= Solution
The <Chernoff bound> gives, for $0<\lambda<1/c$,
$$
\mathbb P(X\geq t)
\leq\exp\left\{-\lambda t+
\frac{\nu\lambda^2}{2(1-c\lambda)}\right\}.
$$
Writing $u=ct/\nu$, elementary differentiation shows that the exponent is minimized at
$$
\lambda_*=\frac1c\left(1-\frac1{\sqrt{1+2u}}\right).
$$
Substitution gives
$$
\mathbb P(X\geq t)\leq
\exp\left\{-\frac{\nu}{c^2}h(u)\right\},
\qquad
h(u)=1+u-\sqrt{1+2u}.
$$
Thus $h$ is the <Legendre transform> generated by the sub-Gamma cumulant bound after its natural rescaling.
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