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

Noonan, James W.

doi:10.1090/S0002-9939-1974-0352440-1

Coefficient differences and Hankel determinants of areally mean $p$-valent functions
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by James W. Noonan PDF

Proc. Amer. Math. Soc. 46 (1974), 29-37 Request permission

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 \[ |{H_q}(n,f)|/{n^{2pq - {q^2}}} \sim |{Q_q}(p)|{(\alpha (f)/\Gamma (2p))^q},\] 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 $|{a_n}| - |{a_{n - 1}}|$.

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