  We study Leggett-Garg inequalities (LGIs) for a two level system (TLS)
undergoing Markovian dynamics described by unital maps. We find analytic
expression of LG parameter $K_{3}$ (simplest variant of LGIs) in terms of the
parameters of two distinct unital maps representing time evolution for
intervals: $t_{1}$ to $t_{2}$ and $t_{2}$ to $t_{3}$. We show that the maximum
violation of LGI for these maps can never exceed well known L\"{u}ders bound of
$K_{3}^{L\ddot{u}ders}=3/2$ over the full parameter space. We further show that
if the map for the time interval $t_{1}$ to $t_{2}$ is non-unitary unital then
irrespective of the choice of the map for interval $t_{2}$ to $t_{3}$ we can
never reach L\"{u}ders bound. On the other hand, if the measurement operator
eigenstates remain pure upon evolution from $t_{1}$ to $t_{2}$, then depending
on the degree of decoherence induced by the unital map for the interval $t_{2}$
to $t_{3}$ we may or may not obtain L\"{u}ders bound. Specifically, we find
that if the unital map for interval $t_{2}$ to $t_{3}$ leads to the shrinking
of the Bloch vector beyond half of its unit length, then achieving the bound
$K_{3}^{L\ddot{u}ders}$ is not possible. Hence our findings not only establish
a threshold for decoherence which will allow for $K_{3} =
K_{3}^{L\ddot{u}ders}$, but also demonstrate the importance of temporal
sequencing of the exposure of a TLS to Markovian baths in obtaining L\"{u}ders
bound.
