On a class of meromorphic functions with deficient zeros and poles

Lemma 1. Let $S_1$ be the set of real numbers $$2\pi$$ , $\omega_1$ , $\omega_2$ , $\cdots$ , $\omega_m$ and S₂ the set consisting of the real numbers $$\psi_1, \psi_2, \cdots, \psi_n$$ . Assume - (i) that the set of real numbers $2\pi$ , $\omega_1$ , $\cdots$ , $\omega_k$ , $(0 \le k \le m)$ is a maximal linearly independent subset of $S_1$ , - (ii) that the real numbers $$2\pi$$ , $\omega_1$ , $\omega_2$ , $\cdots$ , $\omega_k$ , $\psi_1$ , $\psi_2$ , $\cdots$ , $\psi_n$ are linearly independent. Then given $\varepsilon(>0)$ there exists an increasing sequence of positive integers $\{L_s\}_{s=1}^{\infty}$ and sequences of positive integers $\{M_{s,i}\}_{s=1}^{\infty}$ , $(i=1,2,\cdots,m)$ ; $\{N_s,i\}_{s=1}^{\infty}$ $(j=1,2,\cdots,n)$ ; such that for $s=1,2,3,\cdots$ $$|L_s\omega_i-2\pi M_{s,i}|<\varepsilon, \qquad \qquad (i=1,2,\cdots,m)\;,$$ (3.2) $$|L_s\psi_j - (2N_{s,j}+1)\pi| < \varepsilon, \qquad (j=1,2,\cdots,n)$$ , and all $s \geq s_0$ $$\frac{L_{s+1}}{L_s} \leqq 2.$$