## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### An extension of necessary and sufficient conditions for concave functions

Rintaro Ohno

#### Abstract

In the present article we discuss necessary and sufficient conditions for concave functions, i.e. meromorphic functions which map the unit disk conformally on a domain whose complement is convex. The conditions will be given with respect to an arbitrary point $p\in (-1,1)$. We will also look at representation formulas for the related functions as well as an application of the derived formula.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 1 (2015), 1-4.

Dates
First available in Project Euclid: 5 January 2015

https://projecteuclid.org/euclid.pja/1420466269

Digital Object Identifier
doi:10.3792/pjaa.91.1

Mathematical Reviews number (MathSciNet)
MR3296590

Zentralblatt MATH identifier
1317.30034

#### Citation

Ohno, Rintaro. An extension of necessary and sufficient conditions for concave functions. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 1, 1--4. doi:10.3792/pjaa.91.1. https://projecteuclid.org/euclid.pja/1420466269

#### References

• F. G. Avkhadiev, Ch. Pommerenke and K.-J. Wirths, Sharp inequalities for the coefficients of concave schlicht functions, Comment. Math. Helv. 81 (2006), no. 4, 801–807.
• F. G. Avkhadiev and K.-J. Wirths, A proof of the Livingston conjecture, Forum Math. 19 (2007), no. 1, 149–157.
• B. Bhowmik, S. Ponnusamy and K.-J. Wirths, Domains of variability of Laurent coefficients and the convex hull for the family of concave univalent functions, Kodai Math. J. 30 (2007), no. 3, 385–393.
• A. E. Livingston, Convex meromorphic mappings, Ann. Polon. Math. 59 (1994), no. 3, 275–291.
• R. Ohno, Characterizations for concave functions and integral representations, in Topics in finite or infinite dimensional complex analysis, Tohoku University Press, Sendai, 2013, pp. 203–216.
• R. Ohno and H. Yanagihara, On a coefficient body for concave functions, Comput. Methods Funct. Theory 13 (2013), no. 2, 237–251.
• J. A. Pfaltzgraff and B. Pinchuk, A variational method for classes of meromorphic functions, J. Analyse Math. 24 (1971), 101–150.