  We give a new proof of the boundedness of bilinear Schur multipliers of
second order divided difference functions, as obtained earlier by Potapov,
Skripka and Sukochev in their proof of Koplienko's conjecture on the existence
of higher order spectral shift functions. Our proof is based on recent methods
involving bilinear transference and the H\"ormander-Mikhlin-Schur multiplier
theorem. Our approach provides a significant sharpening of the known asymptotic
bounds of bilinear Schur multipliers of second order divided difference
functions. Furthermore, we give a new lower bound of these bilinear Schur
multipliers, giving again a fundamental improvement on the best known bounds
obtained by Coine, Le Merdy, Potapov, Sukochev and Tomskova.
  More precisely, we prove that for $f \in C^2(\mathbb{R})$ and $1 < p, p_1,
p_2 < \infty$ with $\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}$ we have \[
\Vert M_{f^{[2]}}: S_{p_1} \times S_{p_2} \rightarrow S_p \Vert \lesssim \Vert
f'' \Vert_\infty D(p, p_1, p_2), \] where the constant $D(p, p_1, p_2)$ is
specified in Theorem 7.1 and $D(p, 2p, 2p) \approx p^4 p^\ast$ with $p^\ast$
the H\"older conjugate of $p$. We further show that for $f(\lambda) = \lambda
\vert \lambda \vert$, $\lambda \in \mathbb{R}$, for every $1 < p < \infty$ we
have \[ p^2 p^\ast \lesssim \Vert M_{f^{[2]}}: S_{2p} \times S_{2p} \rightarrow
S_p \Vert. \] Here $f^{[2]}$ is the second order divided difference function of
$f$ with $M_{f^{[2]}}$ the associated Schur multiplier. In particular it
follows that our estimate $D(p, 2p, 2p)$ is optimal for $p \searrow 1$.
