  We introduce the notion of Krein-operator convexity in the setting of Krein
spaces. We present an indefinite version of the Jensen operator inequality on
Krein spaces by showing that if $(\mathscr{H},J)$ is a Krein space,
$\mathcal{U}$ is an open set which is symmetric with respect to the real axis
such that $\mathcal{U}\cap\mathbb{R}$ consists of a segment of real axis and
$f$ is a Krein-operator convex function on $\mathcal{U}$ with $f(0)=0$, then
\begin{eqnarray*} f(C^{\sharp}AC)\leq^{J}C^{\sharp}f(A)C \end{eqnarray*} for
all $J$-positive operators $A$ and all invertible $J$-contractions $C$ such
that the spectra of $A$, $C^{\sharp}AC$ and $D^{\sharp}AD$ are contained in
$\mathcal{U}$, where $D$ is a defect operator for $C^{\sharp}$.\\ We also show
that in contrast with usual operator convex functions the converse of this
implication is not true, in general.

Functional Analysis;Operator Algebras