Abstract
In this paper, we study Fekete-Szegö problem for certain subclass of analytic functions with complex
order in the open unit disk by applying the q−analogue of Ruscheweyh operator in conjunction with the principle of
subordination between analytic functions.
Results & Lemmas (11)
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Lemma 1.1.
Lemma 1.1. [11] If p(z) = 1 + c1z + c2z2 +... is a function with positive real part in U and µ is a complex number, then c2 −µc2 1 ≤2 max…
Lemma 1.1. [11] If p(z) = 1 + c1z + c2z2 + ... is a function with positive real part in U and µ is 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 .
Lemma 1.2.
Lemma 1.2. [11] If p (z) = 1 + c1z + c2z2 +... is an analytic function with a positive real part in U, then c2 −νc2 1 ≤ −4ν + 2 if ν…
Lemma 1.2. [11] If p (z) = 1 + c1z + c2z2 + ... is an analytic function with a positive real part in U, then c2 −νc2 1 ≤ −4ν + 2 if ν ≤0 2 if 0 ≤ν ≤1, 4ν −2
Theorem 1.1.
Theorem 1.1. Let φ (z) = 1 + B1z + B2z2 +... with B1 ̸= 0. If f given by (1.1) belongs to the class Kq,b (γ, φ), then (1.7) a3 −µa2 2 ≤…
Theorem 1.1. Let φ (z) = 1 + B1z + B2z2 + ... with B1 ̸= 0. If f given by (1.1) belongs to the class Kq,b (γ, φ), then (1.7) a3 −µa2 2 ≤ |bB1| q[1+γq(q+1)][δ+2]q[δ+1]q max n 1; B2 B1 + 1 − [1+γq(q+1)] [δ+2]q
Corollary 1.1.
Corollary 1.1. Let φ (z) = 1 + B1z + B2z2 +... with B1 ̸= 0. If f given by (1.1) belongs to the class Sδ q (φ), then a3 −µa2 2 ≤ |B1|…
Corollary 1.1. Let φ (z) = 1 + B1z + B2z2 + ... with B1 ̸= 0. If f given by (1.1) belongs to the class Sδ q (φ), then a3 −µa2 2 ≤ |B1| q[δ+2]q[δ+1]q max n 1; B2 B1 + 1 − [δ+2]q
Corollary 1.2.
Corollary 1.2. Let φ (z) = 1 + B1z + B2z2 +... with B1 ̸= 0. If f given by (1.1) belongs to the class Kδ q (φ), then a3 −µa2 2 ≤ |B1|…
Corollary 1.2. Let φ (z) = 1 + B1z + B2z2 + ... with B1 ̸= 0. If f given by (1.1) belongs to the class Kδ q (φ), then a3 −µa2 2 ≤ |B1| q[1+q(q+1)][δ+2]q[δ+1]q max n 1; B2 B1 + 1 − [1+q(q+1)][δ+2]q
Corollary 1.3.
Corollary 1.3. Let φ (z) = 1 + B1z + B2z2 +... with B1 ̸= 0. If f given by (1.1) belongs to the class Sq (φ), then a3 −µa2 2 ≤ |B1| q(q+1)…
Corollary 1.3. Let φ (z) = 1 + B1z + B2z2 + ... with B1 ̸= 0. If f given by (1.1) belongs to the class Sq (φ), then a3 −µa2 2 ≤ |B1| q(q+1) max n 1; B2 B1 + (1 −(q + 1) µ) B1 q
Corollary 1.4.
Corollary 1.4. Let φ (z) = 1 + B1z + B2z2 +... with B1 ̸= 0. If f given by (1.1) belongs to the class Kq (φ), then a3 −µa2 2 ≤ |B1|…
Corollary 1.4. Let φ (z) = 1 + B1z + B2z2 + ... with B1 ̸= 0. If f given by (1.1) belongs to the class Kq (φ), then a3 −µa2 2 ≤ |B1| q(q+1)[1+q(q+1)] max n 1; B2 B1 + 1 −[1+q(q+1)] (1+q) µ
Corollary 1.5.
Corollary 1.5. Let φ (z) = 1 + B1z + B2z2 +... with B1 ̸= 0. If f given by (1.1) belongs to the class Sb (φ), then a3 −µa2 2 ≤|B1b| 2 max …
Corollary 1.5. Let φ (z) = 1 + B1z + B2z2 + ... with B1 ̸= 0. If f given by (1.1) belongs to the class Sb (φ) , then a3 −µa2 2 ≤|B1b| 2 max 1;
Theorem 1.2.
Theorem 1.2. Let φ (z) = 1 + B1z + B2z2 +... with B1 > 0 and B2 ≥0. Let σ1 = (1 + γq)2 [δ + 1]q bB2 1 + q (B2 −B1) [1 + γq (q + 1)] [δ…
Theorem 1.2. Let φ (z) = 1 + B1z + B2z2 + ... with B1 > 0 and B2 ≥0. Let σ1 = (1 + γq)2 [δ + 1]q bB2 1 + q (B2 −B1) [1 + γq (q + 1)] [δ + 2]q bB2 1 , (1.13) σ2 = (1 + γq)2 [δ + 1]q
Corollary 1.6.
Corollary 1.6. Let φ (z) = 1 + B1z + B2z2 +... with B1 > 0 and B2 ≥0. Let χ1 = [δ + 1]q B2 1 + q (B2 −B1) [δ + 2]q B2 1, χ2 = [δ + 1]q
Corollary 1.6. Let φ (z) = 1 + B1z + B2z2 + ... with B1 > 0 and B2 ≥0. Let χ1 = [δ + 1]q B2 1 + q (B2 −B1) [δ + 2]q B2 1 , χ2 = [δ + 1]q
Corollary 1.7.
Corollary 1.7. Let φ (z) = 1 + B1z + B2z2 +... with B1 > 0 and B2 ≥0. Let κ1 = [2]2 q [δ + 1]q B2 1 + q (B2 −B1) [3]q [δ + 2]q B2 1, κ2…
Corollary 1.7. Let φ (z) = 1 + B1z + B2z2 + ... with B1 > 0 and B2 ≥0. Let κ1 = [2]2 q [δ + 1]q B2 1 + q (B2 −B1) [3]q [δ + 2]q B2 1 , κ2 = [2]2
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