  An operator $T$ acting on a Hilbert space is called $(\alpha ,\beta)$-normal
($0\leq \alpha \leq 1\leq \beta $) if \begin{equation*} \alpha ^{2}T^{\ast
}T\leq TT^{\ast}\leq \beta ^{2}T^{\ast}T. \end{equation*} In this paper we
establish various inequalities between the operator norm and its numerical
radius of $(\alpha ,\beta)$-normal operators in Hilbert spaces. For this
purpose, we employ some classical inequalities for vectors in inner product
spaces.
