  The paper present new definitions for weighted DT moduli. Similarly, we a
general outcome in an equivalence of moduli of smoothness are obtained. It is
known that, any $r \in \mathbb{N}_{\circ}$ , $0<p \leq \infty$, $1 \leq \eta
\leq r$ and $\phi(x)=\sqrt{1-x^2}$, the inequalities $\omega^{\phi}_{i+1,r} \;
(f^{(r)}, \| \theta_{\mathcal{N}} \|)_{w_{\alpha, \beta}, p} \sim
\omega^{\phi}_{i,r+1} \; (f^{(r+1)}, \| \theta_{\mathcal{N}} \|)_{w_{\alpha,
\beta}, p}$ and $\omega^{\phi}_{i+\eta} \; (f, \| \theta_{\mathcal{N}}
\|)_{\alpha, \beta, p} \sim \| \theta_{\mathcal{N}} \|^{- \eta}
\omega^{\phi}_{i, 2 \eta} \; (f^{(2 \eta)}, \| \theta_{\mathcal{N}}
\|)_{\alpha+ \eta, \beta+ \eta, p}$ are valid.
