Abstract
AbstractWe find the sharp bound for the third Hankel determinant $$\begin{aligned} H_{3,1}(f):= \left| {\begin{array}{*{20}c} {a_{1} } & {a_{2} } & {a_{3} } \\ {a_{2} } & {a_{3} } & {a_{4} } \\ {a_{3} } & {a_{4} } & {a_{5} } \\ \end{array} } \right| \end{aligned}$$
H
3
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1
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f
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=
a
1
a
2
a
3
a
2
a
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a
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a
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a
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a
5
for analytic functions f with $$a_n:=f^{(n)}(0)/n!,\ n\in \mathbb N,\ a_1:=1,$$
a
n
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=
f
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n
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0
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/
n
!
,
n
∈
N
,
a
1
:
=
1
,
such that $$\begin{aligned} {{\,\mathrm{Re}\,}}f'(z)>0,\quad z\in \mathbb D:=\{z \in \mathbb C: |z|<1\}. \end{aligned}$$
Re
f
′
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z
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>
0
,
z
∈
D
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=
{
z
∈
C
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|
z
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<
1
}
.
Publisher
Springer Science and Business Media LLC
Reference24 articles.
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3. Carathéodory, C.: Über den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene werte nicht annehmen. Math. Ann. 64, 95–115 (1907)
4. Carlitz, L.: Hankel determinants and Bernoulli numbers. Tohoku Math. J. 5, 272–276 (1954)
5. Cho, N.E., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: The bounds of some determinants for starlike functions of order alpha. Bull. Malays. Math. Sci. Soc. 41, 523–535 (2018)
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