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Abstract
In this paper, we obtain the Fekete-Szegö inequality for the generalized bi-subordinate
functions of complex order. The various results, which are presented in this paper, would
generalize those in related works of several earlier authors.
Mathematics Subject Classification (2010). 30C45, 30C80
Results & Lemmas (8)
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Lemma 2.2
Lemma 2.2 ([10]). Let the function w ∈B0 be given by w(z) = c1z + c2z2 + · · · (z ∈D), then for by every complex number s, c2 −sc2 1 ≤1 +…
Lemma 2.2 ([10]). Let the function w ∈B0 be given by w(z) = c1z + c2z2 + · · · (z ∈D) , then for by every complex number s, c2 −sc2 1 ≤1 + (|s| −1) |c1|2 . In the following theorem, we consider functional a3 −µa2 2 for γ nonzero complex num- ber and µ ∈C.
Theorem 2.3.
Theorem 2.3. Let the function f given by (1.1) be in the Sσ(λ, γ; φ). For γ ∈C⧹ 0 and µ ∈C, we have |a2| ≤|γ| B1 1 + λ, (2.2) |a3| ≤ |γ|…
Theorem 2.3. Let the function f given by (1.1) be in the Sσ(λ, γ; φ). For γ ∈C⧹{0} and µ ∈C, we have |a2| ≤|γ| B1 1 + λ , (2.2) |a3| ≤ |γ| |B1| 4 (1 + 2λ) max{2, (|s| + |t|)} (2.3) and a3 −µa2 2 ≤
Theorem 2.4.
Theorem 2.4. Let the function f given by (1.1) be in the Sσ(λ, γ; φ). For γ > 0 and µ ∈R, we have (1) If |B2| ≥B1, then a3 −µa2 2 ≤ …
Theorem 2.4. Let the function f given by (1.1) be in the Sσ(λ, γ; φ). For γ > 0 and µ ∈R, we have (1) If |B2| ≥B1, then a3 −µa2 2 ≤ γ|B2| 2(1+2λ) −(µ −1) γ2B2 1 (1+λ)2 if µ ≤1
Theorem 2.5.
Theorem 2.5. Let the function f given by (1.1) be in the Sσ(λ, γ; φ). For γ ∈C⧹ 0 and µ ∈R, we have (1) If (1+|sin θ|)|B2| 2B1 ≥1, then a3…
Theorem 2.5. Let the function f given by (1.1) be in the Sσ(λ, γ; φ). For γ ∈C⧹{0} and µ ∈R, we have (1) If (1+|sin θ|)|B2| 2B1 ≥1, then a3 −µa2 2 ≤
Corollary 2.6.
Corollary 2.6. If f ∈A is given by (1.1) belongs to the class Sσ [A, B], then (1) For µ ∈C, a3 −µa2 2 ≤ ( A−B 2 if |B| + |4 (1 −µ) (A −B)…
Corollary 2.6. If f ∈A is given by (1.1) belongs to the class Sσ [A, B] , then (1) For µ ∈C, a3 −µa2 2 ≤ ( A−B 2 if |B| + |4 (1 −µ) (A −B) −B| < 2 (A−B) 4 [|B| + |4 (1 −µ) (A −B) −B|] if |B| + |4 (1 −µ) (A −B) −B| ≥2 .
Corollary 2.7.
Corollary 2.7. If f ∈A is given by (1.1) belongs to the class Cσ [A, B], then (1) For µ ∈C, a3 −µa2 2 ≤ ( A−B 6 if |B| + |3 (1 −µ) (A −B)…
Corollary 2.7. If f ∈A is given by (1.1) belongs to the class Cσ [A, B] , then (1) For µ ∈C, a3 −µa2 2 ≤ ( A−B 6 if |B| + |3 (1 −µ) (A −B) −B| < 2 (A−B) 12 [|B| + |3 (1 −µ) (A −B) −B|] if |B| + |3 (1 −µ) (A −B) −B| ≥2 . (2) For µ ∈R, a3 −µa2 2
Corollary 2.8.
Corollary 2.8. If f ∈A is given by (1.1) belongs to the class S∗ σ[γ], then (i) For γ ∈C 0 and µ ∈C, a3 −µa2 2 ≤ ( |γ| if |1 + (1 −µ) 8γ| <…
Corollary 2.8. If f ∈A is given by (1.1) belongs to the class S∗ σ[γ], then (i) For γ ∈C\ {0} and µ ∈C, a3 −µa2 2 ≤ ( |γ| if |1 + (1 −µ) 8γ| < 1 |γ| 2 [|1 + (1 −µ) 8γ| + 1] if |1 + (1 −µ) 8γ| ≥1 .
Corollary 2.9.
Corollary 2.9. If f ∈A is given by (1.1) belongs to the class Cσ[γ], then (i) For γ ∈C 0 and µ ∈C, a3 −µa2 2 ≤ ( |γ| 3 if |1 + (1 −µ) 6γ| <…
Corollary 2.9. If f ∈A is given by (1.1) belongs to the class Cσ[γ], then (i) For γ ∈C\ {0} and µ ∈C, a3 −µa2 2 ≤ ( |γ| 3 if |1 + (1 −µ) 6γ| < 1 |γ| 2 [|1 + (1 −µ) 6γ| + 1] if |1 + (1 −µ) 6γ| ≥1 . (ii) For γ > 0 and µ ∈R,
Function classes studied:
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