Abstract
For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, P. T. Mocanu (Mathematica (Cluj), 11(34) (1969)) have considered Mocanu functions. The object of the present paper is to discuss some sufficient problems for f(z) to be starlike of order α.
Results & Lemmas (5)
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 · coeff
Lemma 1. Let the function w(z) defined by be analytic in with w(0) = 0. If |w(z)| attains its maximum value on the circle |z| = r at a…
Lemma 1. Let the function w(z) defined by
$$w(z) = a_n z^n + a_{n+1} z^{n+1} + a_{n+2} z^{n+2} + \dots$$
$(n = 1, 2, 3, \dots)$
be analytic in $\mathbb{U}$ with w(0) = 0. If |w(z)| attains its maximum value on the circle |z| = r at a point $z_0 \in \mathbb{U}$ , then there exists a real number $k \geq n$ such that
$$\frac{z_0 w'(z_0)}{w(z_0)} = k.$$
<sup>2000</sup> Mathematics Subject Classification. 30C45.
Key words and phrases. Analytic, univalent, starlike, Mocanu function.
Lemma 2 · coeff
Lemma 2. If satisfies for some complex, and some real such that, then <span id="page-1-0"></span>
Lemma 2. If $f(z) \in A_n$ satisfies
$$\left| (\beta - \gamma) \frac{zf'(z)}{f(z)} + \gamma \left( 1 + \frac{zf''(z)}{f'(z)} \right) \right| < \frac{1}{1+\rho} |n\rho\gamma - \beta| \qquad (z \in \mathbb{U})$$
for some complex $\beta$ , $\gamma$ and some real $\rho > 0$ such that $\operatorname{Re}\left(\frac{\beta}{\gamma}\right) < n\rho$ , then
<span id="page-1-0"></span>
$$\left| \frac{f(z)}{zf'(z)} - 1 \right| < \rho \qquad (z \in \mathbb{U}).$$
Theorem 1 · coeff
Theorem 1. If satisfies for some real, some complex and such that, or for some real, some complex and such that, then That is.
Theorem 1. If $f(z) \in A_n$ satisfies
$$\left| (\beta - \gamma) \frac{zf'(z)}{f(z)} + \gamma \left( 1 + \frac{zf''(z)}{f'(z)} \right) \right| < \frac{1}{2} |n\gamma - \beta| \qquad (z \in \mathbb{U})$$
for some real $0 < \alpha \le \frac{1}{2}$ , some complex $\beta$ and $\gamma$ such that $\operatorname{Re}\left(\frac{\beta}{\gamma}\right) < n$ , or
$$\left| (\beta - \gamma) \frac{zf'(z)}{f(z)} + \gamma \left( 1 + \frac{zf''(z)}{f'(z)} \right) \right| < |n\gamma(1 - \alpha) - \alpha\beta| \qquad (z \in \mathbb{U})$$
for some real $\frac{1}{2} \leq \alpha < 1$ , some complex $\beta$ and $\gamma$ such that $\operatorname{Re}\left(\frac{\beta}{\gamma}\right) < n\left(\frac{1}{\alpha} - 1\right)$ , then
$$\left| \frac{f(z)}{zf'(z)} - \frac{1}{2\alpha} \right| < \frac{1}{2\alpha} \qquad (z \in \mathbb{U}).$$
That is $f(z) \in \mathcal{S}^*(\alpha)$ .
Lemma 3 · coeff
Lemma 3. If satisfies for some real, some complex and such that, then <span id="page-4-0"></span>
Lemma 3. If $f(z) \in A_n$ satisfies
$$\left|\beta\left(\frac{zf'(z)}{f(z)} - 1\right) + \gamma \frac{zf''(z)}{f'(z)}\right| < \frac{\rho}{1+\rho}|\beta + \gamma(n+1)| \qquad (z \in \mathbb{U})$$
for some real $\rho > 0$ , some complex $\beta$ and $\gamma$ such that $\operatorname{Re}\left(\frac{\beta}{\gamma}\right) > -(n+1)$ , then
<span id="page-4-0"></span>
$$\left| \frac{f(z)}{zf'(z)} - 1 \right| < \rho \qquad (z \in \mathbb{U}).$$
Theorem 2 · coeff
Theorem 2. If satisfies for some real, some complex and such that, or for some real, some complex and such that, then which shows that.
Theorem 2. If $f(z) \in A_n$ satisfies
$$\left|\beta\left(\frac{zf'(z)}{f(z)} - 1\right) + \gamma \frac{zf''(z)}{f'(z)}\right| < \frac{1}{2}|\beta + \gamma(n+1)| \qquad (z \in \mathbb{U})$$
for some real $0 < \alpha \le \frac{1}{2}$ , some complex $\beta$ and $\gamma$ such that $\operatorname{Re}\left(\frac{\beta}{\gamma}\right) > -(n+1)$ , or
$$\left|\beta\left(\frac{zf'(z)}{f(z)} - 1\right) + \gamma \frac{zf''(z)}{f'(z)}\right| < (1 - \alpha)|\beta + \gamma(n+1)| \qquad (z \in \mathbb{U})$$
for some real $\frac{1}{2} \leq \alpha < 1$ , some complex $\beta$ and $\gamma$ such that $\operatorname{Re}\left(\frac{\beta}{\gamma}\right) > -(n+1)$ , then
$$\left|\frac{f(z)}{zf'(z)} - \frac{1}{2\alpha}\right| < \frac{1}{2\alpha} \qquad (z \in \mathbb{U}),$$
which shows that $f(z) \in \mathcal{S}^*(\alpha)$ .
Function classes studied:
Coefficient bounds & claims (4)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
A_n with Mocanu-type condition: If f(z) in A_n satisfies |(beta-gamma)*zf'/f + gamma*(1+zf''/f')| < 1/(2|n*gamma - beta|) for some real 0 < alpha <= 1/2, some complex beta and gamma with Re(beta/gamma) < n, then |f(z)/(z*f'(z)) - 1/(2*alpha)| < 1/(2*alpha), i.e. f(z) in S*(alpha). [Theorem 1]
coefficient_bound
A_n with Mocanu-type condition: If f(z) in A_n satisfies |beta*(zf'/f - 1) + gamma*zf''/f'| < 1/(2|beta + gamma*(n+1)|) for some real 0 < alpha <= 1/2 and complex beta, gamma with Re(beta/gamma) > -(n+1), then f(z) in S*(alpha). [Theorem 2]
function_family
Class S*(alpha): f in A_n with Re(zf'(z)/f(z)) > alpha, 0 <= alpha < 1
function_family
Class M_alpha: alpha-convex (Mocanu) functions: f in A_n with Re((1-alpha)*zf'(z)/f(z) + alpha*(1+zf''(z)/f'(z))) > 0
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