Article
Kyungpook Mathematical Journal 2020; 60(3): 477-484
Published online September 30, 2020
Copyright © Kyungpook Mathematical Journal.
On the Fekete-SzegÖ Problem for Starlike Functions of Complex Order
Hanan Darwish and Abdel-Moniem Lashin, Bashar Al Saeedi*
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516 Egypt
e-mail : darwish333@yahoo.com and aylashin@mans.edu.eg
Department of Mathematics, Faculty of Science & Engineering, Manchester Metropolitan University, Manchester, M15GD, UK
e-mail : basharfalh@yahoo.com
Received: January 28, 2016; Revised: May 3, 2020; Accepted: May 4, 2020
For a non-zero complex number
Keywords: coefficient estimates, Salagean operator, Fekete Szego problem, starlike functions of complex order
1. Introduction
Fekete and Szegö proved the remarkable result that the estimate
holds with 0 ≤
in the open unit disk
in the unit disk represents various geometric quantities. For example, when
where
This is quite natural to discuss the behavior of
We denote by
and
The notions of
Observe that
Let
For a function
and
With the above operator
for some
We note that
Definition 1.1
Let
By giving specific values to
2. Main Results
We denote by ℘ class of the analytic functions in
Lemma 2.1.([5], p.166)
If
If
and
Theorem 2.2
Denote
By the definition of the class Ψ
which implies the equality
Equating the coefficients of both sides we have
so that, on account of
Taking into account (
and
Thus
Moreover
as asserted.
Remark 2.3
Putting
Corollary 2.4
Remark 2.5
In the above Theorem 2.2 and Corollary 2.4 a special case of Fekete- Szegö problem e.g. for real
Now, we consider functional
Theorem 2.6
Applying (
Then, with the aid of Lemma 2.1, we obtain
An examination of the proof shows that equality is attained for the first case, when
and, for the second case, when
respectively.
Remark 2.7
Putting
Corollary 2.8
We next consider the case, when
Theorem 2.9
First, let
Let, now
Finally, if
Equality is attained for the second case on choosing
Remark 2.10
Put
Corollary 2.11
- M. Darus, and DK. Thomas.
On the Fekete-Szegö theorem for close-to-convex functions . Math Japon.,44 (1996), 507-511. - M. Darus, and DK. Thomas.
On the Fekete-Szegö theorem for close-to-convex functions . Math Japon.,47 (1998), 125-132. - MA. Nasr, and MK. Aouf.
On convex functions of complex order . Mansoura Sci Bull.,(1982), 565-582. - MA. Nasr, and MK. Aouf.
Starlike function of complex order . J Natur Sci Math.,25 (1985), 1-12. - C. Pommerenke.
Univalent Functions . Studia Mathematica/Mathematische Lehrbucher, Vandenhoeck and Ruprecht,, 1975. - GS. Salagean.
Subclasses of univalent functions . Lecture Notes in Math,1013 , Springer-Verlag, Berlin-Heidelberg-New York, 1983:292-310. - T. Sekine.
Generalization of certain subclasses of analytic functions . Internat J Math Math Sci.,10 (4)(1987), 725-732. - P. Wiatrowski.
The coefficients of a certain family of holomorphic functions . Zeszyty Nauk Univ Lodz, Nauki Mat Przyrod Ser II.,39 (1971), 75-85.