Abstract
The logarithmic coefficients $γ_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n$. Recently, D.K. Thomas [On the logarithmic coefficients of close to convex functions, {\it Proc. Amer. Math. Soc.} {\bf 144} (2016), 1681--1687] proved that $|γ_3|\le \frac{7}{12}$ for functions in a subclass of close-to-convex functions (with argument $0$) and
Results & Lemmas (6)
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Theorem 3.8
Theorem 3.8] in (1.5), we can obtain the sharp estimate |γ2| ≤1 2(1 + 2e−2) = 0.635.... For n ≥3, the problem seems much harder, and no…
Theorem 3.8] in (1.5), we can obtain the sharp estimate |γ2| ≤1 2(1 + 2e−2) = 0.635 . . . . For n ≥3, the problem seems much harder, and no significant upper bound for |γn| when f ∈S appear to be known. If f ∈S∗then it is not very difficult to prove that |γn| ≤ 1 n for n ≥1 and equality holds for the Koebe function k(z) = z/(1 −z)2. The inequality |γn| ≤1 n for n ≥2 extends to the class K was claimed in a paper of Elhosh [4]. However, Girela [6] pointed out some error in the proof of Elhosh [4] and
Lemma 2.1.
Lemma 2.1. [2, p. 41] For a function P ∈P of the form (2.1), the sharp inequality |cn| ≤2 holds for each n ≥1. Equality holds for the…
Lemma 2.1. [2, p. 41] For a function P ∈P of the form (2.1), the sharp inequality |cn| ≤2 holds for each n ≥1. Equality holds for the function P(z) = (1+z)/(1−z).
Lemma 2.2.
Lemma 2.2. [10] Let P ∈P be of the form (2.1). Then there exist x, t ∈C with |x| ≤1 and |t| ≤1 such that 2c2 = c2 1 + x(4 −c2 1)
Lemma 2.2. [10] Let P ∈P be of the form (2.1). Then there exist x, t ∈C with |x| ≤1 and |t| ≤1 such that 2c2 = c2 1 + x(4 −c2 1)
Lemma 2.3.
Lemma 2.3. [8, Lemma 3] Let g ∈S∗be of the form g(z) = z + P∞ n=2 bnzn. Then for any λ ∈C, |b3 −λb2 2| ≤max 1, |3 −4λ|. The inequality is…
Lemma 2.3. [8, Lemma 3] Let g ∈S∗be of the form g(z) = z + P∞ n=2 bnzn. Then for any λ ∈C, |b3 −λb2 2| ≤max{1, |3 −4λ|}. The inequality is sharp for k(z) = z/(1 −z)2 if |3 −4λ| ≥1 and for (k(z2))1/2 if |3 −4λ| < 1. For f ∈K0 (close-to-convex functions with argument 0), we obtained the following improved result for |γ3| (compare [12]).
Theorem 2.1.
Theorem 2.1. If f ∈K0 then |γ3| ≤ 1 18(3 + 4 √ 2) = 0.4809.
Theorem 2.1. If f ∈K0 then |γ3| ≤ 1 18(3 + 4 √ 2) = 0.4809.
Theorem 2.2.
Theorem 2.2. Let f ∈CR+ be of the form (1.1) with 1 ≤a2 ≤2. Then (2.15) |γ3| ≤ 1 243(28 + 19 √ 19) = 0.4560. The inequality is sharp.
Theorem 2.2. Let f ∈CR+ be of the form (1.1) with 1 ≤a2 ≤2. Then (2.15) |γ3| ≤ 1 243(28 + 19 √ 19) = 0.4560. The inequality is sharp.
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