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Abstract
In the present paper, the estimate of the third Hankel determinant
H3,1( f ) =
a1 a2 a3
a2 a3 a4
a3 a4 a5
for the class of starlike functions, i.e., for the class of analytic functions f standardly
normalized such that Re(zf ′(z)/ f (z)) > 0, z ∈D := {z ∈C : |z| < 1}, is improved.
Results & Lemmas (5)
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Lemma 2.1
Lemma 2.1 If p ∈P is of the form (1.5) with c1 ≥0, then 2c2 = c2 1 + (4 −c2 1)ζ, (2.1) 4c3 = c3 1 + (4 −c2 1)c1ζ(2 −ζ) + 2(4 −c2 1)(1…
Lemma 2.1 If p ∈P is of the form (1.5) with c1 ≥0, then 2c2 = c2 1 + (4 −c2 1)ζ, (2.1) 4c3 = c3 1 + (4 −c2 1)c1ζ(2 −ζ) + 2(4 −c2 1)(1 −|ζ|2)η (2.2) and 8c4 = c4 1 + (4 −c2 1)ζ
Proposition 2.2
Proposition 2.2 Let : [0, 3] × [0, 1] →R be a function defined by (t, x):= 96θ1(x) −8θ2(x)t + 3θ3(x)t2, (2.4) where for x ∈[0, 1], θ1(x):=…
Proposition 2.2 Let : [0, 3] × [0, 1] →R be a function defined by (t, x) := 96θ1(x) −8θ2(x)t + 3θ3(x)t2, (2.4) where for x ∈[0, 1], θ1(x) := 2 + 8x −x2 −6x3, θ2(x) := 16 + 67x −34x2 −53x3 + 2x4 123
Proposition 2.3
Proposition 2.3 Let : [1, 4] × [0, 1] →R be a function defined by (t, x):= 16ψ1(x) + 8ψ2(x)t + 3ψ3(x)t2, (2.6) where for x ∈[0, 1],…
Proposition 2.3 Let : [1, 4] × [0, 1] →R be a function defined by (t, x) := 16ψ1(x) + 8ψ2(x)t + 3ψ3(x)t2, (2.6) where for x ∈[0, 1], ψ1(x) := −2 + 27x + 21x2 −37x3 + x4, ψ2(x) := 10 −12x −9x2 + 20x3 + x4 and ψ3(x) := x(3 −5x −x2 −x3). Then (t, x) > 0 for 1 ≤t ≤4 and 0 ≤x ≤1.
Proposition 2.4
Proposition 2.4 Let: [3, 4] × [0, 1] →R be a function defined by
Proposition 2.4 Let : [3, 4] × [0, 1] →R be a function defined by
Theorem 2.5
Theorem 2.5 If f ∈S∗is the form (1.1), then |H3,1( f )| ≤8 9. (2.9) 123
Theorem 2.5 If f ∈S∗is the form (1.1), then |H3,1( f )| ≤8 9. (2.9) 123
Function classes studied:
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