AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$
D
=
{
z
∈
C
:
|
z
|
<
1
}
, and $${{\mathcal {S}}}$$
S
be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$
f
(
z
)
=
z
+
∑
n
=
2
∞
a
n
z
n
for $$z\in {\mathbb {D}}$$
z
∈
D
. We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$
H
2
(
2
)
(
f
)
=
a
2
a
4
-
a
3
2
for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$
F
O
(
λ
,
β
)
of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$
1
/
2
≤
λ
≤
1
, and $$0<\beta \le 1$$
0
<
β
≤
1
. Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$
|
H
2
(
2
)
(
f
-
1
)
|
, where $$f^{-1}$$
f
-
1
is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$
|
H
2
(
2
)
(
f
)
|
and $$|H_2(2)(f^{-1})|$$
|
H
2
(
2
)
(
f
-
1
)
|
for strongly convex functions.
Multiplier U-processes: Sharp bounds and applications
