In the article, we establish several inequalities for the Ramanujan constant function
$R(x)=-2\gamma-\psi(x)-\psi(1-x)$
on the interval
$(0, 1/2]$
, where
$\psi(x)$
is the classical psi function and
$\gamma=0.577215\cdots$
is the Euler-Mascheroni constant.
