  We consider three modes $A$, $B$ and $C$ and derive continuous variable
monogamy inequalities that constrain the distribution of bipartite entanglement
amongst the three modes. The inequalities hold for all such tripartite states,
without the assumption of Gaussian states, and are based on measurements of two
conjugate quadrature phase amplitudes $X_{i}$ and $P_{i}$ at each mode $i=A,B$.
The first monogamy inequality is $D_{BA}+D_{BC}\geq1$ where $D_{BA}<1$ is the
widely used symmetric entanglement criterion, for which $D_{BA}$ is the sum of
the variances of $(X_{A}-X_{B})/2$ and $(P_{A}+P_{B})/2$. A second monogamy
inequality is
$Ent_{BA}Ent_{BC}\geq\frac{1}{\left(1+(g_{BA}^{(sym)})^{2}\right)\left(1+(g_{BC}^{(sym)})^{2}\right)}$
where $Ent_{BA}<1$ is the EPR variance product criterion for entanglement. Here
$Ent_{BA}$ is a normalised product of variances of $X_{B}-g_{BA}^{(sym)}X_{A}$
and $P_{B}+g_{BA}^{(sym)}P_{A}$, and $g_{BA}^{(sym)}$ is a parameter that gives
a measure of the symmetry between the moments of $A$ and $B$. We also show that
the monogamy bounds are increased if a standard steering criterion for the
steering of $B$ is not satisfied. We illustrate the monogamy for continuous
variable tripartite entangled states including the effects of losses and noise,
and identify regimes of saturation of the inequalities. The monogamy relations
explain the experimentally observed saturation at $D_{AB}=0.5$ for the
entanglement between $A$ and $B$ when both modes have 50\% losses, and may be
useful to establish rigorous bounds of correlation for the purpose of quantum
key distribution protocols.
