Extreme points and convolution properties of some classes of multivalent functions (Q1912336)

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 \[ G(z)={z^p+{(n+p)x+n+p(2\alpha-1)\over 2p(1-\alpha)} z^{p+1}\over (1-z)^{n+p+1}}\quad \text{and } |x|=1. \]