Abstract
The known Ozaki's condition says that $\mathfrak{Re}\left\{f^{(p)}(z)\right\}>0$ for $|z|<1$ implies that $f(z)=z^p+a_{p+1}z^{p+1}+\cdots$ is at most $p$-valent in $\mathbb D$. In this paper prove an extension of Ozaki's condition. Also, we shall determine the new sufficient conditions for functions to be in the class of $p$-valent starlike of order $α$.
Results & Lemmas (6)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1.1
Lemma 1.1. [4] Let, be an analytic function in with. If there exists a point,, such that for and for some, then we have (1.1) for some,…
Lemma 1.1. [4] Let $q(z) = 1 + \sum_{n=m}^{\infty} c_n z^n$ , $c_m \neq 0$ be an analytic function in $\mathbb{D}$ with $q(z) \neq 0$ . If there exists a point $z_0$ , $|z_0| < 1$ , such that $|\arg\{q(z)\}| < \pi\gamma/2$ for $|z| < |z_0|$ and $|\arg\{q(z_0)\}| = \pi\gamma/2$ for some $\gamma > 0$ , then we have
(1.1)
$$\frac{z_0 q'(z_0)}{q(z_0)} = \frac{2ik \arg\{q(z_0)\}}{\pi},$$
for some $k \ge m(a+a^{-1})/2 \ge (a+a^{-1})/2 \ge 1$ , where $\{q(z_0)\}^{1/\gamma} = \pm ia$ , and a > 0.
Lemma 1.1 was generalized in [14] by considering a hypothesis that $\arg\{q(z)\}\in(\pi\gamma_1/2,\pi\gamma_2/2)$ instead of $|\arg\{q(z)\}|<\pi\gamma/2$ .
In this paper we need also the following lemmas.
Lemma 1.2
Lemma 1.2. [5, Th.5] If, then for all, we have (1.2)
Lemma 1.2. [5, Th.5] If $f(z) \in \mathcal{A}_p$ , then for all $z \in \mathbb{D}$ , we have
(1.2)
$$\Re \left\{ \frac{z f^{(p)}(z)}{f^{(p-1)}(z)} \right\} > 0 \quad \Rightarrow \quad \forall k \in \{1, \dots, p\} : \quad \Re \left\{ \frac{z f^{(k)}(z)}{f^{(k-1)}(z)} \right\} > 0.$$
Lemma 1.3
Lemma 1.3. [5, Th.1] If, then for all, we have (1.3) then f(z) is p-valent in and
Lemma 1.3. [5, Th.1] If $f(z) \in \mathcal{A}_p$ , then for all $z \in \mathbb{D}$ , we have
(1.3)
$$\mathfrak{Re}\left\{p + \frac{zf^{(p+1)}(z)}{f^{(p)}(z)}\right\} > 0 \quad (z \in \mathbb{D}),$$
then f(z) is p-valent in $\mathbb{D}$ and
$$\forall k \in \{1, \dots, p-1\}: \quad \mathfrak{Re}\left\{k + \frac{zf^{(k+1)}(z)}{f^{(k)}(z)}\right\} > 0 \quad (z \in \mathbb{D})..$$
Theorem 2.1 · coeff
Theorem 2.1. Let be analytic in. If (2.1) for some, then (2.2)
Theorem 2.1. Let $f(z) = z^p + \sum_{n=p+1}^{\infty} a_n z^n$ be analytic in $\mathbb{D}$ . If
(2.1)
$$\left| \arg\{f^{(p)}(z)\} \right| < \frac{\pi}{2} \left( \alpha_1 + \frac{2}{\pi} \tan^{-1} \alpha_1 \right), \quad z \in \mathbb{D}$$
for some $\alpha_1 \in (0,1]$ , then
(2.2)
$$\left| \arg \left\{ \frac{f^{(p-1)}(z)}{z} \right\} \right| < \frac{\alpha_1 \pi}{2}, \quad z \in \mathbb{D}.$$
Theorem 2.4 · coeff
Theorem 2.4. Let be analytic in. If (2.11) for some then, we have (2.12) where (2.13) for all.
Theorem 2.4. Let $f(z) = z^p + \sum_{n=p+1}^{\infty} a_n z^n$ be analytic in $\mathbb{D}$ . If
(2.11)
$$\left| \arg\{f^{(p)}(z)\} \right| < \frac{\pi \alpha_0}{2}, \quad z \in \mathbb{D}.$$
for some $\alpha_0 \in (0, 3/2]$ then, we have
(2.12)
$$\left| \arg \left\{ \frac{f^{(p-k)}(z)}{z^k} \right\} \right| < \frac{\pi \alpha_k}{2}, \quad z \in \mathbb{D},$$
where
(2.13)
$$\alpha_0 \in (0, 3/2], \quad \alpha_k + \frac{2}{\pi} \tan^{-1} \frac{\alpha_k}{k} = \alpha_{k-1}$$
for all $k \in \{1, ..., p\}$ .
Theorem 2.5 · coeff
Theorem 2.5. Let be analytic in. Assume that (2.18) for some, and assume that the sequence is defined by <span id="page-6-0"></span> Then…
Theorem 2.5. Let $f(z) = z^p + \sum_{n=p+1}^{\infty} a_n z^n$ be analytic in $\mathbb{D}$ . Assume that
(2.18)
$$\left|\arg\{f^{(p)}(z)\}\right| < \frac{\pi\alpha_0}{2}, \quad z \in \mathbb{D}.$$
for some $\alpha_0 \in (0, 3/2]$ , and assume that the sequence $\{\alpha_k\}_{k=1}^{k=p}$ is defined by
<span id="page-6-0"></span>
$$\alpha_k + \frac{2}{\pi} \tan^{-1} \frac{\alpha_k}{k} = \alpha_{k-1}.$$
Then we have
(2.19)
$$\left| \arg \left\{ \frac{z f^{(p-s+1)}(z)}{f^{(p-s)}(z)} \right\} \right| < \frac{\pi}{2} (\alpha_s + \alpha_{s-1}) \quad z \in \mathbb{D},$$
for all $s \in \{2, ..., p\}$ . Furthermore, if there exists a positive integer $\sigma \in \{2, ..., p\}$ such that $\alpha_{\sigma} + \alpha_{\sigma-1} \leq 1$ , then
(2.20)
$$\left| \arg \left\{ \frac{zf'(z)}{f(z)} \right\} \right| < \frac{\pi(\alpha_{p-1} + \alpha_p)}{2} \le \frac{\pi}{2}, \quad z \in \mathbb{D},$$
or f(z) is p-valently strongly starlike of order $\alpha_{p-1} + \alpha_p$ .
Function classes studied:
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