Tbilisi Mathematical Journal

Coefficient bounds for subclasses of bi-univalent functions defined by fractional derivative operator

Sevtap Sümer Eker, Bilal Şeker, and Sadettin Ece

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, first we introduce new subclasses of analytic and bi-univalent functions defined by fractional derivative operator in the open unit disk and then for the functions belongs to these classes obtain upper bounds for the initial coefficients.

Article information

Tbilisi Math. J., Volume 13, Issue 3 (2020), 1-10.

Received: 18 July 2019
Accepted: 14 May 2020
First available in Project Euclid: 29 September 2020

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Secondary: 30C50: Coefficient problems for univalent and multivalent functions

analytic functions coefficient bounds univalent functions fractional derivative bi-univalent functions


Eker, Sevtap Sümer; Şeker, Bilal; Ece, Sadettin. Coefficient bounds for subclasses of bi-univalent functions defined by fractional derivative operator. Tbilisi Math. J. 13 (2020), no. 3, 1--10. doi:10.32513/tbilisi/1601344894. https://projecteuclid.org/euclid.tbilisi/1601344894

Export citation


  • Ak\in G, Sümer Eker S. Coefficient estimates for a certain class of analytic and bi-univalent functions defined by fractional derivative. Comptes Rendus Mathématique 2014; 352: 1005–1010. doi:10.1016/j.crma.2014.09.022
  • Brannan DA, Clunie JG. 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, London and New York, 1980.
  • Brannan DA, Taha TS. On some classes of bi-univalent functions, in: S.M. Mazhar, A. Hamoui, N.S. Faour (Eds.), Mathematical Analysis and Its Applications, Kuwait; February 18-21, 1985; in: KFAS Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, 1988, pp. 53-60. See also Studia Universitatis Babeş-Bolyai / Mathematica 1986; 31: 70-77.
  • Çağlar M, Orhan H, Yağmur N. Coefficient bounds for new subclasses of bi-univalent functions. Filomat 2013; 27: 1165-1171. doi:10.2298/FIL1307165C
  • Duren PL. Univalent Functions. Grundlehren der Mathematischen Wissenschaften, Vol. 259. Berlin, Germany: Springer, 1983.
  • Hamidi SG, Jahangiri JM. Faber polynomial coefficients of bi-subordinate functions. Comptes Rendus Mathématique 2016; 354: 365-370. doi:10.1016/j.crma.2016.01.013
  • Hussain S, Khan S, Zaighum MA, Darus M, Shareef Z. Coefficients Bounds for Certain Subclass of Bi-univalent Functions Associated with Ruscheweyh q-Differential Operator. Journal of Complex Analysis vol. 2017; Article ID 2826514, 9 pages. doi:10.1155/2017/2826514.
  • Lewin M. On a coefficient problem for bi-univalent functions. Proceedings of the American Mathematical Society 1967; 18: 63-68.
  • 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.
  • Owa S. On the distortion theorems I. Kyungpook Mathematical Journal 1978; 18: 53-59.
  • Owa S, Srivastava HM. Univalent and starlike generalized hypergeometric functions. Canadian Journal of Mathematics 1987; 39: 1057-1077.
  • Pommerenke C. Univalent functions. Göttingen, Vandenhoeck Ruprecht: 1975.
  • Rosihan MA, Lee SK, Ravichandran V, Supramaniam S. Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions. Applied Mathematics Letters 2012; 25: 344-351. doi:10.1016/j.aml.2011.09.012
  • Rudin W. Real and Complex Analysis. McGraw-Hill Education ; 3 edition, 1986.
  • Srivastava HM. Some inequalities and other results associated with certain subclasses of univalent and bi-univalent analytic functions. in: Nonlinear Analysis: Stability; Approximation; and Inequalities (Panos M. Pardalos,Pando G. Georgiev and Hari M. Srivastava, Editors.), Springer Series on Optimization and Its Applications. 2012; Vol. 68, Springer-Verlag, Berlin, Heidelberg and New York, pp. 607-630.
  • Srivastava HM, Mishra AK, Gochhayat P. Certain subclasses of analytic and bi-univalent functions. Applied Mathematics Letters 2010; 23: 1188-1192. doi:10.1016/j.aml.2010.05.009
  • Srivastava HM, Coefficient Estimates for Some Subclasses of $m$-Fold Symmetric Bi-Univalent Functions. Filomat 2018; 32: 9.
  • Srivastava HM, Owa S. Some characterization and distortion theorems involving fractional calculus, linear operators and certain subclasses of analytic functions. Nagoya Mathematical Journal 1987; 106: 1-28.
  • Srivastava HM, Owa S. Univalent Functions, Fractional Calculus, and Their Applications. Halsted Press, Ellis Horwood Limited, Chichester and JohnWiley and Sons, NewYork, Chichester, Brisbane and Toronto, 1989.
  • Srivastava HM, Sümer Eker S, Rosihan MA. Coefficient Bounds for a certain class of analytic and bi-univalent functions. Filomat 2015; 29: 1839–1845. doi: 10.2298/FIL1508839S
  • Srivastava HM, Gaboury S, Ghanim F.Coefficient estimates for a general subclass of analytic and bi-univalent functions of the Ma-Minda type. Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat. (RACSAM) 2018; 112: 1157-1168. doi: 10.1007/s13398-017-0416-5
  • Srivastava HM, Altınkaya Ş, Yalçın S. Certain Subclasses of Bi-Univalent Functions Associated with the Horadam Polynomials. Iran J Sci Technol Trans Sci 2019; 43: 1873-1879. doi: 10.1007/s40995-018-0647-0
  • Srivastava HM, Khan S, Ahmad QZ, Khan N, Hussain S. The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain $q$-integral operator. Stud. Univ. Babeş-Bolyai Math. 2018; 63, 419-436. doi:10.24193/subbmath.2018.4.01
  • Sümer Eker S. Coefficient bounds for subclasses of m-fold symmetric bi-univalent functions. Turkish Journal of Mathematics 2016; 40: 641–646. doi:10.3906/mat-1503-58
  • Sümer Eker S. and Şeker B. On $\lambda$-pseudo bi-starlike and $\lambda$-pseudo bi-convex functions with respect to symmetrical points. Tbilisi Mathematical Journal 2018; 11: 49–57. doi: 10.2478/tmj-2018-0004
  • Şeker B. On a new subclass of bi-univalent functions defined by using Salagean operator. Turkish Journal of Mathematics 2018; 42: 2891-2896. doi:10.3906/mat-1507-100
  • Şeker B, Sümer Eker S. On Subclasses of Bi-Close-to-convex Functions Related to The Odd-Starlike Functions. Palestine Journal of Mathematics 2017; 6 (Special Issue II): 215-221.
  • Taha TS. Topics in Univalent Function Theory. PhD Thesis, University of London, 1981.
  • Tezelci M, Sümer Eker S. On strongly Ozaki bi-close-to-convex functions. Turkish Journal of Mathematics 2019; 43: 862-870. doi:10.3906/mat-1810-104
  • Zaprawa P. On the Fekete-Szegö problem for classes of bi-univalent functions. Bulletin of the Belgian Mathematical Society-Simon Stevin, 2014; 21: 169-178.