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Extreme points and convolution properties of some classes of multivalent functions. (English) Zbl 0858.30013

Throughout each function \(f\) is assumed to be analytic in the open unit disk and have a power series of the form \(f(z)= z^p+\sum^\infty_{k=p+1}a_kz^k\), where \(p\) is a positive integer. For \(0\leq \alpha<1\) let \(S^*_p(\alpha)\) denote the set of \(p\)-valent starlike functions of order \(\alpha\) in the form of \(f\). For \(n\geq-p+1\) let \(D^{n+p-1}f(z)={z^p\over(1-z)^{n+p}}* f(z)\), where \(*\) indicates the Hadamard product of the two series. Let \({\mathcal R}_n(p,\alpha)\) denote the set of functions \(f\) satisfying \(\text{Re }F(z)>p\alpha\) and \[ \int^\pi_{-\pi} \text{Re }F(z)d\theta=2\pi p\;(z=re^{i\theta},\;0<r<1), \] where \(F(z)={z\{D^{n+p-1}f(z)\}'\over D^{n+p-1}f(z)}\). Also, let \(K_n(p)\) denote the \(p\)-valent functions \(f\) such that
\(\text{Re}\{{D^{n+p}f(z)\over D^{n+p-1}f(z)}\}>1/2\) for \(|z|<1\).
In this paper, the extreme points are determined for the closed convex hulls of \({\mathcal R}_n(p,\alpha)\) as well as for \(K_n(p)\). It is also shown that the sets \({\mathcal R}_n(p,\alpha)\) are nondecreasing in \(n\). Another result asserts that each function in \({\mathcal R}_n(p,0)\) is also \(p\)-valent convex of order \(\alpha\) whenever \(n\geq{2(p+1)^4\over p(1-\alpha)}\). Also \(f\in {\mathcal R}_n(p,\alpha)\) if and only if \(f(z)*G(z)\neq 0\) for \(0<|z|<1\), where \[ G(z)={z^p+{(n+p)x+n+p(2\alpha-1)\over 2p(1-\alpha)} z^{p+1}\over (1-z)^{n+p+1}}\quad \text{and } |x|=1. \]

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C99 Geometric function theory
30C55 General theory of univalent and multivalent functions of one complex variable
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