Abstract
Let $\mathcal{A}$ denote the set of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z| < 1\}$ normalized by $f (0) = 0$ and $f'(0) = 1.$ The logarithmic coefficients $γ_n$ of $f \in \mathcal{A}$ are defined by $ \log f(z)/z =2 \sum_{n=1}^{\infty}γ_{n}z^{n}.$ In the present paper, the upper bound of the third logarithmic coefficient in general case of $f''(0)$ was computed when $f$ belongs to some familiar subclasses of close-to-convex functions.
Results & Lemmas (4)
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Lemma 1.1.
Lemma 1.1. [3] Let w(z) = c1z + c2z2 + · · · be a Schwarz function. Then |c1| ≤1, |c2| ≤1 −|c1|2 and |c3| ≤1 −|c1|2 − |c2|2 1 + |c1|. 2.…
Lemma 1.1. [3] Let w(z) = c1z + c2z2 + · · · be a Schwarz function. Then |c1| ≤1, |c2| ≤1 −|c1|2 and |c3| ≤1 −|c1|2 − |c2|2 1 + |c1|. 2. Main Results
Theorem 2.1.
Theorem 2.1. Let f ∈F1. Then |γ3| ≤15.75 48 = 0.328125.
Theorem 2.1. Let f ∈F1. Then |γ3| ≤15.75 48 = 0.328125.
Theorem 2.2.
Theorem 2.2. Let f ∈F2. Then |γ3| ≤0.258765 · · ·.
Theorem 2.2. Let f ∈F2. Then |γ3| ≤0.258765 · · · .
Theorem 2.3.
Theorem 2.3. Let f ∈F3. Then |γ3| ≤17.75 48 = 0.36979 · · ·.
Theorem 2.3. Let f ∈F3. Then |γ3| ≤17.75 48 = 0.36979 · · · .
Function classes studied:
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