Abstract
In this paper, we introduce new and qualitative subclasses \(\mathbf {B} ^{\varepsilon }(\kappa ,\alpha ,\sigma )\), \(\mathbf {B}^{\gamma }(\kappa ,\alpha ,\sigma )\) and \(\mathbf {B}_{s,t}(\kappa ,\alpha ,\sigma )\) of bi-univalent functions. The coefficient bounds and Fekete-Szegö inequalities for functions belonging to these subclasses are obtained. Also, we will get a variety of new results through special cases of our main results.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and preliminaries
Let \({\mathcal {A}}\) be the class of analytic functions in the open unit disk \({\mathcal {U=}}\left\{ z:\left| z\right| <1\right\} \) with conditions \(f(0)=0\) and \(f^{\prime }(0)=1\) having the form:
Further, all functions in \({\mathcal {A}}\) which are univalent in \({\mathcal {U}}\) we will denote it by \({\mathcal {S}}\). So, every function \(f\in {\mathcal {S\ }}\)has an inverse \(f^{-1}\), such that
where
A function \(f\in {\mathcal {A}}\) given by (1.1) is in the class \(\Sigma \) of all bi-univalent functions in \({\mathcal {U}}\) if both f(z) and \(f^{-1}(z)\) are univalent in \({\mathcal {U}}\) (see [1,2,3, 15, 18]).
The class \({\mathcal {S}}^{*}(\varepsilon )\) of starlike functions of order \(\varepsilon \) in \({\mathcal {U}}\) is well-studied and subset of the function class \({\mathcal {S}}\). By definition, we have
Ezrohi [7] introduced the class
Also, Chen [6] introduced the class
It is stated in [4] that a function \(f\in \mathcal {A}\) is in the class \(\mathcal {S}_{\Sigma }^{*}\left[ \varepsilon \right] \) of strongly bi-starlike functions of order \(\varepsilon (0<\varepsilon \le 1)\) if each of the following requirements is met
and
where g \(=f^{-1}\) and given by (1.2).
Also, a function \(f\in \mathcal {A}\) is in the class \(\mathcal {S}_{\Sigma } ^{*}(\gamma )\) of bi-starlike functions of order \(\gamma (0\le \gamma <1)\) if each of the following requirements is met
and
where g \(=f^{-1}\)and given by (1.2).
Now, we introduce the new and Comprehensive subclasses \(\mathbf {B} ^{\varepsilon }(\kappa ,\alpha ,\sigma )\), \(\mathbf {B}^{\gamma }(\kappa ,\alpha ,\sigma )\) and \(\mathbf {B}_{s,t}(\kappa ,\alpha ,\sigma )\).
Definition 1.1
A function f(z) given by (1.1) is said to be in the class \(\mathbf {B}^{\varepsilon }(\kappa ,\alpha ,\sigma )\) where \(\alpha ,\kappa \ge 1,\) \(\sigma \in \mathbb {C}\), \({\mathrm{Re}}\,(\sigma )\ge 0\), and \(0<\varepsilon \le 1\), if the following inequalities are satisfied:
and
where g is given by (1.2).
Definition 1.2
A function f(z) given by (1.1) is said to be in the class \(\mathbf {B}^{\gamma }(\kappa ,\alpha ,\sigma )\) where \(\alpha ,\kappa \ge 1,\) \(\sigma \in \mathbb {C}\), \({\mathrm{Re}}\,(\sigma )\ge 0,\) and \(0\le \gamma <1,\) if the following inequalities are satisfied:
and
where g is given by (1.2).
Definition 1.3
Let the functions \(s,t:\mathcal {U\rightarrow \mathbb {C}}\) such that
Also let \(f\in \mathcal {A}\), defined by (1.1). We say that \(f\in \mathbf {B}_{s,t}(\kappa ,\delta ,\mu )\) where \(\alpha ,\kappa \ge 1,\) \(\sigma \in \mathbb {C},\) \({\mathrm{Re}}\,(\sigma )\ge 0\) if the following inequalities are satisfied:
and
where g is given by (1.2).
Remark 1.4
By taking specific values of the functions s(z) and t(z) in Definition 1.3 we get various well known subclasses of \(\mathcal {A}\), for example, if
or
it is simple to verify that s(z) and t(z) satisfy the Definition 1.3. If \(f\in \mathbf {B}_{s,t}(\kappa ,\alpha ,\sigma ),\) then the function f satisfied the inequalities (1.4) and (1.5) or (1.6) and (1.7), where g is given by (1.2). This means that, \(f\in \mathbf {B}^{\varepsilon }(\kappa ,\alpha ,\sigma )\) or \(f\in \mathbf {B} ^{\gamma }(\kappa ,\alpha ,\sigma ),\) where \(\alpha ,\kappa \ge 1,\) \(0<\varepsilon \le 1,\) \(0\le \gamma <1,\) \(\sigma \in \mathbb {C} \) and \({\mathrm{Re}}\,(\sigma )\ge 0.\)
The purpose of this paper is to introduce qualitative subclasses \(\mathbf {B}^{\varepsilon }(\kappa ,\alpha ,\sigma )\), \(\mathbf {B}^{\gamma } (\kappa ,\alpha ,\sigma )\) and \(\mathbf {B}_{s,t}(\kappa ,\alpha ,\sigma )\) of the function class \(\Sigma .\) Motivated by the earlier work of, Bulut [5], Frasin et al. [8,9,10], Li and Wang [11], Siregar and Raman [14], and Yousef et al. [17, 19,20,21], we find estimates on the coefficients \(|a_{2}|\), \(|a_{3}|\) and \(\left| a_{3}-\varsigma a_{2}^{2}\right| \). Furthermore, variety of new results will follow by specializing cases in our main results.
To proof our theorem we will need the following lemma:
Lemma 1.5
[12] If \(h\in \mathcal {H}\), then \(\left| h_{i}\right| \le 2\) for each i, where \(\mathcal {H}\) is the family of all functions h analytic in \(\mathcal {U}\) for which
2 Coefficient bounds for subclass \(\mathbf {B}_{s,t}(\kappa ,\alpha ,\sigma )\)
In this section we state and prove the main results for subclass\( {\mathbf {B}} _{s,t}(\kappa ,\alpha ,\sigma )\) given by Definition 1.3.
Theorem 2.1
Let f(z) given by (1.1) be in the class \({\mathbf {B}}_{s,t}(\kappa ,\alpha ,\sigma ),~\)where \(\alpha ,\kappa \ge 1,\) \(\sigma \in \mathbb {C}\) and \({\mathrm{Re}}\,(\sigma )\ge 0.\) Then
and
where
Proof
First, we write the equivalent forms for inequalities (1.6) and (1.7) as follows:
and
where s(z) and t(w) are in \(\mathcal {H}\) and satisfy the conditions of Definition 1.3 and have the forms
Now, equating coefficients in (2.2) and (2.3), yields
and
and
Also, adding (2.5) to (2.7), we find that
From Eqs. (2.9) and (2.10), we have
and
respectively. So we get the inequality (2.1).
Next, to find the bound on \(\left| a_{3}\right| \), by subtracting (2.7) from (2.5), we get
Further, in view of (2.9) in Eq. (2.13), it follows that
We thus find that
On other hand, by using (2.10) in (2.13), we get
Consequently, we have
Also, from (2.7) we find that
Consequently, we have
where
Which completes the proof. \(\square \)
3 Corollaries and consequences
Choosing \(\kappa =1\) in Theorem 2.1, we obtain the following Corollary:
Corollary 3.1
Let f(z) given by (1.1) be in the class \(\mathbf {B} _{s,t}(1,\alpha ,\sigma ),~\)where \(\alpha \ge 1,\) \(\sigma \in \mathbb {C} \) and \({\mathrm{Re}}\,(\sigma )\ge 0.\) Then
and
where
Putting \(\alpha =1\) in Corollary 3.1, we obtain the following Corollary:
Corollary 3.2
Let f(z) given by (1.1) be in the class \(\mathbf {B} _{s,t}(1,1,\sigma ),~\)where \(\sigma \in \mathbb {C} \) and \({\mathrm{Re}}\,(\sigma )\ge 0.\) Then
and
where
Putting \(\sigma =0\) in Corollary 3.1, we obtain the following Corollary:
Corollary 3.3
Let f(z) given by (1.1) be in the class \(\mathbf {B} _{s,t}(1,\alpha ,0),~\)where \(\alpha \ge 1.\) Then
and
where
Putting \(\alpha =1\) in Corollary 3.3, we obtain the following Corollary:
Corollary 3.4
Let f(z) given by (1.1) be in the class \(\mathbf {B} _{s,t}(1,1,0).\) Then
and
where
Remark 3.5
The estimates for coefficients \(|a_{2}|\) and \(|a_{3}|\) in Corollary 3.4 obtained by Bulut [5]
Putting \(\sigma =1\) in Corollary 3.2, we obtain the following Corollary:
Corollary 3.6
Let f(z) given by (1.1) be in the class \(\mathbf {B} _{s,t}(1,1,1).\) Then
and
Remark 3.7
The estimates for coefficients \(|a_{2}|\) and \(|a_{3}|\) in Corollary 3.6 obtained by Xu et al. [16]
References
Amourah, A., Frasin, B.A., Abdeljawad, T.: Fekete-Szegö Inequality for analytic and biunivalent functions subordinate to Gegenbauer polynomials. J. Funct. Sp. Article ID 5574673, 7 (2021)
Amourah, A., Frasin, B.A., Murugusundaramoorthy, G., Al-Hawary, T.: Bi-Bazilevic̆ functions of order \(\vartheta + i\delta \) associated with \((p, q)-\) Lucas polynomials. AIMS Math. 6(5), 4296–4305 (2021)
Amourah, A., Illafe, M.: A comprehensive subclass of analytic and Bi-univalent functions associated with subordination. Pal. J. Math. 9, 187–193 (2020)
Brannan, D.A., Taha, T.S.: On some classes of bi-univalent functions. Stud. Univ. Babeş-Bolyai Math. 31(2), 70–77 (1986)
Bulut, S.: Coefficient estimates for a class of analytic and bi-univalent functions. Novi Sad J. Math. 43(2), 59–65 (2013)
Chen, M.P.: On functions satisfying \(Re\Big \lbrace \frac{f(z)}{z}\Big \rbrace >\alpha \). Tamk. J. Math. 5, 231–234 (1974)
Ezrohi, T.G.: Certain estimates in special classes of univalent functions in the unitcircle. Doporidi Akademii Nauk Ukrains, RSR 2, 984–988 (1965)
Frasin, B.A., Al-Hawary, T., Yousef, F.: Necessary and sufficient conditions for hypergeometric functions to be in a subclass of analytic functions. Afr. Mat. 30, 223–230 (2019)
Frasin, B.A., Al-Hawary, T., Yousef, F.: Some properties of a linear operator involving generalized Mittag–Leffler function. Stud. Univ. Babeş-Bolyai Math. 65(1), 67–75 (2020)
Frasin, B.A., Yousef, F., Al-Hawary, T., Aldawish, I.: Application of generalized Bessel functions to classes of analytic functions. Afr. Mat. 32(3), 431–439 (2021)
Li, X.F., Wang, A.P.: Two new subclasses of bi-univalent functions. Int. Math. Forum 7(30), 1495–1504 (2012)
Pommerenke, Ch.: Univalent Functions. Vandenhoeck and Rupercht, Göttingen (1975)
Singh, R.: On Bazilevic̆ functions. Proc. Am. Math. Soc. 38, 261–271 (1973)
Siregar, S., Raman, S.: Certain subclasses of analytic and bi-univalent functions involving double zeta functions. Int. J. Adv. Sci. Eng. Inf. Tech. 2(5), 16–18 (2012)
Srivastava, H.M., Mishra, A.K., Gochhayat, P.: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23, 1188–1192 (2010)
Xu, Q.-H., Gui, Y.-C., Srivastava, H.M.: Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Appl. Math. Lett. 25, 990–994 (2012)
Yousef, F., Alroud, S., Illafe, M.: A comprehensive subclass of bi-univalent functions associated with Chebyshev polynomials of the second kind. Bol. Soc. Mat. Mex. 26, 329–339 (2020)
Yousef, F., Alroud, S., Illafe, M.: New subclasses of analytic and bi-univalent functions endowed with coefficient estimate problems. Anal. Math. Phys. 11, 58 (2021)
Yousef, F., Amourah, A.A., Darus, M.: Differential sandwich theorems for p-valent functions associated with a certain generalized differential operator and integral operator. Ital. J. Pure Appl. Math. 36, 543–556 (2016)
Yousef, F., Frasin, B.A., Al-Hawary, T.: Fekete–Szegö inequality for analytic and bi-univalent functions subordinate to Chebyshev polynomials. Filomat 32(9), 3229–3236 (2018)
Yousef, F., Al-Hawary, T., Murugusundaramoorthy, G.: Fekete–Szegö functional problems for some subclasses of bi-univalent functions defined by Frasin differential operator. Afr. Mat. 30(3–4), 495–503 (2019)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Al-Hawary, T. Coefficient bounds and Fekete–Szegö problem for qualitative subclass of bi-univalent functions. Afr. Mat. 33, 28 (2022). https://doi.org/10.1007/s13370-021-00934-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13370-021-00934-1
Keywords
- Bi-univalent functions
- Analytic function
- Univalent functions
- Coefficient inequalities
- Fekete–Szegö problems