• Renu P. Pathak, Santosh D. Jadhav, Ranjan S. Khatu, Amol B. Patil


Analytic function, univalent function, bi-univalent function, Taylor- Maclaurin series expansion, coefficient bounds


In the present paper, we investigate and introduce generalized integral operator J α m,n defined on the unit disc U. Further, we introduce subclass Rh,pP (λ, δ, γ, m, n) of bi-univalent function using this integral operator and obtain estimates on the initial coefficients |a2| and |a3| for the functions belong to this subclass. Also connection with some results of the earlier known subclasses are mentioned.


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A. Akg¨ul and S¸. Altmkaya, Coefficient estimates associated with a new subclass of bi-univalent functions, Acta Universitatis Apulensis, 52, (2017), pp. 121-128. DOI:10.17114/j.aua.2017.52.10.

A. Akg¨ul, Certain inequalities for a general class of analytic and bi-univalent functions, Sahand Communications in Mathematical Analysis, 14(1), (2019), pp. 1-13. DOI:10.22130/scma.2018.70945.277

D. A. Brannan and J. G. Clunie (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).

D. A. Brannan, J. G. Clunie, W. E. Kirwan, Coefficient estimates for a class of starlike functions, Canad. J. Math., 22, (1970), pp. 476–485.

D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, Studia Univ. Babes-Bolyai Math, 31(2), (1986), pp. 70-77.

P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York, (1983).

A. W. Goodman, Univalent Functions, Marinee Publ.Co., Inc., Tampa, FL., vol.I., xvii+246 pp. ISBN:0-936166-10-x, (1983).

A. W. Goodman, An invitation to the study of univalent and multivalent functions, Internat. J. Math. Math. Sci., 2, (1979), pp. 163–186.

S. D. Jadhav, A. B. Patil, I. A. Wani, Initial Taylor-Maclaurin coefficient bounds and the Fekete-Szego problem for subclasses of m-fold symmetric analytic bi-univalent functions, TWMS Journal of Applied and Engineering Mathematics, 14 (1), (1979), pp. 185–196.

E. Jensen, H. Waadeland, A coefficient inequality for biunivalent functions, Skrifter Norske Vid. Selskab (Trondheim), 15, (1972), pp. 1–11.

A. W. Kedzierawski, Some remarks on bi-univalent functions, Ann. Univ. Mariae Curie- Sklodowska Lublin-Polonia, XXXIX 10, (1985), pp. 77–81.

R. S. Khatu, U. H. Naik and A. B. Patil, Estimation on initial coefficient bounds of generalized subclasses of bi-univalent functions, Int. J. Nonlinear Anal. Appl., 13, (2022), pp. 989–997.

M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18, (1967), pp. 63-68.

Z. Nehari, Conformal Mapping, McGraw-Hill Book Co., New York, (1953).

E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1, Arch. Rational Mech. Anal., 32, (1969), pp. 100-112.

A. B. Patil and U. H. Naik, Estimates on initial coefficients of certain sunclasses of bi-univalent functions associated with quasi-subordination, Global Journal of Mathematical Analysis, 5(1), (2017), pp. 6-10.

D. Styer, D. J. Wright, Results on bi-univalent functions, Proc. Amer. Math. Soc., 82 (2), (1981), pp. 243–248.

T. S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, (1981).

Q. H. Xu, H. G. Xiao and H. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput., 218 (23), (2012), pp. 11461-11465.

Q. H. Xu, Y. C. Gui and H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett., 25, (2012), pp. 990-994.




How to Cite

Renu P. Pathak. (2024). INEQUALITIES FOR A CERTAIN SUBCLASS OF BI-UNIVALENT FUNCTIONS INVOLVING A NEW INTEGRAL OPERATOR. International Journal of Intelligent Systems and Applications in Engineering, 12(21s), 3091 –. Retrieved from



Research Article