Diagnosing $\Lambda$HDE model with statefinder hierarchy and fractional
  growth parameter
  Recently, a new dark energy model called $\Lambda$HDE was proposed. In this
model, dark energy consists of two parts: cosmological constant $\Lambda$ and
holographic dark energy (HDE). Two key parameters of this model are the
fractional density of cosmological constant $\Omega_{\Lambda0}$, and the
dimensionless HDE parameter $c$. Since these two parameters determine the
dynamical properties of DE and the destiny of universe, it is important to
study the impacts of different values of $\Omega_{\Lambda0}$ and $c$ on the
$\Lambda$HDE model. In this paper, we apply various DE diagnostic tools to
diagnose $\Lambda$HDE models with different values of $\Omega_{\Lambda0}$ and
$c$; these tools include statefinder hierarchy \{$S_3^{(1)}, S_4^{(1)}$\},
fractional growth parameter $\epsilon$, and composite null diagnostic (CND),
which is a combination of \{$S_3^{(1)}, S_4^{(1)}$\} and $\epsilon$. We find
that: (1) adopting different values of $\Omega_{\Lambda0}$ only has
quantitative impacts on the evolution of the $\Lambda$HDE model, while adopting
different $c$ has qualitative impacts; (2) compared with $S_3^{(1)}$,
$S_4^{(1)}$ can give larger differences among the cosmic evolutions of the
$\Lambda$HDE model associated with different $\Omega_{\Lambda0}$ or different
$c$; (3) compared with the case of using a single diagnostic, adopting a CND
pair has much stronger ability to diagnose the $\Lambda$HDE model.
