Faber polynomial coefficients for certain subclasses of analytic and biunivalent functions

In this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions defined in the open unit disc. We use the Faber polynomial expansions to find upper bounds for the $n$th$~ n\geq 3 $ Taylor-Maclaurin coefficients $\left\vert a_{n}\right\vert $ of functions belong to these new subclasses with $a_{k}=0$ for $2\leq k\leq n-1$, also we find non-sharp estimates on the first two coefficients $\left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $. The results, which are presented in this paper, would generalize those in related earlier works of several authors.

___

  • [1] Airault H, Bouali A. Differential calculus on the Faber polynomials. Bulletin des Sciences Mathématiques 2006; 130: 179-222.
  • [2] Airault H, Ren J. An algebra of differential operators and generating functions on the set of univalent functions. Bulletin des Sciences Mathématiques 2002; 126: 343-367.
  • [3] 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.
  • [4] Altınkaya S. Bounds for a new subclass of bi-univalent functions subordinate to the Fibonacci numbers. Turkish Journal of Mathematics 2020; 44 (2): 553-560.
  • [5] Altınkaya S, Yalcın S. Faber polynomial coefficient bounds for a subclass of bi-univalent functions. Comptes Rendus Mathematique 2015; 353 (12): 1075-1080.
  • [6] Aouf MK, El-Ashwah RM, Abd-Eltawab AM. New subclasses of bi-univalent functions involving Dziok–Srivastava operator. ISRN Mathematical Analysis 2013; Article ID 387178: 5 p.
  • [7] Bazilevic IE. On a case of integrability in quadratures of the Loewner-Kufarev equation. Matematicheskii Sbornik 1955; 37 (79): 471-476.
  • [8] Bouali A. Faber polynomials, Cayley-Hamilton equation and Newton symmetric functions. Bulletin des Sciences Mathématiques 2006; 130: 49-70.
  • [9] De Branges L. A proof of the Bieberbach conjecture. Acta Mathematica 1985; 154: 137-152.
  • [10] Brannan DA, Clunie JG (Eds). Aspects of Contemporary Complex Analysis (Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham; July 1–20, 1979), Academic Press, New York and London, 1980.
  • [11] Brannan DA, Clunie J, Kirwan WE. Coefficient estimates for a class of starlike functions. Canadian Journal of Mathematics 1970; 22: 476-485.
  • [12] Brannan DA, Taha TS. On some classes of bi-univalent functions. Studia Universitatis Babes-Bolyai Mathematica 1986; 31 (2): 70-77.
  • [13] 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; Article ID 171039: 6 p.
  • [14] 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; Article ID 181932: 7 p.
  • [15] Bulut S. Coefficient estimates for a class of analytic and bi-univalent functions. Novi Sad Journal of Mathematics 2013; 43 (2): 59-65.
  • [16] Bulut S. Coefficient estimates for a new subclass of analytic and bi-univalent functions defined by Hadamard product. Journal of Complex Analysis 2014; 2014 Article ID 302019: 7 p.
  • [17] Caglar M, Orhan H, Yagmur N. Coefficient bounds for new subclasses of bi-univalent functions. Filomat 2013; 27 ( 7): 1165-1171.
  • [18] Darwish HE, Lashin AY, Soileh SM. On Certain Subclasses of Starlike p-valent Functions. Kyungpook Mathematical Journal 2016; 56: 867-876.
  • [19] Deniz E. Certain subclasses of bi-univalent functions satisfying subordinate conditions. Journal of Classical Analysis 2013; 2(1): 49-60.
  • [20] Deniz E, Jahangiri JM, Hamidi SG, Kina SK. Faber polynomial coefficients for generalized bi–subordinate functions of complex order. Journal of Mathematical Inequalities 2018; 12 (3): 645-653.
  • [21] Duren PL. Univalent Functions. Grundlehren der mathematischen Wissenschaften, Band 259. New York, NY, USA: Springer-Verlag, 1983.
  • [22] Faber G. Uber polynomische Entwicklungen. Mathematische Annalen 1903; 57: 385-408.
  • [23] Frasin BA, Aouf MK. New subclasses of bi-univalent functions. Applied Mathematics Letters 2011; 24: 1569-1573.
  • [24] Goyal SP, Goswami P. Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives. Journal of the Egyptian Mathematical Society 2012; 20: 179-182.
  • [25] Goyal SP, Kumar R. Coefficient estimates and quasi-subordination properties associated with certain subclasses of analytic and bi-univalent functions. Mathematica Slovaca 2015; 65 (3): 533-544.
  • [26] Hamidi SG, Jahangiri JM. Faber polynomial coefficients of bi-subordinate functions. Comptes Rendus Mathematique 2016; 354 (4): 365-370.
  • [27] Hayami T, Owa S. Coefficient bounds for bi-univalent functions. Pan-American Mathematical Journal 2012; 22 (4): 15-26.
  • [28] Jahangiri JM, Hamidi SG. Faber polynomial coefficient estimates for analytic bi-bazilevic functions. Matematicki Vesnik 2015; 67 (2): 123-129.
  • [29] Lashin AY. On certain subclasses of analytic and bi-univalent functions. Journal of the Egyptian Mathematical Society 2016; 24 (2): 220-225.
  • [30] Lashin AY. Coefficient estimates for two subclasses of analytic and bi-univalent functions. Ukrainian Mathematical Journal 2019; 70 (9): 1484-1492.
  • [31] Lewin M. On a coefficient problem for bi-univalent functions. Proceedings of the American Mathematical Society 1967; 18: 63-68.
  • [32] Li X-F, Wang A-P. Two new subclasses of bi-univalent functions. International Mathematical Forum 2012; 7: 1495- 1504.
  • [33] Liu M. On certain subclass of p-valent functions. Soochow Journal of Mathematics 2000; 26 (2): 163-171.
  • [34] Magesh N, Rosy T, Varma S. Coefficient estimate problem for a new subclass of bi-univalent functions. Journal of Complex Analysis 2013; 2013 Article ID 474231: 3p.
  • [35] Magesh N, Yamini J. Coefficient bounds for certain subclasses of bi-univalent functions. International Mathematical Forum 2013; 8: 1337-1344.
  • [36] Murugusundaramoorthy G, Magesh N, Prameela V. Coefficient bounds for certain subclasses of bi-univalent function. Abstract and Applied Analysis 2013; Article ID 573017: 3 p.
  • [37] Peng Z-G, Han Q-Q. On the coefficients of several classes of bi-univalent functions. Acta Mathematica Sinica, English Series 2014; 34: 228-240.
  • [38] Ponnusamy S. Polya-Schoenberg conjecture by Caratheodory functions. Journal of the London Mathematical Society 1995; 51 (2): 93-104.
  • [39] Ponnusamy S, Rajasekaran S. New sufficient conditions for starlike and univalent functions. Soochow Journal of Mathematics 1995; 21 (2): 193-201.
  • [40] Porwal S, Darus M. On a new subclass of bi-univalent functions. Journal of the Egyptian Mathematical Society 2013; 21 (3): 190-193.
  • [41] Prajapat JK, Agarwal R. Some results on certain class of analytic functions based on differential subordination. Bulletin of the Korean Mathematical Society 2013; 50 (1): 1-10.
  • [42] Srivastava HM, Bulut S, Caglar M, Yagmur N. Coefficient estimates for a general subclass of analytic and biunivalent functions. Filomat 2013; 27 (5): 831-842.
  • [43] Srivastava HM, Mishra AK, Gochhayat P. Certain subclasses of analytic and bi-univalent functions. Applied Mathematics Letters 2010; 23: 1188-1192.
  • [44] Srivastava HM, Murugusundaramoorthy G, Magesh N. Certain subclasses of bi-univalent functions associated with the Hohlov operator. Global Journal of Mathematical Analysis 2013; 1 (2): 67-73.
  • [45] Srivastava HM, Murugusundaramoorthy G, Vijaya K. Coefficient estimates for some families of bi-Bazilevic functions of the Ma–Minda type involving the Hohlov operator. Journal of Classical Analysis 2013; 2: 167-181.
  • [46] Taha TS. Topics in univalent function theory. PhD, University of London, London, UK, 1981.
  • [47] Tang H, Deng G-T, Li S-H. Coefficient estimates for new subclasses of Ma–Minda bi-univalent functions. Journal of Inequalities and Applications 2013; 2013 Article ID 317: 10 p.
  • [48] Xu Q-H, Gui Y-C, Srivastava HM. Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Applied Mathematics Letters 2012; 25: 990-994.
  • [49] Xu Q-H, Xiao H-G, Srivastava HM. A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems. Applied Mathematics and Computation 2012; 218 (23): 11461-11465.
  • [50] Yang D. Some multivalent starlikeness conditions for analytic functions. Bulletin of the Institute of Mathematics Academia Sinica 2005; 33 (1): 55-67.
  • [51] Zireh A, Adegani EA, Bidkham M. Faber polynomial coefficient estimates for subclass of bi-univalent functions defined by quasi-subordinate. Mathematica Slovaca 2018; 68 (2): 369-378.