The authors show the following: If two meromorphic functions \(f(z)\) and \(g(z)\) have the same \(a\) – points with the same multiplicities, we denote this by \(f=a \leftrightarrows g=a\). The following theorems are proved:
Theorem 1. Let \(a_ 1,a_ 2, \dots, a_ p\) be \(p\) distinct finite complex numbers, and \(a_ i \neq 0\), \(1(i=1,2,\dots,p)\). Suppose that \(f(z)\) and \(g(z)\) are nonconstant meromorphic functions, and \(f(z) \not\equiv g(z)\). If \[ f=0 \leftrightarrows g=0,\;f=1 \leftrightarrows g=1,\;f=\infty \leftrightarrows g=0 \infty, \] and \[ \sum^ p_{i=1} \delta(a_ i,f)>{2p \over p+3}, \] then there exists one \(a_ j\) in \(a_ 1,a_ 2,\dots,a_ p\) such that \(a_ j\) is a Picard exceptional value of \(f(z)\), and \(f(z)\) and \(g(z)\) must satisfy exactly one of the following relations: (1) \((f-a_ j)\) \((g+a_ j-1) \equiv a_ j(1-a_ j)\), (2) \(f-(1-a_ j)g \equiv a_ j\), (3) \(f \equiv a_ jg\).
Theorem 2. Let \(a_ 1,a_ 2 ,\dots, a_ p\) be \(p\) distinct finite complex numbers, and \(a_ i \neq 0\), \(1(i=1,2,\dots,p)\). Suppose that \(f(z)\) and \(g(z)\) are nonconstant entire functions, and \(f(z) \not\equiv g(z)\). If \[ f=0 \leftrightarrows g=0,\;f=1 \leftrightarrows g=1, \] and \[ \sum^ p_{i=1} \delta(a_ i,f)>{p \over p+2}, \] then there exists one \(a_ j\) in \(a_ 1,a_ 2,\dots,a_ p\) such that \(a_ j\) and \(1- a_ j\) are Picard exceptional values of \(f(z)\) and \(g(z)\) respectively, and \[ (f-a_ j)(g+a_ j-1) \equiv a_ j (1-a_ j). \]