On the Fekete-Szegö problem for a class of analytic functions. (English) Zbl 1286.30015

Summary: Let \(\mathcal A\) denote the class of functions which are analytic in the unit disk \(D=\{z:|z|<1\}\) and given by the power series \(f(z)=z+\sum^\infty_{n=2}a_nz^n\). Let \(C\) be the class of convex functions. In this paper, we give the upper bounds of \(|a_3-\mu a^2_2|\) for all real number \(\mu\) and for any \(f(z)\) in the family \(\mathcal V=\{f(z):f \in \mathcal A, \mathrm{Re}(f(z)/g(z))>0 \text{ for some } g\in C\}\).


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
Full Text: DOI


[1] M. Fekete and G. Szegö, “Eine bemerkung über ungerade schlichte funktionen,” The Journal of the London Mathematical Society, vol. 8, no. 2, pp. 85-89, 1933. · Zbl 0006.35302
[2] F. R. Keogh and E. P. Merkes, “A coefficient inequality for certain classes of analytic functions,” Proceedings of the American Mathematical Society, vol. 20, pp. 8-12, 1969. · Zbl 0165.09102
[3] H. R. Abdel-Gawad and D. K. Thomas, “The Fekete-Szegö problem for strongly close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 114, no. 2, pp. 345-349, 1992. · Zbl 0741.30008
[4] R. R. London, “Fekete-Szegö inequalities for close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 117, no. 4, pp. 947-950, 1993. · Zbl 0771.30007
[5] N. E. Cho, “On the Fekete-Szegö problem and argument inequality for strongly quasi-convex functions,” Bulletin of the Korean Mathematical Society, vol. 38, no. 2, pp. 357-367, 2001. · Zbl 0979.30008
[6] S. Kanas and H. E. Darwish, “Fekete-Szegö problem for starlike and convex functions of complex order,” Applied Mathematics Letters, vol. 23, no. 7, pp. 777-782, 2010. · Zbl 1189.30021
[7] M. H. Al-Abbadi and M. Darus, “The Fekete-Szegö theorem for a certain class of analytic functions,” Sains Malaysiana, vol. 40, no. 4, pp. 385-389, 2011. · Zbl 1225.30010
[8] H. Orhan, E. Deniz, and D. Raducanu, “The Fekete-Szegö problem for subclasses of analytic functions defined by a differential operator related to conic domains,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 283-295, 2010. · Zbl 1189.30049
[9] K. Al-Shaqsi and M. Darus, “On Fekete-Szegö problems for certain subclass of analytic functions,” Applied Mathematical Sciences, vol. 2, no. 9-12, pp. 431-441, 2008. · Zbl 1151.30311
[10] B. Bhowmik, S. Ponnusamy, and K.-J. Wirths, “On the Fekete-Szegö problem for concave univalent functions,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 432-438, 2011. · Zbl 1202.30015
[11] H. M. Srivastava, A. K. Mishra, and M. K. Das, “The Fekete-Szegö problem for a subclass of close-to-convex functions,” Complex Variables and Elliptic Equations, vol. 44, no. 2, pp. 145-163, 2001. · Zbl 1021.30014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.