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A class of analytic functions with missing coefficients. (English) Zbl 1220.30028

Summary: Let \(T_n(A, B, \gamma, \alpha)\) (\(-1 \leq B < 1\), \( B < A\), \(0 < \gamma \leq 1\), and \(\alpha > 0 \)) denote the class of functions of the form \(f(z) = z + \sum^{\infty}_{k=n+1} a_kz^k\) (\(n \in \mathbb N^* = \{ 1, 2, 3, \dots \}\)), which are analytic in the open unit disk \(U\) and satisfy the following subordination condition:
\[ f'(z) + \alpha zf''(z) \prec \bigg(\frac{1 + Az}{1 + Bz}\bigg)^{\gamma}\quad\text{for}\quad z \in U, A \leq 1, 0 < \gamma < 1. \]
For the functions \(f(z)\) belonging to the class \(T_n(A, B, \gamma, \alpha)\), we obtain coefficient estimates and sharp bounds for
\[ \text{Re\,} f'(z), \quad \text{Re\,} \frac{f(z)}{z},\quad\text{and}\quad |f(z)|. \] Conditions for univalence and starlikeness, convolution properties, and the radius of convexity are also considered.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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[1] R. M. Ali, “On a subclass of starlike functions,” The Rocky Mountain Journal of Mathematics, vol. 24, no. 2, pp. 447-451, 1994. · Zbl 0816.30010 · doi:10.1216/rmjm/1181072410
[2] P. N. Chichra, “New subclasses of the class of close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 62, no. 1, pp. 37-43, 1977. · Zbl 0355.30013 · doi:10.2307/2041942
[3] H. Silverman, “A class of bounded starlike functions,” International Journal of Mathematics and Mathematical Sciences, vol. 17, no. 2, pp. 249-252, 1994. · Zbl 0796.30006 · doi:10.1155/S0161171294000360
[4] R. Singh and S. Singh, “Convolution properties of a class of starlike functions,” Proceedings of the American Mathematical Society, vol. 106, no. 1, pp. 145-152, 1989. · Zbl 0672.30007 · doi:10.2307/2047386
[5] S. Ponnusamy and V. Singh, “Criteria for strongly starlike functions,” Complex Variables, vol. 34, no. 3, pp. 267-291, 1997. · Zbl 0892.30005
[6] C. Y. Gao and S. Q. Zhou, “Certain subclass of starlike functions,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 176-182, 2007. · Zbl 1130.30010 · doi:10.1016/j.amc.2006.08.113
[7] H. M. Srivastava, N. E. Xu, and D. G. Yang, “Inclusion relations and convolution properties of a certain class of analytic functions associated with the Ruscheweyh derivatives,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 686-700, 2007. · Zbl 1111.30011 · doi:10.1016/j.jmaa.2006.09.019
[8] D. G. Yang and J. L. Liu, “On a class of analytic functions with missing coefficients,” Applied Mathematics and Computation, vol. 215, no. 9, pp. 3473-3481, 2010. · Zbl 1182.30025 · doi:10.1016/j.amc.2009.10.043
[9] Y. C. Kim, “Mapping properties of differential inequalities related to univalent functions,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 272-279, 2007. · Zbl 1115.30004 · doi:10.1016/j.amc.2006.08.124
[10] Y. C. Kim and H. M. Srivastava, “Some applications of a differential subordination,” International Journal of Mathematics and Mathematical Sciences, vol. 22, no. 3, pp. 649-654, 1999. · Zbl 0960.30009 · doi:10.1155/S016117129922649X
[11] H. M. Srivastava, D. G. Yang, and N. E. Xu, “Some subclasses of meromorphically multivalent functions associated with a linear operator,” Applied Mathematics and Computation, vol. 195, no. 1, pp. 11-23, 2008. · Zbl 1175.30028 · doi:10.1016/j.amc.2007.04.065
[12] P. L. Duren, Univalent Functions, vol. 259 of Fundamental Principles of Mathematical Sciences, Springer, New York, NY, USA, 1983. · Zbl 0514.30001
[13] S. S. Miller and P. T. Mocanu, “Second order differential inequalities in the complex plane,” Journal of Mathematical Analysis and Applications, vol. 65, no. 2, pp. 289-305, 1978. · Zbl 0367.34005 · doi:10.1016/0022-247X(78)90181-6
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