I would like to prove a real $C^{\infty}$(polynomial) multivariable function $F : (a_1,a_2,...a_n) \rightarrow (b_1,b_2,...b_n) $ is lipchitz of parameter $l$

is it sufficient to prove the norm of dominant eigenvalue of $Jacobian(F)$ is less than $l$.

In other words, is the "spectral radius of Jaobian matrix is less than 1" a sufficient condition for $F$ to be lipschitz of paramter $l$?

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