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 normalised 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
. Let F be the inverse function of f defined in some set $$|\omega |\le r_{0}(f)$$
|
ω
|
≤
r
0
(
f
)
, and be given by $$F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$$
F
(
ω
)
=
ω
+
∑
n
=
2
∞
A
n
ω
n
. We prove the sharp inequalities $$-1/3 \le |A_4|-|A_3| \le 1/4$$
-
1
/
3
≤
|
A
4
|
-
|
A
3
|
≤
1
/
4
for the class $${\mathcal {K}}\subset {\mathcal {S}}$$
K
⊂
S
of convex functions, thus providing an analogue to the known sharp inequalities $$-1/3 \le |a_4|-|a_3| \le 1/4$$
-
1
/
3
≤
|
a
4
|
-
|
a
3
|
≤
1
/
4
, and giving another example of an invariance property amongst coefficient functionals of convex functions.