On the order of a starlike function
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- by F. Holland and D. K. Thomas
- Trans. Amer. Math. Soc. 158 (1971), 189-201
- DOI: https://doi.org/10.1090/S0002-9947-1971-0277705-5
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Abstract:
It is shown that if $f \in S$, the class of normalised starlike functions in the unit $\operatorname {disc} \Delta$, then \[ ({\text {i}})\quad \quad \lim \limits _{r \to 1 - } \frac {{\log {P_\lambda }(r)}}{{ - \log (1 - r)}} = \alpha \lambda {\text { for }}\lambda > 0;\] \[ ({\text {ii}})\quad \quad \lim \limits _{r \to 1 - } \frac {{\log ||{f_r}|{|_p}}}{{ - \log (1 - r)}} = \alpha p - 1{\text { for }}\alpha p > 1;\] and \[ ({\text {iii}})\quad \lim \limits _{r \to 1 - } \frac {{\log ||f’_r||_{p}}}{{ - \log (1 - r)}} = (1 + \alpha )p - 1\quad {\text {for (1 + }}\alpha )p > 1,\] where ${P_\lambda }(r) = \Sigma _{n = 1}^\infty {n^{\lambda - 1}}|{a_n}{|^\lambda }{r^n},({a_n})$ is the sequence of coefficients and $\alpha$ the order of $f$, and where \[ ||{f_r}|{|_p} = \frac {1}{{2\pi }}\int _0^{2\pi } {|f(r{e^{i\theta }})} {|^p}d\theta .\] The results extend work of Pommerenke. The methods of the paper yield various other results, one in particular being \[ \lim \sup \limits _{n \to \infty } \frac {{{{\log }^ + }n|{a_n}|}}{{\log n}} = \alpha \], a result which has an analogy in the theory of entire functions.References
- J. Clunie and F. R. Keogh, On starlike and convex schlicht functions, J. London Math. Soc. 35 (1960), 229–233. MR 110814, DOI 10.1112/jlms/s1-35.2.229
- J. Clunie and Ch. Pommerenke, On the coefficients of close-to-convex univalent functions, J. London Math. Soc. 41 (1966), 161–165. MR 190321, DOI 10.1112/jlms/s1-41.1.161
- G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
- W. K. Hayman, Multivalent functions, Cambridge Tracts in Mathematics and Mathematical Physics, No. 48, Cambridge University Press, Cambridge, 1958. MR 0108586
- W. K. Hayman, On functions with positive real part, J. London Math. Soc. 36 (1961), 35–48. MR 150310, DOI 10.1112/jlms/s1-36.1.35
- R. R. London and D. K. Thomas, An area theorem for starlike functions, Proc. London Math. Soc. (3) 20 (1970), 734–748. MR 262481, DOI 10.1112/plms/s3-20.4.734
- Ch. Pommerenke, On starlike and convex functions, J. London Math. Soc. 37 (1962), 209–224. MR 137830, DOI 10.1112/jlms/s1-37.1.209
- Christian Pommerenke, On starlike and close-to-convex functions, Proc. London Math. Soc. (3) 13 (1963), 290–304. MR 145061, DOI 10.1112/plms/s3-13.1.290 T. B. Sheil-Small, On starlike univalent functions, Ph.D. Thesis, Imperial College, London, 1965. E. C. Titchmarsh, The theory of functions, Oxford Univ. Press, Oxford, 1960.
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 158 (1971), 189-201
- MSC: Primary 30.42
- DOI: https://doi.org/10.1090/S0002-9947-1971-0277705-5
- MathSciNet review: 0277705