# Integral representations of bounded starlike functions

Annales Polonici Mathematici (1995)

- Volume: 60, Issue: 3, page 289-297
- ISSN: 0066-2216

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topFrode Rønning. "Integral representations of bounded starlike functions." Annales Polonici Mathematici 60.3 (1995): 289-297. <http://eudml.org/doc/262370>.

@article{FrodeRønning1995,

abstract = {For α ≥ 0 let $ℱ_α$ denote the class of functions defined for |z| < 1 by integrating $1/(1-xz)^α$ if α > 0, and log(1/(1-xz)) if α = 0, against a complex measure on |x| = 1. We study families of starlike functions where zf’(z)/f(z) ranges over a parabola with given focus and vertex. We prove a number of properties of these functions, among others that they are bounded and that they belong to $ℱ_0$. In general, it is only known that bounded starlike functions belong to $ℱ_α$ for α > 0.},

author = {Frode Rønning},

journal = {Annales Polonici Mathematici},

keywords = {Cauchy-Stieltjes integrals; starlike functions; bounded starlike functions},

language = {eng},

number = {3},

pages = {289-297},

title = {Integral representations of bounded starlike functions},

url = {http://eudml.org/doc/262370},

volume = {60},

year = {1995},

}

TY - JOUR

AU - Frode Rønning

TI - Integral representations of bounded starlike functions

JO - Annales Polonici Mathematici

PY - 1995

VL - 60

IS - 3

SP - 289

EP - 297

AB - For α ≥ 0 let $ℱ_α$ denote the class of functions defined for |z| < 1 by integrating $1/(1-xz)^α$ if α > 0, and log(1/(1-xz)) if α = 0, against a complex measure on |x| = 1. We study families of starlike functions where zf’(z)/f(z) ranges over a parabola with given focus and vertex. We prove a number of properties of these functions, among others that they are bounded and that they belong to $ℱ_0$. In general, it is only known that bounded starlike functions belong to $ℱ_α$ for α > 0.

LA - eng

KW - Cauchy-Stieltjes integrals; starlike functions; bounded starlike functions

UR - http://eudml.org/doc/262370

ER -

## References

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- [8] W. Ma and D. Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165-175. Zbl0760.30004
- [9] T. H. MacGregor, Analytic and univalent functions with integral representation involving complex measures, Indiana Univ. Math. J. 36 (1987), 109-130. Zbl0644.30009
- [10] F. Rønning, A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A, to appear. Zbl0879.30004
- [11] F. Rønning, On starlike functions associated with parabolic regions, ibid. 45 (14) (1991), 117-122. Zbl0769.30011
- [12] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189-196. Zbl0805.30012

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