  The aim of this paper is twofold. First, it offers a novel formula to
calculate the inner product of the bounded variation function in the Hilbert
space $\mathcal{H}$ associated with the fractional Brownian motion with Hurst
parameter $H\in (0,\frac12)$. This formula is based on a kind of decomposition
of the Lebesgue-Stieljes measure of the bounded variation function and the
integration by parts formula of the Lebesgue-Stieljes measure. Second, as an
application of the formula, we explore that as $T\to\infty$, the asymptotic
line for the square of the norm of the bivariate function
$f_T(t,s)=e^{-\theta|t-s|}1_{\{0\leq s,t\leq T\}}$ in the symmetric tensor
space $\mathcal{H}^{\odot 2}$ (as a function of $T$), and improve the
Berry-Ess\'{e}en type upper bound for the least squares estimation of the drift
coefficient of the fractional Ornstein-Uhlenbeck processes with Hurst parameter
$H\in (\frac14,\frac12)$. The asymptotic analysis of the present paper is much
more subtle than that of Lemma 17 in Hu, Nualart, Zhou(2019) and the improved
Berry-Ess\'{e}en type upper bound is the best improvement of the result of
Theorem 1.1 in Chen, Li (2021). As a by-product, a second application of the
above asymptotic analysis is given, i.e., we also show the Berry-Ess\'{e}en
type upper bound for the moment estimation of the drift coefficient of the
fractional Ornstein-Uhlenbeck processes where the method is obvious different
to that of Proposition 4.1 in Sottinen, Viitasaari(2018).
