Let $G$ be a connected graph with order $n$ and size $m$. Let $D(G)$ and $Tr(G)$ be the distance matrix and diagonal matrix with vertex transmissions of $G$, respectively. For any real $α\in[0,1]$, the generalized distance matrix $D_α(G)$ of $G$ is defined as $$D_α(G)=αTr(G)+(1-α)D(G).$$ The largest eigenvalue of $D_α(G)$ is called the $D_α$ spectral radius or generalized distance spectral radius of $G$, denoted by $μ_α(G)$. In this paper, we establish a lower bound on the difference between the maximum vertex transmission and the $D_α$ spectral radius of non-transmission regular graphs, and we also characterize the extremal graphs attaining the bound.