Fixed points of meromorphic functions and of their differences,divided differences and shifts

Let f(z) be a finite order meromorphic function and let c ∈ C {0} be a constant. If f(z) has a Borel exceptional value a ∈ C, it is proved that

$$\max \left\{ {\tau \left( {f\left( z \right)} \right),\tau \left( {{\Delta _c}f\left( z \right)} \right)} \right\} = \max \left\{ {\tau \left( {f\left( z \right)} \right),\tau \left( {f\left( {z + c} \right)} \right)} \right\} = \max \left\{ {\tau \left( {{\Delta _c}f\left( z \right)} \right),\tau \left( {f\left( {z + c} \right)} \right)} \right\} = \sigma \left( {f\left( z \right)} \right)$$

If f(z) has a Borel exceptional value b ∈ (C {0}) ∪ {∞}, it is proved that

$$\max \left\{ {\tau \left( {f\left( z \right)} \right),\tau \left( {\frac{{{\Delta _c}f\left( z \right)}}{{f\left( z \right)}}} \right)} \right\} = \max \left\{ {\tau \left( {\frac{{{\Delta _c}f\left( z \right)}}{{f\left( z \right)}}} \right),\tau \left( {f\left( {z + c} \right)} \right)} \right\} = \sigma \left( {f\left( z \right)} \right)$$

unless f(z) takes a special form. Here τ (g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z), and σ(g(z)) denotes the order of growth of g(z).