An unified approach to the Fekete–Szegö problem S. Kanas Department of Mathematics, Rzeszow University of Technology, Institute of Mathematics, Maria Curie-Sklodowska University in Lublin, Poland article info Keywords: Keywords and phrases: univalent functions Fekete–Szegö problem Carathéodory class Conic sections abstract An unifying approach to the estimates of Fekete–Szegö functional ja 3  la 2 2 j are presented. Sharp bounds for that functional are found. References to a functions related to conic sec- tions are given. Ó 2012 Elsevier Inc. All rights reserved. 1. Background and bounds for functions with positive real part Assume that p ¼ 1 þ p 1 z þ is holomorphic and univalent functions with positive real part in the unit disk U, pðUÞ is symmetric with respect to real axis and p 0 ð0Þ¼ p 1 > 0. Next, let PðpÞ be a family of functions u, holomorphic in U such that uð0Þ¼ 1 and uðUÞ  pðUÞ, that is u  p. It is clear that PðpÞ  P ¼ Pðð1 þ zÞ=ð1  zÞÞ, the well known class of Carathéodory functions. Here and hereafter, we use the notation h  g (or hðzÞ gðzÞ) in U for analytic functions to mean the subordina- tions, namely that there exists the Schwarz function x of the unit disk onto itself with xð0Þ¼ 0 such that hðzÞ¼ gðxðzÞÞ. We note that if g is univalent, h  g is equivalent to hð0Þ¼ gð0Þ and hðUÞ  gðUÞ. Ma and Minda proved the following result for the class P. Lemma 1.1 [18]. If pðzÞ¼ 1 þ p 1 z þ p 2 z 2 þ is a function with positive real part, then for any real number l, jp 2  l p 2 1 j 6 2 maxf1; j2l  1jg ð1:1Þ and the result is sharp for the functions given by pðzÞ¼ 1 þ z 2 1  z 2 pðzÞ¼ 1 þ z 1  z : In the case, when l < 0 or l > 1, inequality holds if and only if pðzÞ is ð1 þ zÞ=ð1  zÞ or one of its rotations. For 0 < l < 1 the equality holds if and only if pðzÞ is ð1 þ z 2 Þ=ð1  z 2 Þ or one of its rotations. Inequality becomes equality when l ¼ 0 if and only if pðzÞ¼ 1 þ k 2 1 þ z 1  z þ 1  k 2 1  z 1 þ z ; 0 6 k 6 1 or one of its rotations. While for l ¼ 1, equality holds if and only if pðzÞ is the reciprocal of one of the functions such that equality holds in the case of l ¼ 0. Remark 1.1. Although the above upper bounds are sharp, the authors ascertain that (1.1) can be improved in some cases, for instance jp 2  lp 2 1 jþ ljp 1 j 2 6 2; 0 < l 6 1=2 ð1:2Þ 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.01.070 E-mail address: skanas@prz.rzeszow.pl Applied Mathematics and Computation 218 (2012) 8453–8461 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc