For a continuous and positive function
$w(\lambda )$
,
$\lambda>0$
and
$\mu $
a positive measure on
$(0,\infty )$
, we consider the integral transform
$$ \begin{align*} \mathcal{D}( w,\mu ) ( T) :=\int_{0}^{\infty }w(\lambda) ( \lambda +T) ^{-1}\,d\mu ( \lambda ) , \end{align*} $$
where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among other things that if B,
$A>0,$
then
$\mathcal {D}( w,\mu ) $
is operator subadditive on
$(0,\infty ) $
, that is,
$$ \begin{align*} \mathcal{D}( w,\mu ) ( A) +\mathcal{D}( w,\mu) ( B) \geq \mathcal{D}( w,\mu )(A+B). \end{align*} $$
From this, we derive that if
$f:[0,\infty )\rightarrow \mathbb {R}$
is an operator monotone function on
$[0,\infty )$
, then the function
$[ f( t) -f( 0) ] t^{-1}$
is operator subadditive on
$( 0,\infty ) .$
Also, if
$f:[0,\infty )\rightarrow \mathbb {R}$
is an operator convex function on
$[0,\infty )$
, then the function
$[ f( t) -f( 0) -f_{+}^{\prime }( 0) t ] t^{-2}$
is operator subadditive on
$( 0,\infty ) .$