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Ma-Minda φ-classes studied in this paper:
Abstract
In this research article we consider two well known subclasses of starlike and
bounded turning functions associated with nephroid domain. Our aims to
find third Hankel determinant for these classes.
1
Results & Lemmas (16)
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Lemma 1.
Lemma 1. If p(z) ∈P and it is of the form (1.3), then |cn| ≤ 2 for n ≥1, (2.1) |cn+k −δcnck| ≤ 2 for 0 ≤δ ≤1, 2 |2δ −1| elsewhere.,
Lemma 1. If p(z) ∈P and it is of the form (1.3), then |cn| ≤ 2 for n ≥1, (2.1) |cn+k −δcnck| ≤ 2 for 0 ≤δ ≤1, 2 |2δ −1| elsewhere. ,
Lemma 2.
Lemma 2. [46]. If p(z) ∈P and is represented by (1.3), then c2 −νc2 1 ≤ −4ν + 2 (ν ≤0), 2 (0 ≤ν ≤1), 4ν −2 (ν ≥1).
Lemma 2. [46]. If p(z) ∈P and is represented by (1.3), then c2 −νc2 1 ≤ −4ν + 2 (ν ≤0), 2 (0 ≤ν ≤1), 4ν −2 (ν ≥1).
Lemma 3.
Lemma 3. [47]. If p(z) ∈P and is represented by (1.3), then ac3 1 −bc1c2 + dc3 ≤2 |a| + 2 |b −2a| + 2 |a −b + d|. 3 Bounds of |H3,1 (f)|…
Lemma 3. [47]. If p(z) ∈P and is represented by (1.3), then ac3 1 −bc1c2 + dc3 ≤2 |a| + 2 |b −2a| + 2 |a −b + d| . 3 Bounds of |H3,1 (f)| for class S∗ Ne
Theorem 1.
Theorem 1. Let f ∈S∗ Ne of the form (1.1). Then |a2| ≤ 1, |a3| ≤ 1 2, |a4| ≤ 7 18, |a5| ≤
Theorem 1. Let f ∈S∗ Ne of the form (1.1). Then |a2| ≤ 1, |a3| ≤ 1 2, |a4| ≤ 7 18, |a5| ≤
Theorem 2.
Theorem 2. Let f(z) ∈S∗ Ne be of the form (1.1). Then a3 −λa2 2 ≤ 1−2λ 2, λ ≤0 1
Theorem 2. Let f(z) ∈S∗ Ne be of the form (1.1). Then a3 −λa2 2 ≤ 1−2λ 2 , λ ≤0 1
Theorem 3.
Theorem 3. Let f(z) ∈S∗ Ne be of the form (1.1). Then for ξ ∈C, we have a3 −ξa2 2 ≤1 2 max 1, |2ξ −1|.
Theorem 3. Let f(z) ∈S∗ Ne be of the form (1.1). Then for ξ ∈C, we have a3 −ξa2 2 ≤1 2 max {1, |2ξ −1|} .
Theorem 4.
Theorem 4. Let f(z) ∈S∗ Ne be of the form (1.1). Then a3 −a2 2 ≤1 2. This results is sharp. Earthline J. Math. Sci. Vol. 6 No. 2 (2021),…
Theorem 4. Let f(z) ∈S∗ Ne be of the form (1.1). Then a3 −a2 2 ≤1 2. This results is sharp. Earthline J. Math. Sci. Vol. 6 No. 2 (2021), 293-308
Theorem 5.
Theorem 5. Let f(z) ∈S∗ Ne be of the form (1.1). Then |a2a3 −a4| ≤7 18.
Theorem 5. Let f(z) ∈S∗ Ne be of the form (1.1). Then |a2a3 −a4| ≤7 18.
Theorem 6.
Theorem 6. Let f(z) ∈S∗ Ne be of the form (1.1). Then a2a4 −a2 3 ≤4 9.
Theorem 6. Let f(z) ∈S∗ Ne be of the form (1.1). Then a2a4 −a2 3 ≤4 9.
Theorem 7.
Theorem 7. Let f(z) ∈S∗ Ne be of the form (1.1). Then |H3,1 (f)| ≤377 648 ≃0.581 79.
Theorem 7. Let f(z) ∈S∗ Ne be of the form (1.1). Then |H3,1 (f)| ≤377 648 ≃0.581 79.
Theorem 8.
Theorem 8. Let f ∈RNe of the form (1.1). Then |a2| ≤ 1 2, |a3| ≤ 1 3, |a4| ≤ 1 4, |a5| ≤
Theorem 8. Let f ∈RNe of the form (1.1). Then |a2| ≤ 1 2, |a3| ≤ 1 3, |a4| ≤ 1 4, |a5| ≤
Theorem 9.
Theorem 9. Let f(z) ∈RNe be of the form (1.1). Then for ξ ∈C, we have a3 −ξa2 2 ≤1 3 max 1, 3 |ξ| 4 .
Theorem 9. Let f(z) ∈RNe be of the form (1.1). Then for ξ ∈C, we have a3 −ξa2 2 ≤1 3 max 1, 3 |ξ| 4 .
Theorem 10.
Theorem 10. Let f(z) ∈RNe be of the form (1.1). Then a3 −a2 2 ≤1 3.
Theorem 10. Let f(z) ∈RNe be of the form (1.1). Then a3 −a2 2 ≤1 3.
Theorem 11.
Theorem 11. Let f(z) ∈RNe be of the form (1.1). Then |a2a3 −a4| ≤1 4.
Theorem 11. Let f(z) ∈RNe be of the form (1.1). Then |a2a3 −a4| ≤1 4.
Theorem 12.
Theorem 12. Let f(z) ∈RNe be of the form (1.1). Then a2a4 −a2 3 ≤11 72.
Theorem 12. Let f(z) ∈RNe be of the form (1.1). Then a2a4 −a2 3 ≤11 72.
Theorem 13.
Theorem 13. Let f(z) ∈RNe be of the form (1.1). Then |H3,1 (f)| ≤677 2160 ≃0.313 43.
Theorem 13. Let f(z) ∈RNe be of the form (1.1). Then |H3,1 (f)| ≤677 2160 ≃0.313 43.
Function classes studied:
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