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Abstract
For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, R. Singh and S. Singh (Coll. Math. 47(1982), 309-314) have considered some sufficient problems for f(z) to be univalent in U. The object of the present paper is to discuss some sufficient problems for f(z) to be some classes of analytic functions in U.
Results & Lemmas (7)
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
Lemma 1. Let w(z) be analytic in the open unit disk with w(0) = 0. Then if |w(z)| attains its maximum value on the circle |z| = r at a…
Lemma 1. Let w(z) be analytic in the open unit disk $\mathbb{U}$ with w(0) = 0. Then if |w(z)| attains its maximum value on the circle |z| = r at a point $z_0 \in \mathbb{U}$ , then we have $z_0w'(z_0) = kw(z_0)$ , where $k \ge 1$ is a real number.
<sup>2000</sup> Mathematics Subject Classification. 30C45.
Key words and phrases. Analytic function, univalent function, starlike function.
Theorem 1
Theorem 1. If satisfies for some real, and, then.
Theorem 1. If $f(z) \in A$ satisfies
$$|f'(z) - 1|^{\beta} \left| \delta + \frac{zf''(z)}{f'(z)} \right|^{\gamma} < \left( \frac{1 + 2\delta}{2} \right)^{\gamma} \qquad (z \in \mathbb{U})$$
for some real $\beta$ , $\gamma \geq 0$ and $\delta > -\frac{1}{2}$ , then $f(z) \in \mathcal{C}$ .
Corollary 1
Corollary 1. If satisfies for some real, then. Making in Theorem 1, we see
Corollary 1. If $f(z) \in A$ satisfies
$$|f'(z)-1|^{1-\lambda}\left|1+\frac{zf''(z)}{f'(z)}\right|^{\lambda}<\left(\frac{3}{2}\right)^{\lambda}$$
$(z\in\mathbb{U})$
for some real $\lambda \geq 0$ , then $f(z) \in \mathcal{C}$ .
Making $\delta = 0$ in Theorem 1, we see
Corollary 2
Corollary 2. If satisfies for some real and, then. Remark 1. If we take in Corollary 2, then we have that, for some real, implies
Corollary 2. If $f(z) \in A$ satisfies
$$|f'(z) - 1|^{\beta} \left| \frac{zf''(z)}{f'(z)} \right|^{\gamma} < \left(\frac{1}{2}\right)^{\gamma} \qquad (z \in \mathbb{U})$$
for some real $\beta$ and $\gamma \geq 0$ , then $f(z) \in \mathcal{C}$ .
Remark 1. If we take $\gamma = 0$ in Corollary 2, then we have that, for some real $\beta$ ,
$$|f'(z) - 1|^{\beta} < 1 \qquad (z \in \mathbb{U})$$
implies
$$|f'(z) - 1| < 1 \qquad (z \in \mathbb{U}).$$
Theorem 2
Theorem 2. If satisfies (2) or (3) for some real, with, then.
Theorem 2. If $f(z) \in A$ satisfies
$$\left| \frac{zf'(z)}{f(z)} - 1 \right|^{\beta} \left| z \left( \frac{zf'(z)}{f(z)} \right)' \right|^{\gamma} < \left( \frac{1}{2} \right)^{\gamma} \qquad (z \in \mathbb{U})$$
(2)
or
$$\left| \frac{zf'(z)}{f(z)} + 1 \right|^{\beta} \left| z \left( \frac{zf'(z)}{f(z)} \right)' \right|^{\gamma} < \left( \frac{1}{2} \right)^{\gamma} \qquad (z \in \mathbb{U})$$
(3)
for some real $\beta$ , $\gamma$ with $\beta + 2\gamma \ge 0$ , then $f(z) \in \mathcal{S}^*$ .
Theorem 3
Theorem 3. If satisfies for some real, and with, then.
Theorem 3. If $f(z) \in A$ satisfies
$$\left| \frac{zf'(z)}{f(z)} - 1 \right|^{\beta} \left| z \left( \frac{zf'(z)}{f(z)} \right)' \right|^{\gamma} < \left( \frac{1}{2} \right)^{\gamma} (1 - \alpha)^{\beta + \gamma} \qquad (z \in \mathbb{U})$$
for some real $0 \le \alpha < 1$ , $\beta$ and $\gamma$ with $\beta + 2\gamma \ge 0$ , then $f(z) \in \mathcal{S}^*(\alpha)$ .
Theorem 4
Theorem 4. If satisfies for some real, and, then.
Theorem 4. If $f(z) \in A$ satisfies
$$\left|\frac{zf'(z)}{f(z)}\right|^{\alpha} \left|z\left(\frac{zf'(z)}{f(z)}\right)'\right|^{\beta} < \left(\frac{1}{2}\mu\right)^{\beta} \qquad (z \in \mathbb{U})$$
for some real $\alpha \geq 0$ , $\beta > 0$ and $\mu = \frac{\beta}{\alpha + \beta}$ , then $f(z) \in \mathcal{STS}(\mu)$ .
Coefficient bounds & claims (3)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
function_family
Class C(alpha): f in A with |f'(z) - 1| < 1 - alpha, 0 <= alpha < 1; convex functions of order alpha
function_family
Class S*(alpha): f in A with Re(z f'(z)/f(z)) > alpha, 0 <= alpha < 1
function_family
Class STS(mu): f in A with Re((z f'(z)/f(z))^(1/mu)) > 0, 0 < mu <= 1
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