Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Coefficient differences and Hankel determinants of areally mean $ p$-valent functions


Author: James W. Noonan
Journal: Proc. Amer. Math. Soc. 46 (1974), 29-37
MSC: Primary 30A36
DOI: https://doi.org/10.1090/S0002-9939-1974-0352440-1
MathSciNet review: 0352440
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: With $ p > 0$, denote by $ {S_p}$ the class of functions analytic and areally mean $ p$-valent in the open unit disc. If $ f \in {S_p}$, it is well known that $ \alpha (f) = {\lim _{r \to 1}}{(1 - r)^{2p}}M(r,f)$ exists and is finite. If $ q \geq 1$ is an integer, denote the $ q$ Hankel determinant of $ f$ by $ {H_q}(n,f)$. In this paper the asymptotic behavior of $ {H_q}(n,f)$, as $ n \to \infty $, is related to $ \alpha (f)$. A typical result is: if $ \alpha (f) > 0$, and if $ p > q - 3/4$, then

$\displaystyle \vert{H_q}(n,f)\vert/{n^{2pq - {q^2}}} \sim \vert{Q_q}(p)\vert{(\alpha (f)/\Gamma (2p))^q},$

where $ {Q_q}$ is a polynomial of degree at most $ {q^2} - q$. In the course of the proof, asymptotic results are proved concerning certain coefficient differences, and in particular concerning $ \vert{a_n}\vert - \vert{a_{n - 1}}\vert$.

References [Enhancements On Off] (What's this?)

  • [1] B. G. Eke, The asymptotic behaviour of areally mean valent functions, J. Analyse Math. 20 (1967), 147-212. MR 36 #5331. MR 0222279 (36:5331)
  • [2] W. K. Hayman, The asymptotic behavior of $ p$-valent functions, Proc. London Math. Soc. (3) 5 (1955), 257-284. MR 17, 142. MR 0071536 (17:142j)
  • [3] -, On successive coefficients of univalent functions, J. London Math. Soc. 38 (1963), 228-243. MR 26 #6382. MR 0148885 (26:6382)
  • [4] -, On the second Hankel determinant of mean univalent functions, Proc. London Math. Soc. (3) 18 (1968), 77-94. MR 36 #2794. MR 0219715 (36:2794)
  • [5] -, Multivalent functions, Cambridge Tracts in Math. and Math. Phys., no. 48, Cambridge Univ. Press, Cambridge, 1958. MR 21 #7302. MR 0108586 (21:7302)
  • [6] -, Research problems in function theory, Athlone Press, London, 1967. MR 36 #359. MR 0217268 (36:359)
  • [7] K. W. Lucas, On successive coefficients of areally mean $ p$-valent functions, J. London Math. Soc. 44 (1969), 631-642. MR 39 #4379. MR 0243055 (39:4379)
  • [8] J. W. Noonan and D. K. Thomas, On the Hankel determinants of areally mean $ p$-valent functions, Proc. London Math. Soc. (3) 25 (1972), 503-524. MR 46 #5605. MR 0306479 (46:5605)
  • [9] Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. 41 (1966), 111-122. MR 32 #2575. MR 0185105 (32:2575)
  • [10] -, On the Hankel determinants of univalent functions, Mathematika 14 (1967), 108-112. MR 35 #6811. MR 0215976 (35:6811)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30A36

Retrieve articles in all journals with MSC: 30A36


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0352440-1
Keywords: Areally mean $ p$-valent, Hankel determinants, coefficient differences
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society