In the class of weighted spaces of entire functions $$ B_{\Phi(x,y)}=\Bigl\{f(z)\in A_\infty:\sup_{z\in C}\frac{|f(z)|}{\Phi(x,y)}<\infty\Bigr\}\quad(z=x+iy), $$ where $\Phi(x,y)$ is a continuous function on $R^2$ possessing certain additional properties, estimates are obtained for the norms of derivatives and norms of functions involving a translation of the independent variable in terms of the norm of the original function. These estimates are then used to prove the existence and uniqueness of solutions in the spaces $B_{\Phi(x,y)}$ of linear differential-difference equations of infinite order with constant coefficients.