  We compute the 't Hooft coupling correction to the infinite coupling
expression for the second order transport coefficient $\lambda_2$ in ${\cal
N}=4$ $SU(N_c)$ supersymmetric Yang-Mills theory at finite temperature in the
limit of infinite $N_c$, which originates from the $R^4$ terms in the low
energy effective action of the dual type IIB string theory. Using this result,
we show that the identity involving the three second order transport
coefficients, $2 \eta \tau_\Pi - 4 \lambda_1 - \lambda_2 =0$, previously shown
by Haack and Yarom to hold universally in relativistic conformal field theories
with string dual descriptions to leading order in supergravity approximation,
holds also at next to leading order in this theory. We also compute corrections
to transport coefficients in a (hypothetical) strongly interacting conformal
fluid arising from the generic curvature squared terms in the corresponding
dual gravity action (in particular, Gauss-Bonnet action), and show that the
identity holds to linear order in the higher-derivative couplings. We discuss
potential implications of these results for the near-equilibrium entropy
production rate at strong coupling.
