The Loewner theory is used to obtain the sharp upper bound for the functional
Re
{
e
2
i
θ
(
a
3
−
a
2
2
)
+
4
σ
e
i
θ
a
2
}
\operatorname {Re} \{ {e^{2i\theta }}({a_3} - a_2^2) + 4\sigma {e^{i\theta }}{a_2}\}
over the class of univalent functions
f
(
z
)
=
b
(
z
+
a
2
z
2
+
a
3
z
3
+
…
)
f(z) = b(z + {a_2}{z^2} + {a_3}{z^3} + \ldots )
which map the unit disc into itself;
θ
∈
R
,
σ
∈
[
0
,
1
]
\theta \in {\mathbf {R}},\sigma \in [0,1]
and
b
∈
(
0
,
1
]
b \in (0,1]
are fixed parameters.