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Abstract
In this study, we initially established bounds for the logarithmic coefficients concerning a
specific subclass of bounded turning functions ℛ℘ that are linked to the cardioid domain. For
the functions belonging to this class, we identified sharp bounds for the second Hankel
determinant of logarithmic coefficients, denoted as 𝐻2,1(𝐹𝑓/2). In conclusion, we computed the
bounds of the third Hankel determinant of logarithmic coefficients 𝐻3,1(𝐹𝑓/2).
Results & Lemmas (9)
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Lemma 1.1
Lemma 1.1 ([4]). If 𝑝∈𝒫 is of the form (2) with 𝑐1 ≥0, then 𝑐1 = 2𝑑1, 𝑐2 = 2𝑑1 2 + 2(1 −𝑑1 2)𝑑2, 𝑐3 = 2𝑑1 3 + 4(1 −𝑑1 2)𝑑1𝑑2 −2(1 −𝑑1…
Lemma 1.1 ([4]). If 𝑝∈𝒫 is of the form (2) with 𝑐1 ≥0, then 𝑐1 = 2𝑑1, 𝑐2 = 2𝑑1 2 + 2(1 −𝑑1 2)𝑑2, 𝑐3 = 2𝑑1 3 + 4(1 −𝑑1 2)𝑑1𝑑2 −2(1 −𝑑1 2)𝑑1𝑑2 2 + 2(1 −𝑑1 2)(1 −|𝑑2|2)𝑑3
Lemma 1.2.
Lemma 1.2. If 𝑝∈𝒫 takes the form (2) then the following inequalities are hold |𝑐𝑛| ≤2 𝑓𝑜𝑟 𝑛≥1, (16) |𝑐𝑛+𝑘−𝜇𝑐𝑛𝑐𝑘| ≤2 𝑓𝑜𝑟 0 ≤𝜇 ≤1, (17)
Lemma 1.2. If 𝑝∈𝒫 takes the form (2) then the following inequalities are hold |𝑐𝑛| ≤2 𝑓𝑜𝑟 𝑛≥1, (16) |𝑐𝑛+𝑘−𝜇𝑐𝑛𝑐𝑘| ≤2 𝑓𝑜𝑟 0 ≤𝜇 ≤1, (17)
Lemma 1.3
Lemma 1.3 ([1]). Let 𝑝∈𝒫 and has form (2), then |𝐾𝑐1 3 −𝐿𝑐1𝑐2 + 𝑀𝑐3| ≤2|𝐾| + 2|𝐿−2𝐾| + 2|𝐾−𝐿+ 𝑀| ⋅
Lemma 1.3 ([1]). Let 𝑝∈𝒫 and has form (2), then |𝐾𝑐1 3 −𝐿𝑐1𝑐2 + 𝑀𝑐3| ≤2|𝐾| + 2|𝐿−2𝐾| + 2|𝐾−𝐿+ 𝑀| ⋅
Lemma 1.4
Lemma 1.4 ([2]). Given real numbers 𝐴, 𝐵, 𝐶, let 𝑌(𝐴, 𝐵, 𝐶) ≔max |𝐴+ 𝐵𝑧+ 𝐶𝑧2| + 1 −|𝑧|2 ∶𝑧∈𝕌̅. If 𝐴𝐶≥0, then 𝑌(𝐴, 𝐵, 𝐶) = |𝐴| + |𝐵| + |𝐶|,…
Lemma 1.4 ([2]). Given real numbers 𝐴, 𝐵, 𝐶, let 𝑌(𝐴, 𝐵, 𝐶) ≔max{|𝐴+ 𝐵𝑧+ 𝐶𝑧2| + 1 −|𝑧|2 ∶𝑧∈𝕌̅}. If 𝐴𝐶≥0, then 𝑌(𝐴, 𝐵, 𝐶) = { |𝐴| + |𝐵| + |𝐶|, |𝐵| ≥2(1 −|𝐶|), 1 + |𝐴| + 𝐵2 4(1 −|𝐶|), |𝐵| < 2(1 −|𝐶|) ⋅ If 𝐴𝐶< 0, then 𝑌(𝐴, 𝐵, 𝐶) = {
Theorem 2.1.
Theorem 2.1. If 𝑓∈ℛ℘ and it has the form given in (1), then
Theorem 2.1. If 𝑓∈ℛ℘ and it has the form given in (1), then
Theorem 3.1.
Theorem 3.1. If 𝑓∈ℛ℘, then |𝐻2,1 (𝐹𝑓 2 )| ≤1 36 ⋅ (40) The inequality in (40) is sharp.
Theorem 3.1. If 𝑓∈ℛ℘, then |𝐻2,1 (𝐹𝑓 2 )| ≤1 36 ⋅ (40) The inequality in (40) is sharp.
Theorem 4.1.
Theorem 4.1. If 𝑓∈ℛ℘, then |𝛾2𝛾4 −𝛾3 2| ≤4877 69120 ∙ (44)
Theorem 4.1. If 𝑓∈ℛ℘ , then |𝛾2𝛾4 −𝛾3 2| ≤4877 69120 ∙ (44)
Theorem 4.2.
Theorem 4.2. If 𝑓∈ℛ℘, then |𝛾1𝛾4 −𝛾2𝛾3| ≤41 576 ∙ (45)
Theorem 4.2. If 𝑓∈ℛ℘ , then |𝛾1𝛾4 −𝛾2𝛾3| ≤41 576 ∙ (45)
Theorem 4.3.
Theorem 4.3. If 𝑓∈ℛ℘, then |𝐻3,1(𝑓)| ≤ 77869 2211840 ≈0.035 ⋅ (46)
Theorem 4.3. If 𝑓∈ℛ℘ , then |𝐻3,1(𝑓)| ≤ 77869 2211840 ≈0.035 ⋅ (46)
Function classes studied:
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