekete-Szegö inequality for analytic and bi-univalent functions subordinate to Chebyshev polynomials, Filomat, 32(9)(2018), 3229-3236.