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Abstract
We investigate the third Hankel determinant problem for
some starlike functions in the open unit disc, that are related to shell-
like curves and connected with Fibonacci numbers. For this, firstly, we
prove a conjecture, posed in [17], for sharp upper bound of second Hankel
determinant. In the sequel, we obtain another sharp coefficient bound
which we apply in solving the problem of the third Hankel determinant
for these functions.
AMS Subject Classification: 30C45, 30C50.
Results & Lemmas (8)
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Lemma 1.1.
Lemma 1.1. ([12]) Let p ∈P with p(z) = 1 + c1z + c2z2 + · · ·, then |cn| ≤2, for n ≥1. (1.4) If |c1| = 2, then p(z) ≡p1(z) ≡(1 + xz)/(1…
Lemma 1.1. ([12]) Let p ∈P with p(z) = 1 + c1z + c2z2 + · · · , then |cn| ≤2, for n ≥1. (1.4) If |c1| = 2, then p(z) ≡p1(z) ≡(1 + xz)/(1 −xz) with x = c1 2 . Conversely, if p(z) ≡p1(z) for some |x| = 1, then c1 = 2x. Furthermore, we have c2 −c2 1 2 ≤2 −|c1|2 2
Lemma 1.2.
Lemma 1.2. ([14]) Let p ∈P with coefficients cn as above, then |c1c2 −c3| ≤2. (1.6)
Lemma 1.2. ([14]) Let p ∈P with coefficients cn as above, then |c1c2 −c3| ≤2. (1.6)
Lemma 1.3.
Lemma 1.3. ([9]) Let p ∈P with coefficients cn as above, then |c3 −2c1c2 + c3 1| ≤2. (1.7)
Lemma 1.3. ([9]) Let p ∈P with coefficients cn as above, then |c3 −2c1c2 + c3 1| ≤2. (1.7)
Lemma 1.4.
Lemma 1.4. ([16]) If f(z) = z + P∞ n=2 anzn belongs to the class SL, then |an| ≤|τ|n−1un, (1.8) where un is the sequence of Fibonacci…
Lemma 1.4. ([16]) If f(z) = z + P∞ n=2 anzn belongs to the class SL, then |an| ≤|τ|n−1un, (1.8) where un is the sequence of Fibonacci numbers and τ = 1− √ 5 2 . Equality holds in (1.8) for the function f0(z) = z 1−αz−α2z2 .
Lemma 1.5.
Lemma 1.5. ([13]) If f(z) = z + P n=2 ∞anzn belongs to the class SL, then |a3 −λa2 2| ≤τ 2(2 + λ) for all λ ∈C. (1.9) The above estimation…
Lemma 1.5. ([13]) If f(z) = z + P n=2 ∞anzn belongs to the class SL, then |a3 −λa2 2| ≤τ 2(2 + λ) for all λ ∈C. (1.9) The above estimation is sharp. If λ > 0, then the equality in (1.9) is attained by the function f0(z) = z 1−αz−α2z2 while by the function −f0(−z), when λ ≤0. Especially, when λ = 1 in (1.9), we obtain |a3 −a2 2| ≤3τ 2. In this study, we use ideas and techniques used in geometric function theory. The central problem considered here is the sharp upper bounds for the functionals |H2
Theorem 2.1.
Theorem 2.1. If f(z) = z + a2z2 +... belongs to SL, then |H2(2)| = |a2a4 −a2 3| ≤τ 4. (2.1) The bound is sharp.
Theorem 2.1. If f(z) = z + a2z2 + . . . belongs to SL, then |H2(2)| = |a2a4 −a2 3| ≤τ 4. (2.1) The bound is sharp.
Theorem 2.2.
Theorem 2.2. If f(z) = z + a2z2 +... belongs to SL, then |a2a3 −a4| ≤|τ|3. (2.12) The bound is sharp.
Theorem 2.2. If f(z) = z + a2z2 + . . . belongs to SL, then |a2a3 −a4| ≤|τ|3. (2.12) The bound is sharp.
Theorem 2.3.
Theorem 2.3. If f(z) = z + a2z2 +... belongs to SL, then |H3(1)| ≤20τ 6. (2.16)
Theorem 2.3. If f(z) = z + a2z2 + . . . belongs to SL, then |H3(1)| ≤20τ 6. (2.16)
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