Ma-Minda φ-classes studied in this paper:
Abstract
The article aims to determine the sharp bounds of coefficients,
Fekete-Szegö, Zalcman inequalities for the family SS∗
tanh of starlike function with
respect to symmetric points linked with tan hyperbolic function. We also estimate
determinant of
H2,2
f
is also obtained for the same class. Further, we study
the logarithmic and inverse coefficients for the same class.
1
Results & Lemmas (19)
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Lemma 1.
Lemma 1. ([21])Let w (z) = P∞ n=1 wnzn be a Schwarz function. Then for any real numbers α and β with (α, β) ∈(α, β) ∈D1 ∪D2 ∪D3 ∪D4, we…
Lemma 1. ([21])Let w (z) = P∞ n=1 wnzn be a Schwarz function. Then for any real numbers α and β with (α, β) ∈(α, β) ∈D1 ∪D2 ∪D3 ∪D4, we have the following sharp estimate given by w3 + αw1w2 + βw3 1 ≤1. where D1 = |α| ≤1 2, – 1 ≤β ≤1 , D2
Lemma 2.
Lemma 2. ([34])If w ∈B0 is in the form of (2), then |w2| ≤ 1 – |w1|2, (3) |wn| ≤ 1, n ≥1. (4) Furthermore, the inequality of (3) can be…
Lemma 2. ([34])If w ∈B0 is in the form of (2) , then |w2| ≤ 1 – |w1|2 , (3) |wn| ≤ 1, n ≥1. (4) Furthermore, the inequality of (3) can be improved in the manner of w2 + ηw2 1 ≤max 1, |η|
Lemma 3.
Lemma 3. ([7])Let w (z) = w1z + w2z2 +... be a Schwarz function. Then |w3| ≤ 1 – |w1|2 – |w2|2 1 + |w1|, (6) |w4| ≤ 1 – |w1|2 – |w2|2. (7)
Lemma 3. ([7])Let w (z) = w1z + w2z2 + ... be a Schwarz function. Then |w3| ≤ 1 – |w1|2 – |w2|2 1 + |w1|, (6) |w4| ≤ 1 – |w1|2 – |w2|2 . (7)
Lemma 4.
Lemma 4. Let frakw(z) = w1z + w2z2 +...be a Schwarz function. Then w1w3 – w2 2 ≤1 – |w1|2.
Lemma 4. Let frakw(z) = w1z + w2z2 + ...be a Schwarz function. Then w1w3 – w2 2 ≤1 – |w1|2 .
Theorem 1.
Theorem 1. Let f ∈SS∗ tanh. Then |a2| ≤ 1 2, |a3| ≤ 1 2, |a4| ≤ 1 4, |a5|
Theorem 1. Let f ∈SS∗ tanh. Then |a2| ≤ 1 2, |a3| ≤ 1 2, |a4| ≤ 1 4, |a5|
Theorem 2.
Theorem 2. Let f ∈SS∗ tanh. Then a3 – ηa2 2 ≤max 1 2, –η 4
Theorem 2. Let f ∈SS∗ tanh. Then a3 – ηa2 2 ≤max 1 2, –η 4
Corollary 1.
Corollary 1. If f ∈SS∗ tanh is of the form (1), then a3 – a2 2 ≤1 2. The result is sharp with the extremal function given by (17). Now we…
Corollary 1. If f ∈SS∗ tanh is of the form (1) , then a3 – a2 2 ≤1 2. The result is sharp with the extremal function given by (17) . Now we consider the zalcman functionals for f ∈SS∗ tanh.
Theorem 3.
Theorem 3. Suppose that f ∈SS∗ tanh be the form of (1), then |a4 – a2a3| ≤1 4, (20) and a5 – a2 3 ≤1 4. (21) These inequalities (20) and…
Theorem 3. Suppose that f ∈SS∗ tanh be the form of (1) , then |a4 – a2a3| ≤1 4, (20) and a5 – a2 3 ≤1 4. (21) These inequalities (20) and (21) are sharp for the extremal function given by (18) and (19) .
Theorem 4.
Theorem 4. Let f ∈SS∗ tanh be of the form (1), then H2,2 = a2a4 – a2 3 ≤1 4. This inequality is sharp with the extremal function given by…
Theorem 4. Let f ∈SS∗ tanh be of the form (1) , then H2,2 = a2a4 – a2 3 ≤1 4. This inequality is sharp with the extremal function given by (17).
Theorem 5.
Theorem 5. Let f ∈SS∗ tanh. Then |γ1| ≤ 1 4, |γ2| ≤ 1 4, |γ3| ≤ 1 8. All these bounds are sharp.
Theorem 5. Let f ∈SS∗ tanh. Then |γ1| ≤ 1 4, |γ2| ≤ 1 4, |γ3| ≤ 1 8. All these bounds are sharp.
Theorem 6.
Theorem 6. Let f ∈SS∗ tanh be of the form (1). Then γ2 – ηγ2 1 ≤max 1 4,
Theorem 6. Let f ∈SS∗ tanh be of the form (1) . Then γ2 – ηγ2 1 ≤max 1 4,
Corollary 2.
Corollary 2. Let f ∈SS∗ tanh be of the form (1), then γ2 – γ2 1 ≤1 4. Equality is determined by using (24), (25) and (17).
Corollary 2. Let f ∈SS∗ tanh be of the form (1) , then γ2 – γ2 1 ≤1 4. Equality is determined by using (24) , (25) and (17) .
Theorem 7.
Theorem 7. Let f ∈SS∗ tanh. Then |γ3 – γ1γ2| ≤1 8. This result is sharp. Equality is determined by using (24), (25), (26) and (18).
Theorem 7. Let f ∈SS∗ tanh. Then |γ3 – γ1γ2| ≤1 8. This result is sharp. Equality is determined by using (24) , (25) , (26) and (18) .
Theorem 8.
Theorem 8. If f ∈SS∗ tanh is of the form (1), then H2,1 Ff /2 ≤1 16. This result is sharp. Equality is determined by using (24), (25),…
Theorem 8. If f ∈SS∗ tanh is of the form (1) , then H2,1 Ff /2 ≤1 16. This result is sharp. Equality is determined by using (24) , (25) , (26) and (17) .
Theorem 9.
Theorem 9. Let f ∈SS∗ tanh. Then |A2| ≤ 1 2, |A3| ≤ 1 2, |A4| ≤ 1 4. These bounds are sharp.
Theorem 9. Let f ∈SS∗ tanh. Then |A2| ≤ 1 2, |A3| ≤ 1 2, |A4| ≤ 1 4. These bounds are sharp.
Theorem 10.
Theorem 10. Let f ∈SS∗ tanh be of the form (1). Then A3 – ηA2 2 ≤max 1 2,
Theorem 10. Let f ∈SS∗ tanh be of the form (1) . Then A3 – ηA2 2 ≤max 1 2,
Corollary 3.
Corollary 3. If f ∈SS∗ tanh is of the form (1), then A3 – A2 2 ≤1 2. Equality is determined by using (31), (32) and (17).
Corollary 3. If f ∈SS∗ tanh is of the form (1) , then A3 – A2 2 ≤1 2. Equality is determined by using (31) , (32) and (17) .
Theorem 11.
Theorem 11. If f ∈SS∗ tanh is of the form (1), then |A4 – A2A3| ≤1 4. This result is sharp. Equality is determined by using (31), (32),…
Theorem 11. If f ∈SS∗ tanh is of the form (1) , then |A4 – A2A3| ≤1 4. This result is sharp. Equality is determined by using (31) , (32) , (33) and (18) . VFAST Transactions on Mathematics volume 12, Issue 2 2024 109
Theorem 12.
Theorem 12. If f ∈SS∗ tanh is of the form (1), then H2,2 f –1 ≤1 4. This result is sharp. Equality is determined by using (31), (32),…
Theorem 12. If f ∈SS∗ tanh is of the form (1) , then H2,2 f –1 ≤1 4. This result is sharp. Equality is determined by using (31) , (32) , (33) and (17) .