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Results & Lemmas (2)
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Theorem 1
Theorem 1. A function f belongs to the class Co(p) if and only if f satisfies (ii) and there exists a function holomorphic in D such that…
Theorem 1. A function f belongs to the class Co(p) if and only if f satisfies (ii) and there exists a function $\omega$ holomorphic in D such that $\omega(D) \subset \overline{D}$ and
(7)
$$\frac{f''(z)}{f'(z)} = \frac{2p}{1-zp} + \frac{2}{p(1-z/p)} + \frac{2z\omega(z) - \alpha(1+\omega(z))}{1-\alpha z(1+\omega(z)) + z^2\omega(z)}, \quad z \in D,$$
where
$$\alpha := \frac{2p}{1 + p^2} \in (0, 1).$$
Theorem 2 · coeff
Theorem 2. Let, and. Then the domain of variability of is described by the inequality (6). A point on the boundary of this domain is…
Theorem 2. Let $f \in Co(p)$ , $p \in (0,1)$ and $n \in \{2,3,4,5\}$ . Then the domain of variability of $a_n(f)$ is described by the inequality (6). A point on the boundary of this domain is attained if and only if there exists $a \theta \in [0,2\pi)$ such that $f = f_{\theta}$ , where $f_{\theta}$ is defined as in (4).