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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 $f(0)=0=f'(0)-1$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n$. In the present paper, we consider close-to-convex functions (with argument $0$) with respect to odd starlike functions and determine the sharp upper bound of $|γ_n|$, $n=1,2,3$ for such functions $f$.
Results & Lemmas (5)
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Lemma 2.1.
Lemma 2.1. [5, 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. [5, 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. [12] Let P ∈P be of the form (2.1) and µ be a complex number. Then (2.2) |c2 −µc2 1| ≤2 max 1, |2µ −1|. The result is sharp for…
Lemma 2.2. [12] Let P ∈P be of the form (2.1) and µ be a complex number. Then (2.2) |c2 −µc2 1| ≤2 max{1, |2µ −1|}. The result is sharp for the functions given by P(z) = 1+z2 1−z2 and P(z) = 1+z 1−z.
Lemma 2.3.
Lemma 2.3. [11] 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) and 4c3 =…
Lemma 2.3. [11] 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) and 4c3 = c3 1 + 2(4 −c2 1)c1x −c1(4 −c2 1)x2 + 2(4 −c2 1)(1 −|x|2)t.
Theorem 2.1.
Theorem 2.1. Let f ∈F be of the form (1.1). Then |γ1| ≤ 1 2, |γ2| ≤ 1 2 and |γ3| ≤ 1 972(95 + 23 √ 46). The inequalities are sharp.
Theorem 2.1. Let f ∈F be of the form (1.1). Then |γ1| ≤ 1 2, |γ2| ≤ 1 2 and |γ3| ≤ 1 972(95 + 23 √ 46). The inequalities are sharp.
Theorem 3
Theorem 3, page 35]). Using Lemma 2.2, it follows from (2.6) that |γ2| ≤1 6|b3| + 1 6 c2 −3 8c2 1 ≤1 6 + 1 3 = 1 2 and equality holds for a…
Theorem 3, page 35]). Using Lemma 2.2, it follows from (2.6) that |γ2| ≤1 6|b3| + 1 6 c2 −3 8c2 1 ≤1 6 + 1 3 = 1 2 and equality holds for a function f defined by zf ′(z) = g(z)P(z), where g(z) = z/(1 −z2) and P(z) = (1 + z2)/(1 −z2). Writing c2 and c3 in terms of c1 with the help of Lemma 2.3, it follows from (2.7) that
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