  It has been conjectured by Maldacena, Shenker, and Stanford [J. High Energy
Phys.~08 (2016) 106] that the exponential growth rate of the
out-of-time-ordered correlator (OTOC) $F(t)$ has a universal upper bound $2\pi
k_B T/\hbar$. Here we introduce a one-parameter family of out-of-time-ordered
correlators $F_\gamma(t)$ ($0\leq\gamma\leq 1$), which has as good properties
as $F(t)$ as a regularization of the out-of-time-ordered part of the squared
commutator $\langle [A(t), B(0)]^2\rangle$ that diagnoses quantum many-body
chaos, and coincides with $F(t)$ at $\gamma=1/2$. We rigorously prove that if
$F_\gamma(t)$ shows a transient exponential growth for all $\gamma$ in
$0\leq\gamma\leq 1$, that is, if the OTOC shows an exponential growth
regardless of the choice of the regularization, then the growth rate $\lambda$
does not depend on the regularization parameter $\gamma$, and satisfies the
inequality $\lambda\leq 2\pi k_B T/\hbar$.

Statistical Mechanics;High Energy Physics - Theory;Quantum Physics