General coefficient estimates for bi-univalent functions: a new approach
We prove for univalent functions f(z)=z+∑∞k=nakzk;(n≥2) in the unit disk U={z:|z|
___
- [1] Airault H. Remarks on Faber polynomials. International Mathematical Forum 2008; 3: 449–456.
- [2] Airault H, Bouali A. Differential calculus on the Faber polynomials. Bulletin des Sciences Mathématiques 2006; 130
(3): 179–222.
- [3] Airault H, Ren J. An algebra of differential operators and generating functions on the set of univalent functions.
Bulletin des Sciences Mathématiques 2002; 126 (5): 343–367.
- [4] Ali RM, Lee SK, Ravichandran V, Supramaniam S. Coefficient estimates for bi-univalent Ma-Minda starlike and
convex functions. Applied Mathematics Letters 2012; 25: 344–351.
- [5] Altınkaya Ş, Yalçın S, Çakmak S. A subclass of bi-univalent functions based on the Faber polynomial expansions
and the Fibonacci numbers. Mathematics 2019; 7: 160.
- [6] Aouf MK, El-Ashwah RM, Abd-Eltawab AM. New subclasses of biunivalent functions involving Dziok-Srivastava
operator. ISRN Mathematical Analysis 2013; 2013: 387178.
- [7] Brannan DA, Clunie JG (editors). Aspects of Contemporary Complex Analysis (Proceedings of the NATO Advanced
Study Institute Held at the University of Durham; July 20, 1979). New York, NY, USA: Academic Press, 1980.
- [8] Brannan DA, Taha TS. On some classes of bi-univalent functions. In: Mazhar SM, Hamoui A, Faour NS (editors).
Mathematical Analysis and Its Applications; Kuwait; 1985. KFAS Proceedings Series, Vol. 3. Oxford, UK: Pergamon
Press, 1988, pp. 53–60.
- [9] Bulut S. Coefficient estimates for initial Taylor-Maclaurin coefficients for a subclass of analytic and bi-univalent
functions defined by Al-Oboudi differential operator. Scientific World Journal 2013; 2013: 171039.
- [10] Bulut S. Coefficient estimates for new subclasses of analytic and bi-univalent functions defined by Al-Oboudi
differential operator. Journal of Function Spaces and Applications 2013; 2013: 181932.
249
- [11] Bulut S. Coefficient estimates for a class of analytic and bi-univalent functions. Novi Sad Journal of Mathematics
2013; 43 (2): 59–65.
- [12] Bulut S. Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions.
Comptes rendus de l’Académie des Sciences Paris Series I 2014; 352 (6): 479–484.
- [13] Bulut S. Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions. Filomat 2016; 30
(6): 1567–1575.
- [14] Caglar M, Orhan H, Ya N. Coefficient bounds for new subclasses of bi-univalent functions. Filomat 2013; 27 (7):
1165–1171.
- [15] Duren PL. Univalent Functions. Grundlehren der Mathematischen Wissenschaften, Vol. 259. New York, NY, USA:
Springer, 1983.
- [16] Faber G. Uber polynomische Entwickelungen. Mathematische Annalen 1903; 57 (3): 389–408 (in German).
- [17] Frasin BA, Aouf MK. New subclasses of bi-univalent functions. Applied Mathematics Letter 2011; 24: 1569–1573.
- [18] Goodman AW. Univalent Functions, Vol. I. Tampa, FL, USA: Mariner Publishing Company, 1983.
- [19] Goyal SP, Goswami P. Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by
fractional derivatives. Journal of Egyptian Mathematical Society 2012; 20: 179–182.
- [20] Hamidi SG, Halim SA, Jahangiri JM. Coefficient estimates for a class of meromorphic bi-univalent functions.
Comptes rendus de l’Académie des Sciences Paris Series I 2013; 351 (9-10): 349–352.
- [21] Hamidi SG, Janani T, Murugusundaramoorthy G, Jahangiri JM. Coefficient estimates for certain classes of meromorphic bi-univalent functions. Comptes rendus de l’Académie des Sciences Paris Series I 2014; 352 (4): 277–282.
- [22] Hayami T, Owa S. Coefficient bounds for bi-univalent functions. Pan-American Mathematical Journal 2012; 22 (4):
15–26.
- [23] Hayman WK. Multivalent Functions, Second Edition. Cambridge, UK: Cambridge University Press, 1994.
- [24] Jahangiri JM, Hamidi SG. Coefficient estimates for certain classes of bi-univalent functions. International Journal
of Mathematics and Mathematical Sciences 2013; 2013: 190560.
- [25] Jahangiri JM, Hamidi SG, Halim SA. Coefficients of bi-univalent functions with positive real part derivatives.
Bulletin of the Malaysian Mathematical Sciences Society 2014; (2) 37: 633–640.
- [26] Lewin M. On a coefficient problem for bi-univalent functions. Proceedings of the American Mathematical Society
1967; 18: 63–68.
- [27] Loewner C. Untersuchungen uber schlichte konforme Abbildungen des Einheitskreises. Mathematische Annalen
1923; 89: 103-121 (in German).
- [28] Murugusundaramoorthy G, Magesh N, Prameela V. Coefficient bounds for certain subclasses of bi-univalent functions. Abstract and Applied Analysis 2013; 2013: 573017.
- [29] Netanyahu E. The minimal distance of the image boundary from the origin and the second coefficient of a univalent
function in |z| < 1. Archive for Rational Mechanics and Analysis 1969; 32: 100–112.
- [30] Porwal S, Darus M. On a new subclass of bi-univalent functions. Journal of Egyptian Mathematical Society 2013;
21 (3): 190–193.
- [31] Srivastava HM, Mishra AK, Gochhayat P. Certain subclasses of analytic and bi-univalent functions. Applied
Mathematics Letters 2010; 23: 1188-1192.
- [32] Taha TS. Topics in univalent function theory. PhD, University of London, London, UK, 1981.
- [33] Todorov PG. On the Faber polynomials of the univalent functions of class Σ. Journal of Mathematical Analysis
and Applications 1991; 162: 268-276.
- [34] Wang ZG, Bulut S. A note on the coefficient estimates of bi-close-to-convex functions. Comptes Rendus Mathematique 2017; 355 (8): 876-880.
250
- [35] Xu QH, Gui YC, Srivastava HM. Coefficient estimates for a certain subclass of analytic and bi-univalent functions.
Applied Mathematics Letters 2012; 25: 990-994.
- [36] Xu QH, Xiao HG, Srivastava HM. A certain general subclass of analytic and bi-univalent functions and associated
coefficient estimate problems. Applied Mathematics and Computation 2012; 218: 11461–11465.