Kyoto Journal of Mathematics

Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions

Rintaro Ohno and Toshiyuki Sugawa

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this article, we discuss the coefficient regions of analytic self-maps of the unit disk with a prescribed fixed point. As an application, we solve the Fekete–Szegő problem for normalized concave functions with a pole in the unit disk.

Article information

Kyoto J. Math., Volume 58, Number 2 (2018), 227-241.

Received: 11 December 2015
Accepted: 17 June 2016
First available in Project Euclid: 9 June 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

Fekete–Szegő problem fixed point concave functions variability region


Ohno, Rintaro; Sugawa, Toshiyuki. Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions. Kyoto J. Math. 58 (2018), no. 2, 227--241. doi:10.1215/21562261-2017-0015.

Export citation


  • [1] F. G. Avkhadiev, C. Pommerenke, and K.-J. Wirths, On the coefficients of concave univalent functions, Math. Nachr. 271 (2004), 3–9.
  • [2] F. G. Avkhadiev, C. Pommerenke, and K.-J. Wirths, Sharp inequalities for the coefficients of concave schlicht functions, Comment. Math. Helv. 81 (2006), 801–807.
  • [3] F. G. Avkhadiev and K.-J. Wirths, Convex holes produce lower bounds for coefficients, Complex Var. Theory Appl. 47 (2002), 553–563.
  • [4] F. G. Avkhadiev and K.-J. Wirths, Concave schlicht functions with bounded opening angle at infinity, Lobachevskii J. Math. 17 (2005), 3–10.
  • [5] F. G. Avkhadiev and K.-J. Wirths, A proof of the Livingston conjecture, Forum Math. 19 (2007), 149–157.
  • [6] B. Bhowmik, S. Ponnusamy, and K.-J. Wirths, On the Fekete-Szegő problem for concave univalent functions, J. Math. Anal. Appl. 373 (2011), 432–438.
  • [7] J. H. Choi, Y. C. Kim, and T. Sugawa, A general approach to the Fekete-Szegö problem, J. Math. Soc. Japan 59 (2007), 707–727.
  • [8] J. Dieudonné, Recherches sur quelques problèmes relatifs aux polynômes et aux fonctions bornées d’une variable complexe, Ann. Sci. Éc. Norm. Supér. (3) 48 (1931), 247–358.
  • [9] P. L. Duren, Univalent Functions, Grundlehren Math. Wiss. 259, Springer, New York, 1983.
  • [10] J. A. Jenkins, On a conjecture of Goodman concerning meromorphic univalent functions, Michigan Math. J. 9 (1962), 25–27.
  • [11] R. Ohno, “Characterizations for concave functions and integral representations” in Topics in Finite or Infinite Dimensional Complex Analysis, Tohoku Univ. Press, Sendai, 2013, 203–216.
  • [12] R. Ohno and T. Sugawa, On the second Hankel determinant of concave functions, J. Anal. 23 (2015), 99–109.