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Estimates for the Koebe Constant and the Second Coefficient for Some Classes of Univalent Functions

Published online by Cambridge University Press:  20 November 2018

D. Bshouty ,
W. Hengartner  and
G. Schober
D. Bshouty
Affiliation:
The Technion, Haifa, Israel
W. Hengartner
Affiliation:
Université Laval, Québec, Québec
G. Schober
Affiliation:
Indiana University, Bloomington, Indiana
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Let S be the set of all normalized univalent analytic functions ƒ(z) = z + a2z2 + … in the open unit disk U. Then ƒ(U) contains the disk . Here is the best possible constant and is referred to as the Koebe constant for S. On the other extreme, ƒ(U) cannot contain the disk {|w| < 1}; unless ƒ is the identity mapping.

In order to interpolate between the class S and the identity mapping, one may introduce the families , of functions ƒ ∈ S such that ƒ(U) contains the disk {|w| < d};. Then S(d1) ⊃ S(d2) for d1 < d2, and S(1) contains only the identity mapping. It is obvious that d is the “Koebe constant” for S(d). The relation between d and the second coefficient a2 has been studied by E. Netanyahu [5, 6].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1311 - 1324
Copyright
Copyright © Canadian Mathematical Society 1980

References

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