Lower Bound Estimates of the Order of Meromorphic Solutions to Non-Homogeneous Linear Differential-Difference Equations

In this article, we deal with the order of growth of solutions of non-homogeneous linear differential-difference equation \begin{equation*} \sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=F(z), \end{equation*} where $A_{ij},$ $F\left(z\right)$ are entire or meromorphic functions and $c_i$ $(0,1,\mathrm{.}\mathrm{.}\mathrm{.},n)$ are non-zero distinct complex numbers. Under the sufficient condition that there exists one coefficient having the maximal lower order or having the maximal lower type strictly greater than the order or the type of other coefficients, we obtain estimates of the lower bound of the order of meromorphic solutions of the above equation.