The authors treat functional equations of the form \[ \sum^p_{j=1} a_j(z) f^{k_j}_j(z)\equiv 1,\tag{1} \] where \(p\geq 2\) an integer, and \(a_j(z)\), \(j= 1,\dots,p\) are meromorphic functions. They consider the solution \((f_1,\dots, f_p)\) of (1) satisfying a growth condition \(T(r, a_j)= o(\max_{1\leq k\leq p}T(r, f_k))\), \(1\leq j\leq p\), as \(r\to\infty\) and \(r\in E\), where \(E\) is an exceptional set of finite linear measure. Define a constant \(A_p\) as \(A_2= 1/2\), \(A_p= (2p- 3)/3\) if \(p= 3,4,5\), \(A+p= (2p+ 1 -2\sqrt{2p})/2\) if \(p\geq 6\). The main result is the following. Suppose that (1) possesses the solution satisfying the growth condition above. Then we have \[ \sum^p_{j=1} {1\over k_j}\geq {1\over p-1+ A_p}. \] Methods in the proofs are careful estimates for the counting functions in the Nevanlinna theory. This theorem is an improvement of the result in [\textit{K. W. Yu} and \textit{C. C. Yang} [Indian J. Pure Appl. Math. 33, No. 10, 1495--1502 (2002; Zbl 1023.30030)].