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UNIVALENCE CRITERIA AND ANALOGUES OF THE JOHN CONSTANT
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 3 / December 2013
- Published online by Cambridge University Press:
- 12 December 2012, pp. 423-434
- Print publication:
- December 2013
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- Article
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- You have access
- Export citation
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Let
$p(z)= z{f}^{\prime } (z)/ f(z)$ for a function 
$f(z)$ analytic on the unit disc 
$\mid z\mid \lt 1$ in the complex plane and normalised by 
$f(0)= 0, {f}^{\prime } (0)= 1$. We provide lower and upper bounds for the best constants 
${\delta }_{0} $ and 
${\delta }_{1} $ such that the conditions 
${e}^{- {\delta }_{0} / 2} \lt \mid p(z)\mid \lt {e}^{{\delta }_{0} / 2} $ and 
$\mid p(w)/ p(z)\mid \lt {e}^{{\delta }_{1} } $ for 
$\mid z\mid , \mid w\mid \lt 1$ respectively imply univalence of 
$f$ on the unit disc.
