  It is known that the famous Heins Theorem (also known as the de Branges
Lemma) about the minimum of two entire functions of minimal type does not
extend to functions of finite exponential type. We study in detail pairs of
entire functions $f, g$ of finite exponential type satisfying
$\sup_{z\in\mathbb{C}}\min\{|f(z)|,|g(z)|\}<\infty.$ It turns out that $f$ and
$g$ have to be bounded on some rotating half-planes. We also obtain very close
upper and lower bounds for possible rotation functions of these half-planes.
