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Abstract

Let $f(z)$ be a finite order transcendental meromorphic function such that $\lambda (1/f(z))<\sigma (f(z))$, and let $c\in \mathbb {C}\setminus \{0\}$ be a constant such that $f(z+c)\not \equiv f(z)+c$. We mainly prove that $$\eqalign {\max\{\tau (f(z)), \tau (\Delta _c f(z))\}&=\max\{\tau (f(z)), \tau (f(z+c))\} \cr &=\max\{\tau (\Delta _c f(z)), \tau (f(z+c))\}=\sigma (f(z)), }$$where $\tau (g(z))$ denotes the exponent of convergence of fixed points of the meromorphic function $g(z)$, and $\sigma (g(z))$ denotes the order of growth of $g(z).$