Abstract
For an analytic and univalent function $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$, the logarithmic coefficients $γ_n$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n$. In the present paper, we consider the class of close-to-convex functions (with argument $0$), and determine the sharp upper bound of $|γ_3|$ for such functions $f$, which proves a recent conjecture of the first and third authors [1].
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 2.1
Lemma 2.1. [3, p. 41] For a function of the form (2.1), the sharp inequality holds for each. Equality holds for the function p(z) =…
Lemma 2.1. [3, p. 41] For a function $p \in \mathcal{P}$ of the form (2.1), the sharp inequality $|c_n| \leq 2$ holds for each $n \geq 1$ . Equality holds for the function p(z) = (1+z)/(1-z).
Lemma 2.2
Lemma 2.2. [10] Let be of the form (2.1). Then there exist with and, such that and
Lemma 2.2. [10] Let $p \in \mathcal{P}$ be of the form (2.1). Then there exist $x, t \in \mathbb{C}$ with $|x| \leq 1$ and $|t| \leq 1$ , such that
$$2c_2 = c_1^2 + x(4 - c_1^2),$$
and
$$4c_3 = c_1^3 + 2(4 - c_1^2)c_1x - c_1(4 - c_1^2)x^2 + 2(4 - c_1^2)(1 - |x|^2)t.$$
Lemma 2.3
Lemma 2.3. [8, Lemma 3] Let and be given by. Then for any, The inequality is sharp when g(z) = k(z) if, and when if. We now present our…
Lemma 2.3. [8, Lemma 3] Let $g \in S^*$ and be given by $g(z) = z + \sum_{n=2}^{\infty} b_n z^n$ . Then for any $\lambda \in \mathbb{C}$ ,
$$|b_3 - \lambda b_2^2| \le \max\{1, |3 - 4\lambda|\}.$$
The inequality is sharp when g(z) = k(z) if $|3 - 4\lambda| \ge 1$ , and when $g(z) = (k(z^2))^{1/2}$ if $|3 - 4\lambda| < 1$ .
We now present our results, beginning with
Theorem 2.1
Theorem 2.1. If is of the form (1.1), then The inequality is sharp.
Theorem 2.1. If $f \in \mathcal{K}_0$ is of the form (1.1), then
$$\operatorname{Re} \gamma_3 \le \frac{1}{243} (28 + 19\sqrt{19}).$$
The inequality is sharp.
Theorem 2.2
Theorem 2.2. If is of the form (1.1), then The inequality is sharp.
Theorem 2.2. If $f \in \mathcal{K}_0$ is of the form (1.1), then
$$|\gamma_3| \le \frac{1}{243}(28 + 19\sqrt{19}) = 0.4560\dots$$
The inequality is sharp.
Function classes studied:
Coefficient bounds & claims (3)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
Re(gamma_3) ≤ (28 + 19*sqrt(19))/243 for class K_0 (sharp) [Theorem 2.1]
coefficient_bound
|gamma_3| ≤ (28 + 19*sqrt(19))/243 for class K_0 (sharp) [Theorem 2.2]
function_family
Class K_0: close-to-convex functions with argument 0: Re(zf'(z)/g(z)) > 0 for some g in S*
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