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HOLOMORPHIC FUNCTIONS WITH IMAGE OF GIVEN LOGARITHMIC OR ELLIPTIC CAPACITY
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 94 / Issue 2 / April 2013
- Published online by Cambridge University Press:
- 08 March 2013, pp. 145-157
- Print publication:
- April 2013
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- Article
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For holomorphic functions
$f$ in the unit disk 
$ \mathbb{D} $ with 
$f(0)= 0$, we prove a modulus growth bound involving the logarithmic capacity (transfinite diameter) of the image. We show that the pertinent extremal functions map the unit disk conformally onto the interior of an ellipse. We prove a modulus growth bound for elliptically schlicht functions in terms of the elliptic capacity 
${\mathrm{d} }_{\mathrm{e} } f( \mathbb{D} )$ of the image. We also show that the function 
${\mathrm{d} }_{\mathrm{e} } f(r \mathbb{D} )/ r$ is increasing for 
$0\lt r\lt 1$.
