On a class of \(p\)-valent analytic functions (Q2732373)

Let \(A(p)\) denote the class of functions \(f\) analytic in \(E=\{z: |z|<1\}\) of the form \(f(z)=z^p +\sum^\infty_{m=p+1} a_mz^m\) \((p\in \mathbb{N} =\{1,2,3, \dots\})\). Let \(I^\sigma\) \((\sigma\) real) be the operator on \(f\) in \(A(p)\) defined by NEWLINE\[NEWLINEI^\sigma f(z)=z^p+ \sum^\infty_{m=p+1} \left({m+1 \over p+1} \right)^{-\sigma}a_mz^m,NEWLINE\]NEWLINE and let \(T_\sigma (p,\alpha)\) denote the subclass \(q\) functions \(f\) in \(A(p)\) satisfying NEWLINE\[NEWLINE\text{Re} \biggl\{z \bigl( I^\sigma (f(z) \bigr)/I^\sigma f(z) \biggr\}> p\alpha, \quad 0\leq \alpha<1,\;z \in E.NEWLINE\]NEWLINE In this paper the author mainly determines the extreme points of the closed convex hull of \(T_\sigma(p,\alpha)\), which are, then, used to derive coefficient bounds. Among certain other properties of \(T_\sigma(p, \alpha)\) proved the following is worth mentioning: If \(\beta\neq 0\) is a complex number satisfying either \(|2\beta|p(1-\alpha) -1|\leq 1\) or \(|2\beta p(1- \alpha)+1 |\leq 1\) \(({I^\sigma f(z) \over z^p})^\beta \prec{1 \over (1-z)^{2p \beta(1-\alpha)}}\), \(z\in E\) and this is the best dominant. \((\prec\) denotes subordination).