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Coefficient estimates for a certain subclass of analytic and bi-univalent functions. (English) Zbl 1244.30033

Summary: We introduce and investigate an interesting subclass \(\mathcal{H}_{\Sigma}^{h,p}\) of analytic and bi-univalent functions in the open unit disk \(\mathbb{U}\). For functions belonging to the class \(\mathcal{H}_{\Sigma}^{h,p}\), we obtain estimates on the first two Taylor-Maclaurin coefficients \(|a_{2}|\) and \(|a_{3}|\). The results presented in this paper generalize and improve some recent work of the last author, A.K. Mishra and P. Gochhayat [ibid. 23, No. 10, 1188–1192 (2010; Zbl 1201.30020)].

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable

Citations:

Zbl 1201.30020
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References:

[1] Duren, P. L., (Univalent Functions. Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band 259 (1983), Springer-Verlag: Springer-Verlag New York, Berlin, Heidelberg and Tokyo)
[2] (Srivastava, H. M.; Owa, S., Current Topics in Analytic Function Theory (1992), World Scientific Publishing Company: World Scientific Publishing Company Singapore, New Jersey, London and Hong Kong) · Zbl 0976.00007
[3] Altıntaş, O.; Irmak, H.; Owa, S.; Srivastava, H. M., Coefficient bounds for some families of starlike and convex functions of complex order, Appl. Math. Lett., 20, 1218-1222 (2007) · Zbl 1139.30005
[4] Breaz, D.; Breaz, N.; Srivastava, H. M., An extension of the univalent condition for a family of integral operators, Appl. Math. Lett., 22, 41-44 (2009) · Zbl 1163.30304
[5] Owa, S.; Nunokawa, M.; Saitoh, H.; Srivastava, H. M., Close-to-convexity, starlikeness, and convexity of certain analytic functions, Appl. Math. Lett., 15, 63-69 (2002) · Zbl 1038.30011
[6] Srivastava, H. M.; Eker, S. S., Some applications of a subordination theorem for a class of analytic functions, Appl. Math. Lett., 21, 394-399 (2008) · Zbl 1138.30014
[7] Srivastava, H. M.; Xu, Q.-H.; Wu, G.-P., Coefficient estimates for certain subclasses of spiral-like functions of complex order, Appl. Math. Lett., 23, 763-768 (2010) · Zbl 1189.30041
[8] Xu, Q.-H.; Srivastava, H. M.; Li, Z., A certain subclass of analytic and close-to-convex functions, Appl. Math. Lett., 24, 396-401 (2011) · Zbl 1206.30035
[9] Lewin, M., On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18, 63-68 (1967) · Zbl 0158.07802
[10] (Brannan, D. A.; Clunie, J. G., Aspects of Contemporary Complex Analysis. Aspects of Contemporary Complex Analysis, (Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham; July 1-20, 1979) (1980), Academic Press: Academic Press New York and London)
[11] Netanyahu, E., The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in \(\mid z \mid < 1\), Arch. Rational Mech. Anal., 32, 100-112 (1969) · Zbl 0186.39703
[12] Brannan, D. A.; Taha, T. S., On some classes of bi-univalent functions, (Mazhar, S. M.; Hamoui, A.; Faour, N. S., Mathematical Analysis and Its Applications. Mathematical Analysis and Its Applications, Kuwait; February 18-21, 1985. Mathematical Analysis and Its Applications. Mathematical Analysis and Its Applications, Kuwait; February 18-21, 1985, KFAS Proceedings Series, vol. 3 (1988), Pergamon Press, Elsevier Science Limited: Pergamon Press, Elsevier Science Limited Oxford), 53-60, See also Studia Univ. Babeş-Bolyai Math. 31 (2) (1986) 70-77 · Zbl 0614.30017
[13] T.S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, 1981.; T.S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, 1981.
[14] Brannan, D. A.; Clunie, J.; Kirwan, W. E., Coefficient estimates for a class of star-like functions, Canad. J. Math., 22, 476-485 (1970) · Zbl 0197.35602
[15] Srivastava, H. M.; Mishra, A. K.; Gochhayat, P., Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23, 1188-1192 (2010) · Zbl 1201.30020
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