  In this article, we refined the error term of some results of Levinson and
Montgomery \cite{LM}, Ki and Lee \cite{KL} on zero density estimates of
$\zeta^{(k)}$. We also have shown that all most all zeros of
Matsumoto-Tanigawa's $\eta_k$-function are near the critical line
$\frac{1}{2}$. Moreover, we obtain an asymptotic expression for the mean square
of the product of higher derivatives of Hardy's $Z$-function with Dirichlet
polynomials whose $n$-th coefficient is arbitrary and of size $\ll
n^{\epsilon}$, for $\epsilon >0$. For $k=0$ our result recovers the result of
Balasubramanian, Conrey and Heath-Brown \cite{BCH} %and also we for shorter
intervals with relatively small mollifier length. More generally, we consider
the mean square of the product of two arbitrary order derivatives of Hardy
$Z$-function with Dirichlet polynomials and by using this we obtain a similar
type of mean square result for the product of two arbitrary order derivatives
of the Riemann zeta function with mollifiers.
