Abstract
Let f be analytic and f′(z) ≠ 0 in D and let \(A_{f}(z)={1-\mid z \mid^{2}\over 2}{f^{\prime \prime}(z)\over f\prime(z)}-{\overline z}\} {\rm for}\ z \ \epsilon\ D \) Many properties of f can be described by the (linear-invariant) order \({\rm sup}\mid A_{f}(z)\mid\atop \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!z\in {\rm D}\) The work of Avkhadiev and Wirths led to the introduction of the lower order of f defined by infz∈D ¦A f(z)¦. It is perhaps a surprise that there are many (necessarily unbounded) functions of positive lower order. This paper studies some properties of these functions.
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Dedicated to Walter Hayman FRS on his eightieth birthday
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Pommerenke, C. On the Lower Order of Locally Univalent Functions. Comput. Methods Funct. Theory 8, 373–384 (2008). https://doi.org/10.1007/BF03321694
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DOI: https://doi.org/10.1007/BF03321694