Initial seven coefficient estimates for a subclass of bi-starlike functions

T.  Janani; S. Yalçın

doi:10.37863/umzh.v73i11.1046

T. Janani School Comput. Sci. and Eng., Vellore Inst. Technology, India https://orcid.org/0000-0001-6174-8360
S. Yalçın Bursa Uludag Univ., Turkey https://orcid.org/0000-0002-0243-8263

DOI: https://doi.org/10.37863/umzh.v73i11.1046

Keywords: Analytic functions, Univalent functions, Bi-univalent functions, Bi-starlike,

Abstract

UDC 517.5

In the present article, a subclass of bi-starlike functions is studied and initial seven Taylor–Maclaurin coefficient estimates $|a_{2}|, |a_{3}|, \ldots , |a_{7}|$ for functions in the subclass of the function class $ \Sigma $ are obtained for the first time in the literature.
Few new or known consequences of the results are also pointed out.

References

A. K. Bakhtin, G. P. Bakhtina, Yu. B. Zelinskii, Topological-algebraic structures and geometric methods in complex analysis, Proc. Inst. Math. NAS Ukraine, 73 (2008).

A. K. Bakhtin, I. V. Denega, Weakened problem on extremal decomposition of the complex plane, Mat. Stud., 51, № 1, 35 – 40 (2019), https://doi.org/10.15330/ms.51.1.35-40 DOI: https://doi.org/10.15330/ms.51.1.35-40

D. A. Brannan, J. G. Clunie (Eds), Aspects of contemporary complex analysis, Proc. NATO Adv. Study Inst., Univ. Durham, Durham, July, 1979, Acad. Press, New York, London (1980), p. 1 – 20.

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

I. Denega, Estimates of the inner radii of non-overlapping domains, J. Math. Sci., № 6, 1 – 9 (2019). DOI: https://doi.org/10.1007/s10958-019-04516-2

I. V. Denega, Ya. V. Zabolotnii, Estimates of products of inner radii of non-overlapping domains in the complex plane, Complex Var. and Elliptic Equat., 62, № 11, 1611 – 1618 (2017)б https://doi.org/10.1080/17476933.2016.1265952 DOI: https://doi.org/10.1080/17476933.2016.1265952

E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Class. Anal., 2, № 1, 49 – 60 (2013), https://doi.org/10.7153/jca-02-05 DOI: https://doi.org/10.7153/jca-02-05

V. Ya. Gutlyanskii, V. I. Ryazanov, Geometric and topological theory of functions and mappings, Naukova Dumka, Kyiv (2011).

T. Janani, G. Murugusundaramoorthy, K. Vijaya, New subclass of pseudo-type meromorphic bi-univalent functions of complex order, Novi Sad J. Math., 48, № 1, 93 – 102 (2018), https://doi.org/10.30755/nsjom.06870 DOI: https://doi.org/10.30755/NSJOM.06870

M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18, 63 – 68 (1967), https://doi.org/10.2307/2035225 DOI: https://doi.org/10.1090/S0002-9939-1967-0206255-1

G. Murugusundaramoorthy, Subclasses of bi-univalent functions of complex order based on subordination conditions involving wright hypergeometric functions, J. Math. and Fundam. Sci., 47, 60 – 75 (2015), https://doi.org/10.5614/j.math.fund.sci.2015.47.1.5 DOI: https://doi.org/10.5614/j.math.fund.sci.2015.47.1.5

E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $| z| < 1$, Arch. Ration. Mech. and Anal., 32, 100 – 112 (1969), https://doi.org/10.1007/BF00247676 DOI: https://doi.org/10.1007/BF00247676

Z. Peng, G. Murugusundaramoorthy, T. Janani, Coefficient estimate of bi-univalent functions of complex order associated with the Hohlov operator, J. Complex Anal., 2014, Article 693908, (2014), p. 1 – 6, https://doi.org/10.1155/2014/693908 DOI: https://doi.org/10.1155/2014/693908

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

T. S. Taha, Topics in univalent function theory, Ph. D. Thesis, Univ. London (1981).

Q.-H. Xu, Y.-C. Gui, H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett., 25, № 6, 990 – 994 (2012), https://doi.org/10.1016/j.aml.2011.11.013 DOI: https://doi.org/10.1016/j.aml.2011.11.013

Q.-H. Xu, H.-G. Xiao, H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. and Comput., 218, 11461 – 11465 (2012), https://doi.org/10.1016/j.amc.2012.05.034 DOI: https://doi.org/10.1016/j.amc.2012.05.034

C. Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, G¨ottingen (1975).

W. C. Ma, D. Minda, A unified treatment of some special classes of functions, Proc. Conf. Complex Anal., Tianjin, 1992, Conf. Proc. Lect. Notes Anal., Int. Press, Cambridge, MA (1994), p. 157 – 169.