  For a multipartite quantum state, the maximal violation of all Bell
inequalities constitutes a measure of its nonlocality [Loubenets, J. Math.
Phys. 53, 022201 (2012)]. In the present article, for the maximal violation of
Bell inequalities by a pure bipartite state, possibly infinite-dimensional, we
derive a new upper bound expressed in terms of the Schmidt coefficients of this
state. This new upper bound allows us also to specify general analytical
relations between the maximal violation of Bell inequalities by a bipartite
quantum state, pure or mixed, and such entanglement measures for this state as
"negativity" and "concurrence". To our knowledge, no any general analytical
relations between measures for bipartite nonlocality and entanglement have been
reported in the literature though, for a general bipartite state, specifically
such relations are important for the entanglement certification and
quantification scenarios. As an example, we apply our new results to finding
upper bounds on nonlocality of bipartite coherent states intensively discussed
last years in the literature in view of their experimental implementations.
