Abstract
Let $\mathcal{S}$ denote the class of functions analytic and univalent (i.e. one-to-one) in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:\, |z|<1\}$ normalized by $f(0)=0=f'(0)-1$. The logarithmic coefficients $γ_n$ of $f\in\mathcal{S}$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n.$ In the present paper, we determine the sharp upper bounds for $|γ_1|$, $|γ_2|$ and $|γ_3|$ when $f$ belongs to some familiar subclasses of close-to-convex functions.
Results & Lemmas (9)
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Lemma 1.3.
Lemma 1.3. [12] Let h ∈P be of the form (1.2). Then 2c2 = c2 1 + x(4 −c2 1) 4c3 = c3 1 + 2(4 −c2 1)c1x −c1(4 −c2 1)x2 + 2(4 −c2 1)(1…
Lemma 1.3. [12] Let h ∈P be of the form (1.2). Then 2c2 = c2 1 + x(4 −c2 1) 4c3 = c3 1 + 2(4 −c2 1)c1x −c1(4 −c2 1)x2 + 2(4 −c2 1)(1 −|x|2)t. for some complex valued x and t with |x| ≤1 and |t| ≤1.
Lemma 1.4.
Lemma 1.4. [14, pp 166] Let h ∈P be of the form (1.2). Then c2 −c2 1 2 ≤2 −|c1|2 2. The inequality is sharp for functions Lt,θ(z) of the…
Lemma 1.4. [14, pp 166] Let h ∈P be of the form (1.2). Then c2 −c2 1 2 ≤2 −|c1|2 2 . The inequality is sharp for functions Lt,θ(z) of the form Lt,θ(z) = t 1 + eiθz 1 −eiθz + (1 −t) 1 + ei2θz2 1 −ei2θz2
Lemma 1.5.
Lemma 1.5. [13] Let h ∈P be of the form (1.2) and µ be a complex number. Then |c2 −µc2 1| ≤2 max 1, |2µ −1|. The result is sharp for the…
Lemma 1.5. [13] Let h ∈P be of the form (1.2) and µ be a complex number. Then |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.
Theorem 2.14.
Theorem 2.14. Let f ∈F1 be given by (1.1). Then (i) |γ1| ≤3 4, (ii) |γ2| ≤4 9. (iii) If 1/2 ≤a2 ≤3/2 then |γ3| ≤ 1 288 11 + 15 √ 30 . The…
Theorem 2.14. Let f ∈F1 be given by (1.1). Then (i) |γ1| ≤3 4, (ii) |γ2| ≤4 9. (iii) If 1/2 ≤a2 ≤3/2 then |γ3| ≤ 1 288 11 + 15 √ 30 . The inequalities are sharp.
Lemma 1.3
Lemma 1.3 we obtain c2 = 1 12 76 −13 √ 30 and c3 = 1 72 554 −75 √ 30 . It is not
Lemma 1.3 we obtain c2 = 1 12 76 −13 √ 30 and c3 = 1 72 554 −75 √ 30 . It is not
Theorem 2.29.
Theorem 2.29. Let f ∈F2 be given by (1.1). Then (i) |γ1| ≤1 4, (ii) |γ2| ≤1 2. (iii) If 0 ≤a2 ≤1 then |γ3| ≤ 1 972 95 + 23 √ 46 . The…
Theorem 2.29. Let f ∈F2 be given by (1.1). Then (i) |γ1| ≤1 4, (ii) |γ2| ≤1 2. (iii) If 0 ≤a2 ≤1 then |γ3| ≤ 1 972 95 + 23 √ 46 . The inequalities are sharp.
Lemma 1.3
Lemma 1.3, we obtain c2 = 1 27 134 −19 √ 46 and c3 = 2 243 721 −71 √ 46 . It is not
Lemma 1.3, we obtain c2 = 1 27 134 −19 √ 46 and c3 = 2 243 721 −71 √ 46 . It is not
Theorem 2.40.
Theorem 2.40. Let f ∈F3 be given by (1.1). Then (i) |γ1| ≤3 4, (ii) |γ2| ≤2 5. (iii) If 1/2 ≤a2 ≤3/2 then |γ3| ≤743+131 √ 262 7776. The…
Theorem 2.40. Let f ∈F3 be given by (1.1). Then (i) |γ1| ≤3 4, (ii) |γ2| ≤2 5. (iii) If 1/2 ≤a2 ≤3/2 then |γ3| ≤743+131 √ 262 7776 . The inequalities are sharp.
Lemma 1.3
Lemma 1.3, we obtain c2 = 1 108 548 −37 √ 262 and c3 = 47525 √ 262−698926 44712. Therefore for given (c1, c2, c3) =
Lemma 1.3, we obtain c2 = 1 108 548 −37 √ 262 and c3 = 47525 √ 262−698926 44712 . Therefore for given (c1, c2, c3) =
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