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Abstract
Estimates for initial coefficients of Taylor-Maclaurin series of bi-univalent functions belonging to certain classes defined by subordination are obtained. Our estimates improve upon the earlier known estimates for second and third coefficient. The bound for the fourth coefficient is new. In addition, bound for the fifth coefficient is obtained for bi-starlike and strongly bi-starlike functions of order $ρ$ and $β$ respectively.
Results & Lemmas (9)
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Theorem 2.2.
Theorem 2.2. Let the function f given by (1.1) be in the class ST λ σ (φ) for λ ≥0. (a) If (1 + 2λ)2B1 ≤|(1 + 4λ)B2 1 + (B1 −B2)(1 + 2λ)2|,…
Theorem 2.2. Let the function f given by (1.1) be in the class ST λ σ (φ) for λ ≥0. (a) If (1 + 2λ)2B1 ≤|(1 + 4λ)B2 1 + (B1 −B2)(1 + 2λ)2|, then the coefficients a2, a3 and a4 satisfies |a2| ≤ B1 √B1 p |(1 + 4λ)B2 1 + (B1 −B2)(1 + 2λ)2| , |a3| ≤min B1
Corollary 2.3.
Corollary 2.3. Let f given by (1.1) be in the class STσ(ϕ). (a) For B1 ≤|B2 1 + B1 −B2| |a2| ≤ B1 √B1 p |B2 1 + B1 −B2|, (2.35) |a3| ≤min…
Corollary 2.3. Let f given by (1.1) be in the class STσ(ϕ). (a) For B1 ≤|B2 1 + B1 −B2| |a2| ≤ B1 √B1 p |B2 1 + B1 −B2| , (2.35) |a3| ≤min B1 2|B2 1 + (B1 −B2)|
Theorem 2.1
Theorem 2.1]. Definition 2.7. For λ ≥0, the class Mλ(ϕ) consists of functions f ∈A satisfying λ 1 + zf ′′(z) f ′(z) + (1 −λ)zf ′(z) f…
Theorem 2.1]. Definition 2.7. For λ ≥0, the class Mλ(ϕ) consists of functions f ∈A satisfying λ 1 + zf ′′(z) f ′(z) + (1 −λ)zf ′(z) f ′(z) ≺ϕ (z ∈D). The class Mλ σ (ϕ) consists of f ∈σ such that f, g ∈Mλ(ϕ) where g is the analytic continuation of f −1 to the unit disk D.
Theorem 2.8.
Theorem 2.8. Let f ∈Mλ σ (φ).
Theorem 2.8. Let f ∈Mλ σ (φ).
Corollary 2.9.
Corollary 2.9. Let f ∈CVσ(φ). (a) If 2B1 ≤|B2 1 + 2(B1 −B2)|, then |a2| ≤ B1 √B1 p 2|B2 1 + 2(B1 −B2)|, |a3| ≤min B1 6|B2 1 + 2(B1 −B2)|
Corollary 2.9. Let f ∈CVσ(φ). (a) If 2B1 ≤|B2 1 + 2(B1 −B2)|, then |a2| ≤ B1 √B1 p 2|B2 1 + 2(B1 −B2)| , |a3| ≤min B1 6|B2 1 + 2(B1 −B2)|
Theorem 3.1.
Theorem 3.1. Let f(z) ∈STσ(ρ). For 0 ≤ρ ≤1/2, we have |a5| ≤2 3(1 −ρ) + 3 2(1 −ρ)2 + 2 3 √ 2(1 −ρ)3/2.
Theorem 3.1. Let f(z) ∈STσ(ρ). For 0 ≤ρ ≤1/2, we have |a5| ≤2 3(1 −ρ) + 3 2(1 −ρ)2 + 2 3 √ 2(1 −ρ)3/2.
Corollary 3.2.
Corollary 3.2. Let f(z) = z + P∞ n=2 anzn, (z ∈D) be bi-starlike function. Then |a5| ≤ 13 6 + 2 √ 2 3 ≃3.10947. Now we consider the class…
Corollary 3.2. Let f(z) = z + P∞ n=2 anzn, (z ∈D) be bi-starlike function. Then |a5| ≤ 13 6 + 2 √ 2 3 ≃3.10947. Now we consider the class SSσ(β) of strongly bi-starlike fubctions of order β. Ali and Singh [2] proved that if f is srongly starlike function of order β, then |a5| ≤
Theorem 3.3.
Theorem 3.3. Let the function f ∈SSσ(β). Then for 1/2 ≤β ≤1, |a5| ≤β 9
Theorem 3.3. Let the function f ∈SSσ(β). Then for 1/2 ≤β ≤1, |a5| ≤β 9
Corollary 3.4.
Corollary 3.4. Let f(z) given by (1.1) be in the class SSσ(1/2). Then |a5| ≤1 36 332 27 + √ 6 ≃0.409605. References [1] R.M. Ali, S.K.…
Corollary 3.4. Let f(z) given by (1.1) be in the class SSσ(1/2). Then |a5| ≤1 36 332 27 + √ 6 ≃0.409605. References [1] R.M. Ali, S.K. Lee, V. Ravichandran, S. Supramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett. 25 (2012), no. 3, 344–351. [2] R. M. Ali and V. Singh, On the fourth and fifth coefficients of strongly starlike functions, Results Math. 29 (1996), no. 3-4, 197–202. [3] D. A. Brannan and T. S. Taha, On some classes of bi-univalent
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