  Let $\mathscr{M}$ be a finite von Neumann algebra with a faithful normal
tracial state $\tau$ and $\mathfrak{A}$ be a finite subdiagonal subalgebra of
$\mathscr{M}$ with respect to a $\tau$-preserving faithful normal conditional
expectation $\Phi$ on $\mathscr{M}$. Let $\Delta$ denote the Fuglede-Kadison
determinant corresponding to $\tau$. For $X \in \mathscr{M}$, define $|X| :=
(X^*X)^{\frac{1}{2}}$. In 2005, Labuschagne proved the so-called Jensen's
inequality for finite subdiagonal algebras i.e. $\Delta(\Phi(A)) \le \Delta(A)$
for an operator $A \in \mathfrak{A}$, thus resolving a long-standing open
problem posed by Arveson in 1967. In this article, we prove the following more
general result: $\tau(f( |\Phi(A)|)) \le \tau(f( |A|))$ for $A \in
\mathfrak{A}$ and any increasing continuous function $f : [0, \infty) \to
\mathbb{R}$ such that $f \circ \exp$ is convex on $\mathbb{R}$. Under the
additional hypotheses that $A$ is invertible in $\mathscr{M}$ and $f \circ
\exp$ is strictly convex, we have $\tau(f( |\Phi(A)|)) = \tau(f( |A|))$
$\Longleftrightarrow \Phi(A) = A$. As an application, we show that for $A \in
\mathfrak{A}$ the point spectrum of $A$ is contained in the point spectrum of
$\Phi(A)$, though such a conclusion does not hold in general for their spectra.
