  Given a knot in 3-space, one can associate a sequence of Laurrent
polynomials, whose $n$th term is the $n$th colored Jones polynomial. The
Generalized Volume Conjecture states that the value of the $n$-th colored Jones
polynomial at $\exp(2 \pi i \a/n)$ is a sequence of complex numbers that grows
exponentially, for a fixed real angle $\a$. Moreover the exponential growth
rate of this sequence is proportional to the volume of the 3-manifold obtained
by $(1/\a,0)$ Dehn filling. In this paper we will prove that (a) for every
knot, the limsup in the hyperbolic volume conjecture is finite and bounded
above by an exponential function that depends on the number of crossings. (b)
Moreover, for every knot $K$ there exists a positive real number $\a(K)$ (which
depends on the number of crossings of the knot) such that the Generalized
Volume Conjecture holds for $\a \in [0, \a(K))$. Finally, we point out that a
theorem of Agol-Storm-W.Thurston proves that the bounds in (a) are optimal,
given by knots obtained by closing large chunks of the weave.
