2013
DOI: 10.1017/s1446788712000559
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Holomorphic Functions With Image of Given logarithmic or Elliptic Capacity

Abstract: 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(… Show more

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Cited by 3 publications
(16 citation statements)
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“…The article [6] was a great source of inspiration for several mathematicians to prove monotonicity theorems. Such results can be found in [4], [7], [11], [27] and references therein.…”
Section: Introduction and Statement Of The Resultssupporting
confidence: 58%
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“…The article [6] was a great source of inspiration for several mathematicians to prove monotonicity theorems. Such results can be found in [4], [7], [11], [27] and references therein.…”
Section: Introduction and Statement Of The Resultssupporting
confidence: 58%
“…See for example [3], [4], [5], [6], [7], [8], [9], [11], [16,Chapter 4]. where the at most countably infinite set {z 1 (w), z 2 (w), .…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 99%
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