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Abstract
Let S∗
q (φ) and Cq(φ) denote the classes of normalized functions
f(z) = z +a2z2 +a3z3 +..., which are defined in the open unit disk D and satisfying
zDqf(z)/f(z) ≺φ(z) and Dq(zDqf(z))/Dqf(z) ≺φ(z), where φ is the function
with real part, respectively. In this paper, we investigate new results of Fekete-
Szeg¨o inequalities for the classes S∗
q (φ) and Cq(φ).
2010 Mathematics Subject Classification: 30C45.
Results & Lemmas (7)
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Lemma 1.
Lemma 1. [16] Let p ∈P with p(z) = 1 + c1z + c2z2 +..., then |cn| ≤2 for n ≥1. If |c1| = 2, then p(z) ≡p1(z) = 1+γ1z 1−γ2z with γ1 = c1 2.…
Lemma 1. [16] Let p ∈P with p(z) = 1 + c1z + c2z2 + ..., then |cn| ≤2 for n ≥1. If |c1| = 2, then p(z) ≡p1(z) = 1+γ1z 1−γ2z with γ1 = c1 2 . Conversely, if p(z) ≡p1(z) for some |γ1| = 1, then c1 = 2γ1 and |c1| = 2. Furthermore, we have c2 −c2 1 2 ≤2 −|c1|2 2 . 57
Theorem 2.
Theorem 2. Let φ(z) = 1+B1z +B2z2 +..., where the coefficients Bn are real with B1 ̸= 0. If f belongs to the class S∗ q (φ), then |a2| ≤ |B1|…
Theorem 2. Let φ(z) = 1+B1z +B2z2 +..., where the coefficients Bn are real with B1 ̸= 0 . If f belongs to the class S∗ q (φ), then |a2| ≤ |B1| [2]q −1, (7) |a3| ≤ |B1| [3]q −1max 1,
Theorem 3.
Theorem 3. Let µ be a nonzero complex number and let f ∈S∗ q (φ), then |a3 −µa2 2| ≤ |B1| [3]q −1max 1,
Theorem 3. Let µ be a nonzero complex number and let f ∈S∗ q (φ), then |a3 −µa2 2| ≤ |B1| [3]q −1max 1,
Corollary 4.
Corollary 4. Taking q →1−in Theorem 3, we obtain |a3 −µa2 2| ≤|B1| 2 max 1,
Corollary 4. Taking q →1−in Theorem 3, we obtain |a3 −µa2 2| ≤|B1| 2 max 1,
Theorem 5.
Theorem 5. Let φ(z) = 1+B1z +B2z2 +..., where the coefficients Bn are real with B1 ̸= 0. If f belongs to the class Cq(φ), then |a2| ≤ |B1|…
Theorem 5. Let φ(z) = 1+B1z +B2z2 +..., where the coefficients Bn are real with B1 ̸= 0. If f belongs to the class Cq(φ), then |a2| ≤ |B1| [2]q([2]q −1), (19) |a3| ≤ |B1| [3]q([3]q −1)max 1,
Theorem 6.
Theorem 6. Let µ be a nonzero complex number and let f ∈Cq(φ), then |a3 −µa2 2| ≤ |B1| [3]q([3]q −1)max 1,
Theorem 6. Let µ be a nonzero complex number and let f ∈Cq(φ), then |a3 −µa2 2| ≤ |B1| [3]q([3]q −1)max 1,
Corollary 7.
Corollary 7. Taking q →1−in Theorem 6, we obtain |a3 −µa2 2| ≤|B1| 6 max 1,
Corollary 7. Taking q →1−in Theorem 6, we obtain |a3 −µa2 2| ≤|B1| 6 max 1,
Function classes studied:
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