Abstract
Let \(z_0\) and \(w_0\) be given points in the open unit disk \(\mathbb {D}\) with \(|w_0| < |z_0|\), and \(\mathcal {H}_0\) be the class of all analytic self-maps f of \(\mathbb {D}\) normalized by \(f(0)=0\). In this paper, we establish the third-order Dieudonné’s Lemma and apply it to explicitly determine the variability region \(\{f'''(z_0): f\in \mathcal {H}_0,f(z_0) =w_0, f'(z_0)=w_1\}\) for given \(z_0,w_0,w_1\) and give the form of all the extremal functions.
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Acknowledgements
The author would like to thank Professor Toshiyuki Sugawa for his helpful comments and valuable suggestions. Especially, I would like to thank Professor Hiroshi Yanagihara for his good comments. The author is also grateful to Professor Ming Li for her helpful suggestions. Finally, the author would like to thank the anonymous referee for careful reading and helpful comments.
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Communicated by V. Ravichandran.
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Chen, G. Variability Regions for the Third Derivative of Bounded Analytic Functions. Bull. Malays. Math. Sci. Soc. 44, 4175–4194 (2021). https://doi.org/10.1007/s40840-021-01162-3
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DOI: https://doi.org/10.1007/s40840-021-01162-3