We give a new lower bound in some inequalities for frames in a Hilbert space. If $\{f_i\}_{i\in I}$ is a Parseval frame for the Hilbert space $\mathbb{H}$ with frame operator $ S f =\sum_{i\in I} ⟨f, f_i⟩f_i$, then, for every $J\subset I$ and $f\in \mathbb{H}$, we have \[ eft(\dfrac{1+2lpha}{2+2lpha}\right) \|f\|^{2} eq um_{iı J} |⟨f,f_i⟩|^2+eft\|um_{iı J^{c}}⟨f,f_i⟩f_i\right\|^2, \] where $\alpha=\inf \left\lbrace R\left(\frac{\|S_{J^{c}}f\|}{\|S_{J}f\|}\right)\,:\, f\in\mathbb{H} , J\subset I \right\rbrace$ with Specht's ratio $R$. Also we obtain some improvements of the inequalities for general frames and alternate dual frames under suitable conditions. Our results refine the remarkable results obtained by Balan et al. and Gavruta.