Let
I
⊂
R
I\subset \mathbb {R}
be an interval and let
f
f
,
φ
\varphi
be arbitrary elements of
H
1
(
I
)
H^1(I)
and
B
M
O
(
I
)
BMO(I)
, respectively, with
∫
I
φ
=
0
\int _I\varphi =0
. The paper contains the proof of the estimate
∫
I
f
φ
≤
2
‖
f
‖
H
1
(
I
)
‖
φ
‖
B
M
O
(
I
)
,
\begin{equation*} \int _I f\varphi \leq \sqrt {2}\|f\|_{H^1(I)}\|\varphi \|_{BMO(I)}, \end{equation*}
and it is shown that
2
\sqrt {2}
cannot be replaced by a smaller universal constant. The argument rests on the existence of a special function enjoying appropriate size and concavity requirements.