By applying Nevanlinna’s second fundamental theorem and Borel type inequalities the author has improved some earlier results on unicity of meromorphic functions and obtained the following main result:
Let \(f(z)\) and \(g(z)\) be two distinct non-constant meromorphic functions satisfying for all \(z\in\mathbb{C}\): \[ f(z)=0\Leftrightarrow g(z)=0,\quad f(z)=1\Leftrightarrow g(z)=1,\text{ and }f(z)=\infty\Leftrightarrow g(z)=\infty. \] Assume further that \(\delta(a,f)+\delta(b,f)+\delta(\infty,f)>3/2\) for some \(a\neq 0,1\), \(b\neq 0,1\).
Then either (i) \(a\) and \(1-a\) are deficient values of \(f\) and \(g\), respectively, and \(f\) and \(g\) must satisfy one of the following three relations: 1) \((f-a)(g+a-1)=a(1-a)\), 2) \(f-(1-a)g=a\), 3) \(f=ag\),
or (ii) \(b\) and \((1-b)\) are deficient values of \(f\) and \(g\), respectively, and \(f\) and \(g\) must satisfy one of the following three relations: 1) \((f-b)(g+b-1)=b(1-b)\), 2) \(f-(1-b)g=b\), 3) \(f=bg\).