Hostname: page-component-7c8c6479df-fqc5m Total loading time: 0 Render date: 2024-03-28T09:34:50.219Z Has data issue: false hasContentIssue false

The range of a continuous linear functional over a class of functions defined by subordination

Published online by Cambridge University Press:  18 May 2009

Richard Fournier
Affiliation:
Département de Mathématiques et de StatistiqueUniversité de MontréalMontréal, QuebecCanadaH3C 3J7
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Δ = {z ∈ ℂ ⃒ ⃓z⃓ <1) and H(Δ) the set of analytic functions on Δ. We recall the definition of subordination between two functions, say ƒ and g, analytic on Δ: this means that f(0)= g(0) and there is a function ρ ∈ H (Δ) such that ρ(0) = 0, ⃒ ρ(z)⃒<1 if z ∈ Δ, and f(z) ≡ g(ρ(z)). Subordination between f and g will be denoted by

f<g. The Hadamard product (or convolution) of two functions and in H(Δ)is the function f * g ∈ H(Δ)definedas f * g (z)= .

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

REFERENCES

1.Ahlfors, L., Conformal invariants (McGraw-Hill, 1973).Google Scholar
2.Brickman, L., MacGregor, T. H. and Wilken, D. R., Convex hulls of some classical families of univalent functions, Trans. Amer. Math. Soc., 156 (1971), 91107.CrossRefGoogle Scholar
3.Duren, P. L., Univalent functions (Springer Verlag, 1983).Google Scholar
4.Fournier, R., On integrals of bounded analytic functions in the closed unit disc, Complex Variables, 11 (1989), 125133.Google Scholar
5.Hallenbeck, D. J. and MacGregor, T. H., Linear problems and convexity techniques in geometric function theory (Pitman, 1984).Google Scholar
6.Hallenbeck, D. J. and Ruscheweyh, St., Subordination by convex functions, Proc. Amer. Math. Soc., 52 (1975), 191195.CrossRefGoogle Scholar
7.Ruscheweyh, St. and Sheil-Small, T., Hadamard products of schlichtfunctions and the Pólya-Schoenberg conjecture, Comm. Math. Helvet., 48 (1973), 119135.CrossRefGoogle Scholar