Abstract
Let $\mathcal{U(α, λ)}$, $0<α<1$, $0 < λ<1$ be the class of functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+α}f'(z)-1\right|<λ$$ in the unit disc ${\mathbb D}$. For $f\in \mathcal{U(α, λ)}$ we give sharp bounds of its initial logarithmic coefficients $γ_{1},\,γ_{2},\,γ_{3}.$
Results & Lemmas (3)
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Lemma 1
Lemma 1. [4] Let,,. Then there exists a function, analytic in, such that, for all, and (5) <span id="page-1-1"></span>By we denote the…
Lemma 1. [4] Let $f \in \mathcal{U}(\alpha, \lambda)$ , $0 < \alpha < 1$ , $0 < \lambda < 1$ . Then there exists a function $\omega$ , analytic in $\mathbb{D}$ , such that $\omega(0) = 0$ , $|\omega(z)| < 1$ for all $z \in \mathbb{D}$ , and
(5)
$$\left[\frac{z}{f(z)}\right]^{\alpha} = 1 - \alpha \lambda z^{\alpha} \int_{0}^{z} \frac{\omega(t)}{t^{\alpha+1}} dt.$$
<span id="page-1-1"></span>By $\Omega$ we denote the class of analytic functions in $\mathbb{D}$ :
(6)
$$\omega(z) = c_1 z + c_2 z^2 + c_3 z^3 + \cdots,$$
with $\omega(0) = 0$ , and $|\omega(z)| < 1$ for all $z \in \mathbb{D}$ .
In their paper [8] Prokhorov and Szynal obtained sharp estimates on the functional
<span id="page-1-0"></span>
$$\Psi(\omega) = |c_3 + \mu c_1 c_2 + \nu c_1^3|$$
within the class of all $\omega \in \Omega$ . For our application we need only a part of those results.
Lemma 2
Lemma 2. [8] Let. For and real numbers, let and Then, the sharp estimate holds, where
Lemma 2. [8] Let $\omega(z) = c_1 z + c_2 z^2 + c_3 z^3 + \cdots \in \Omega$ . For $\mu$ and $\nu$ real numbers, let
$$\Psi(\omega) = |c_3 + \mu c_1 c_2 + \nu c_1^3|,$$
and
$$\begin{array}{lcl} D_1 & = & \left\{ (\mu,\nu) : |\mu| \leq \frac{1}{2}, |\nu| \leq 1 \right\}, \\ \\ D_2 & = & \left\{ (\mu,\nu) : \frac{1}{2} \leq |\mu| \leq 2, \frac{4}{27} (|\mu|+1)^3 - (|\mu|+1) \leq \nu \leq 1 \right\}, \\ \\ D_3 & = & \left\{ (\mu,\nu) : |\mu| \leq 2, |\nu| \geq 1 \right\}. \end{array}$$
Then, the sharp estimate $\Psi(\omega) \leq \Phi(\mu, \nu)$ holds, where
$$\Phi(\mu, \nu) = \begin{cases} 1, & (\mu, \nu) \in D_1 \cup D_2 \cup \{(2, 1)\}; \\ |\nu|, & (\mu, \nu) \in D_3. \end{cases}$$
Theorem 1 · coeff
Theorem 1. Let belongs to the class and is defined by (4). Then the following results are best possible. (i) when and. (ii) Let and let be…
Theorem 1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ belongs to the class $\mathcal{U}(\alpha, \lambda)$ and $\lambda_{\star}$ is defined by (4). Then the following results are best possible.
(i)
$$|\gamma_1| \le \frac{\lambda}{2(1-\alpha)}$$
when $0 < \lambda \le \lambda_{\star}$ and $0 < \alpha < 1$ .
(ii) Let
$$\lambda_1 = \frac{2(1-\alpha)^2}{\alpha(2-\alpha)}$$
and let $\alpha_1 = 0.4825...$ be the unique real root of the equation
$$7\alpha^4 - 20\alpha^3 + 24\alpha^2 - 16\alpha + 4 = 0$$
on the interval (0,1). Then
$$|\gamma_2| \leq \frac{\lambda}{2(2-\alpha)} \quad \text{if} \quad 0 < \lambda \leq \begin{cases} \lambda_1, \ \alpha \in [\alpha_1,1), \\ \lambda_\star, \ \alpha \in (0,\alpha_1], \end{cases}$$
and
$$|\gamma_2| \le \frac{\alpha \lambda^2}{4(1-\alpha)^2}$$
if $\lambda_1 \le \lambda \le \lambda_{\star}$ , $\alpha \in [\alpha_1, 1)$ .
(iii) Let $\lambda_{1/2} = \frac{(1-\alpha)(2-\alpha)}{2\alpha(3-\alpha)}$ , $\lambda_{\nu} = \sqrt{\frac{3(1-\alpha)^3}{\alpha^2(3-\alpha)}}$ and $\alpha_{1/2} = 0.2512\dots$ and $\alpha_{\nu} = 0.5337\dots$ are the unique roots of equations
$$4 - 12\alpha - 19\alpha^2 + 14\alpha^3 - 2\alpha^4 = 0$$
and
$$3 - 9\alpha + 9\alpha^2 - 5\alpha^3 = 0,$$
on the interval (0,1), respectively. Then
$$|\gamma_3| \le \frac{\lambda}{2(3-\alpha)} \quad \text{if} \quad 0 < \lambda \le \begin{cases} \lambda_{\star}, & \alpha \in (0, \alpha_{1/2}], \\ \lambda_{1/2}, & \alpha \in [\alpha_{1/2}, \alpha_2], \\ \lambda_{\nu}, & \alpha \in [\alpha_2, 1), \end{cases}$$
where $\alpha_2 = 0.9555...$ is the unique real root of equation $11\alpha^2 - 44\alpha + 32 = 0$ on (0,1). Also,
$$|\gamma_3| \le \frac{\alpha^2 \lambda^3}{6(1-\alpha)^3}$$
if $\lambda_{\nu} \le \lambda \le \lambda_{\star}$ , $\alpha \in [\alpha_{\nu}, 1)$ .
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