Subclasses of \(\lambda\)-bi-pseudo-starlike functions with respect to symmetric points based on shell-like curves

  • H. Özlem Güney Dicle University, Faculty of Science, Department of Mathematics, Diyarbakır, Turkey.
  • G. Murugusundaramoorthy School of Advanced Sciences, Vellore Institute of Technology, Vellore -632014, India.
  • K. Vijaya School of Advanced Sciences, Vellore Institute of Technology, Vellore -632014, India.
Keywords: Analytic functions, bi-univalent, shell-like curve, Fibonacci numbers, starlike functions

Abstract

In this paper we define the subclass \(\mathcal{PSL}^\lambda_{s,\Sigma}(\alpha,\tilde{p}(z))\) of the class \(\Sigma\) of bi-univalent functions defined in the unit disk, called \(\lambda\)-bi-pseudo-starlike, with respect to symmetric points, related to shell-like curves connected with Fibonacci numbers. We determine the initial Taylor-Maclaurin coefficients \(|a_2|\) and \(|a_3|\) for functions \(f\in\mathcal{PSL}^\lambda_{s,\Sigma}(\alpha,\tilde{p}(z)).\) Further we determine the Fekete-Szegö result for the function class \(\mathcal{PSL}^\lambda_{s,\Sigma}(\alpha,\tilde{p}(z))\) and for the special cases \(\alpha=0\), \(\alpha=1\) and \(\tau =-0.618\) we state corollaries improving the initial Taylor-Maclaurin coefficients \(|a_2|\) and \(|a_3|\).

References

K. O. Babalola, “On λ-pseudo-starlike functions”, J. Class. Anal., vol. 3, no. 2, pp. 137–147, 2013.

D. A. Brannan, J. Clunie and W. E. Kirwan, “Coefficient estimates for a class of star-like functions”, Canad. J. Math., vol. 22, no. 3, pp. 476-485, 1970.

D. A. Brannan and T. S. Taha, “On some classes of bi-univalent functions”, Studia Univ. Babes-Bolyai Math., vol. 31, no. 2, pp. 70-77, 1986.

P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band 259, New York, Berlin, Heidelberg and Tokyo: Springer-Verlag, 1983.

S. Joshi, S. Joshi and H. Pawar, “On some subclasses of bi-univalent functions associated with pseudo-starlike function”, J. Egyptian Math. Soc., vol. 24, no. 4, pp. 522-525, 2016.

J. Dziok, R. K. Raina and J. Sokól, “On α−convex functions related to a shell-like curve connected with Fibonacci numbers”, Appl. Math. Comput., vol. 218, no. 3, pp. 996–1002, 2011.

M. Fekete and G. Szegö, “Eine Bemerkung über ungerade Schlichte Funktionen”, J. London Math. Soc., vol. 8, no. 2, pp. 85-89, 1933.

S. S. Miller and P. T. Mocanu Differential Subordinations Theory and Applications, Series of Monographs and Text Books in Pure and Applied Mathematics, vol. 225, New York: Marcel Dekker, 2000.

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

Ch. Pommerenke, Univalent Functions, Math. Math, Lehrbucher, Vandenhoeck and Ruprecht, Göttingen, 1975.

R. K. Raina and J. Sokól, “Fekete-Szegö problem for some starlike functions related to shell- like curves”, Math. Slovaca, vol. 66, no. 1, pp. 135-140, 2016.

V. Ravichandran, “Starlike and convex functions with respect to conjugate points”, Acta Math. Acad. Paedagog. Nyházi. (N.S.), vol. 20, no. 1, pp. 31-37, 2004.

K. Sakaguchi, “On a certain univalent mapping”, J. Math. Soc. Japan, vol. 11, no. 1, pp. 72-75, 1959.

J. Sokól, “On starlike functions connected with Fibonacci numbers”, Zeszyty Nauk. Politech. Rzeszowskiej Mat, vol. 23, pp. 111-116, 1999.

H. M. Srivastava, A. K. Mishra and P. Gochhayat, “Certain subclasses of analytic and bi-univalent functions”, Appl. Math. Lett., vol. 23, no. 10, pp. 1188-1192, 2010.

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., vol. 25, no. 6, pp. 990-994, 2012.

X.-F. Li and A.-P. Wang, “Two new subclasses of bi-univalent functions”, Int. Math. Forum, vol. 7, no. 30, pp. 1495-1504, 2012.

G. Wang, C. Y. Gao and S. M. Yuan, “On certain subclasses of close-to-convex and quasi-convex functions with respect to k−symmetric points”, J. Math. Anal. Appl., vol. 322, no. 1, pp. 97–106, 2006.

P. Zaprawa, “On the Fekete-Szegö problem for classes of bi-univalent functions”, Bull. Belg. Math. Soc. Simon Stevin, vol. 21, no. 1, pp. 169-178, 2014.

Published
2021-08-01
How to Cite
[1]
H. Özlem Güney, G. Murugusundaramoorthy, and K. Vijaya, “Subclasses of \(\lambda\)-bi-pseudo-starlike functions with respect to symmetric points based on shell-like curves”, CUBO, vol. 23, no. 2, pp. 299–312, Aug. 2021.
Section
Articles