Sharp initial coefficient bounds and the Fekete–Szegö problem for some certain subclasses of analytic and bi-univalent functions

  • A. B. Patil AISSMS College of Engineering, Pune, India
  • T. G. Shaba Landmark University, Omu-Aran, Nigeria
Keywords: Analytic function, Univalent function, Bi-univalent function, Starlike function, Bazilevic function, Fekete-Szego functional

Abstract

UDC 517.5

We introduce two new subclasses $\mathcal{U}_{\Sigma}(\alpha,\lambda)$ and ${\mathcal{B}_1}_{\Sigma}(\alpha)$ of analytic bi-univalent functions defined in the open unit disk $\mathbb{U}$, which are associated with the Bazilevich functions.  In addition, for functions that belong to these subclasses, we obtain sharp bounds for the initial Taylor–Maclaurin coefficients $a_2$ and $a_3,$ as well as the sharp estimate for the Fekete–Szegö functional $a_3-\mu a_2^2.$

References

R. M. Ali, S. K. Lee, M. Obradović, Sharp bounds for initial coefficients and the second Hankel determinant, Bull. Korean Math. Soc., 57, № 4, 839–850 (2020).

R. M. Ali, V. Ravichandran, N. Seenivasagan, Coefficient bounds for $p$-valent functions, Appl. Math. and Comput., 187, № 1, 35–46 (2007). DOI: https://doi.org/10.1016/j.amc.2006.08.100

Ş. Altinkaya, S. Yalçin, Fekete–Szegö inequalities for certain classes of bi-univalent functions, Int. Sch. Res. Notices, Article ID 327962, 1–6 (2014). DOI: https://doi.org/10.1155/2014/327962

D. A. Brannan, J. G. Clunie (Eds), Aspects of contemporary complex analysis},

Proc. of the NATO Advanced Study Institute Held at the University of Durham, Durham, July 1–20, 1979, Acad. Press, New York, London (1980).

D. A. Brannan, T. S. Taha, On some classes of bi-univalent functions, Stud. Univ. Babeş-Bolyai Math., 31, № 2, 70–77 (1986).

P. L. Duren, Univalent functions, Grundlehren Math. Wiss., 259 (1983).

R. Fournier, S. Ponnusamy, A class of locally univalent functions defined by a differential inequality, Complex Var. and Elliptic Equat., 52, № 1, 1–8 (2007). DOI: https://doi.org/10.1080/17476930600780149

B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24, 1569–1573 (2011). DOI: https://doi.org/10.1016/j.aml.2011.03.048

S. B. Joshi, S. S. Joshi, H. Pawar, On some subclasses of bi-univalent functions associated with pseudo-starlike functions, J. Egyptian Math. Soc., 24, № 4, 522–525 (2016). DOI: https://doi.org/10.1016/j.joems.2016.03.007

M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18, 63–68 (1967). DOI: https://doi.org/10.1090/S0002-9939-1967-0206255-1

U. H. Naik, A. B. Patil, On initial coefficient inequalities for certain new subclasses of bi-univalent functions, J. Egyptian Math. Soc., 25, № 3, 291–293 (2017). DOI: https://doi.org/10.1016/j.joems.2017.04.001

Z. Nehari, Conformal mapping, McGraw-Hill, New York (1952).

M. Obradović, A class of univalent functions, Hokkaido Math. J., 27, № 2, 329–335 (1998). DOI: https://doi.org/10.14492/hokmj/1351001289

M. Obradović, A class of univalent functions II, Hokkaido Math. J., 28, № 3, 557–562 (1999). DOI: https://doi.org/10.14492/hokmj/1351001237

M. Obradović, S. Ponnusamy, K.-J. Wirths, Coefficient characterizations and sections for some univalent functions, Sib. Math. J., 54, 679–696 (2013). DOI: https://doi.org/10.1134/S0037446613040095

M. Obradović, S. Ponnusamy, K.-J. Wirths, Geometric studies on the class $U(lambda)$, Bull. Malays. Math. Sci. Soc., 39, 1259–1284 (2016). DOI: https://doi.org/10.1007/s40840-015-0263-5

A. B. Patil, U. H. Naik, Bounds on initial coefficients for a new subclass of bi-univalent functions, New Trends Math. Sci., 6, № 1, 85–90 (2018). DOI: https://doi.org/10.20852/ntmsci.2018.248

A. B. Patil, T. G. Shaba, On sharp Chebyshev polynomial bounds for a general subclass of bi-univalent functions, Appl. Sci., 23, 109–117 (2021).

C. Pommerenke, Univalent functions, Vandenhoeck and Rupercht, Göttingen (1975).

S. Porwal, M. Darus, On a new subclass of bi-univalent functions, J. Egyptian Math. Soc., 21, № 3, 190–193 (2013). DOI: https://doi.org/10.1016/j.joems.2013.02.007

R. Singh, On Bazilević functions, Proc. Amer. Math. Soc., 38, 261–271 (1973). DOI: https://doi.org/10.1090/S0002-9939-1973-0311887-9

H. M. Srivastava, D. Bansal, Coefficient estimates for a subclass of analytic and bi-univalent functions, J. Egyptian Math. Soc., 23, № 2, 242–246 (2015). DOI: https://doi.org/10.1016/j.joems.2014.04.002

H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23, 1188–1192 (2010). DOI: https://doi.org/10.1016/j.aml.2010.05.009

D. Styer, D. J. Wright, Results on bi-univalent functions, Proc. Amer. Math. Soc., 82, № 2, 243–248 (1981). DOI: https://doi.org/10.1090/S0002-9939-1981-0609659-5

D. L. Tan, Coefficient estimates for bi-univalent functions, Chin. Ann. Math. Ser. A, 5, 559–568 (1984).

A. Vasudevarao, H. Yanagihara, On the growth of analytic functions in the class $U(lambda)$, Comput. Methods Funct. Theory, 13, 613–634 (2013). DOI: https://doi.org/10.1007/s40315-013-0045-8

P. Zaprawa, On the Fekete–Szegö problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin, 21, № 1, 169–178 (2014). DOI: https://doi.org/10.36045/bbms/1394544302

Published
02.03.2023
How to Cite
PatilA. B., and ShabaT. G. “Sharp Initial Coefficient Bounds and the Fekete–Szegö Problem for Some Certain Subclasses of Analytic and Bi-Univalent Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 2, Mar. 2023, pp. 198 -06, doi:10.37863/umzh.v75i2.6602.
Section
Research articles