Articles
Kyungpook Mathematical Journal 2017; 57(4): 613-621
Published online December 23, 2017
Copyright © Kyungpook Mathematical Journal.
Coefficient Estimates for Sãlãgean Type λ-bi-pseudo-starlike Functions
Santosh Joshi1
Sahsene Altinkaya and Sibel Yalçin2
Department of Mathematics, Walchand College of Engineering, Sangli 416415, India1
Department of Mathematics, Faculty of Arts and Science, Uludag University, 16059, Bursa, Turkey2
Received: September 28, 2016; Revised: October 2, 2017; Accepted: October 30, 2017
Abstract
In this paper, we have constructed subclasses of bi-univalent functions associated with λ–bi-pseudo-starlike functions in the unit disc
Keywords: analytic functions, bi-starlike functions, coefficient bounds
1. Introduction
Let
Let
A function
However, the familiar Koebe function is not a member of the bi-univalent function class ∑
are also not members of ∑ (see [15]).
Historically, Lewin [11] studied the class of bi-univalent functions, obtaining the bound 1.51 for the modulus of the second coefficient |
For
We note that
In this paper, motivated by the earlier work of Babalola [3] and Joshi et. al. [10], we aim at introducing two new subclasses of the function class ∑ and find estimate on the coefficients |
We note the following lemma required for obtaining our results.
Lemma 1.1. ([13])
2. Coefficient Bounds for the Function Class S Σ λ ( k , α )
Definition 2.1
A function
and
where the function
Theorem 2.2
Let
where
and
Now, equating the coefficients in (
and
From (
and
Also from (
Therefore, we have
Applying Lemma 1.1 for the coefficients
Next, in order to find the bound on |
Then, in view of (
This completes the proof of Theorem 2.2.
Putting
Remark 2.3. ([10])
Let
and
3. Coefficient Bounds for the Function Class S Σ λ ( k , β )
Definition 3.1
A function
and
where the function
Theorem 3.2
Let
where
It follows from (
and
From (
and
Also from (
Therefore, we have
Appyling Lemma 1.1 for the coefficients
Next, in order to find the bound on |
Then, in view of (
This completes the proof of Theorem 3.2.
Putting
Remark 3.3. ([10])
Let
and
Taking
Corollary 3.4
Corollary 3.5
References
- Altınkaya, Ş, and Yalçın, S (2015). Faber polynomial coefficient bounds for a subclass of biunivalent functions. C R Acad Sci Paris. 353, 1075-1080.
- Altınkaya, Ş, and Yalçın, S (2015). Coefficient estimates for two new subclasses of bi-univalent functions with respect to symmetric points. J Funct Spaces, 5.
- Babalola, KO (2013). On λ-pseudo-starlike functions. J Class Anal. 3, 137-147.
- Brannan, DA, and Clunie, JG 1979. Aspects of contemporary complex analysis., Proceedings of the NATO Advanced Study Instute Held, University of Durham.
- Brannan, DA, and Taha, TS (1986). On some classes of bi-univalent functions. Stud Univ Babeş-Bolyai Math. 31, 70-77.
- Çağlar, M, Deniz, E, and Srivastava, HM (2017). Second Hankel determinant for certain subclasses of bi-univalent functions. Turkish J Math. 41, 694-706.
- Frasin, BA, and Aouf, MK (2011). New subclasses of bi-univalent functions. Appl Math Lett. 24, 1569-1573.
- Hamidi, SG, and Jahangiri, JM (2014). Faber polynomial coefficient estimates for analytic bi-close-to-convex functions. C R Acad Sci Paris. 352, 17-20.
- Hamidi, SG, and Jahangiri, JM (2016). Faber polynomial coefficients of bi-subordinate functions. C R Acad Sci Paris. 354, 365-370.
- Joshi, S, Joshi, S, and Pawar, H (2016). On some subclasses of bi-univalent functions associated with pseudo-starlike functions. J Egyptian Math Soc. 24, 522-525.
- Lewin, M (1967). On a coefficient problem for bi-univalent functions. Proc Amer Math Soc. 18, 63-68.
- Netanyahu, E (1969). The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1. Arch Ration Mech Anal. 32, 100-112.
- Pommerenke, C (1975). Univalent functions. Vandenhoeck. Göttingen: Ruprecht
- Sãlãgean, GS 1983. Subclasses of univalent functions., Complex Analysis - Fifth Romanian Finish Seminar, Bucharest, pp.362-372.
- Srivastava, HM, Mishra, AK, and Gochhayat, P (2010). Certain subclasses of analytic and bi-univalent functions. Appl Math Lett. 23, 1188-1192.
- Srivastava, HM, Bulut, S, Çağlar, M, and Yağmur, N (2013). Coefficient estimates for a general subclass of analytic and bi-univalent functions. Filomat. 27, 831-842.
- Srivastava, HM, Sivasubramanian, S, and Sivakumar, R (2014). Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions. Tbilisi Math J. 7, 1-10.
- Srivastava, HM, Eker, SS, and Ali, RM (2015). Coefficient bounds for a certain class of analytic and bi-univalent functions. Filomat. 29, 1839-1845.
- Srivastava, HM, and Bansal, D (2015). Coefficient estimates for a subclass of analytic and bi-univalent functions. J Egyptian Math Soc. 23, 242-246.
- Srivastava, HM, Joshi, SB, Joshi, SS, and Pawar, H (2016). Coefficient estimates for certain subclasses of meromorphically bi-univalent functions. Palest J Math. 5, 250-258.
- Srivastava, HM, Gaboury, S, and Ghanim, F (2016). Initial coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions. Acta Math Sci Ser B Engl Ed. 36, 863-871.
- Srivastava, HM, Gaboury, S, and Ghanim, F (2017). Coefficient estimates for some general subclasses of analytic and bi-univalent functions. Afr Mat. 28, 693-706.
- Xu, Q-H, Gui, Y-C, and Srivastava, HM (2012). Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Appl Math Lett. 25, 990-994.
- Xu, Q-H, Xiao, H-G, and Srivastava, HM (2012). A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems. Appl Math Comput. 218, 11461-11465.