Applications to Entire Functions

Abstract

Let f(z) be an entire function. The function u(z) ≔ log|f(z)| is subharmonic in ℝ²(= ℂ). Hence the scale of growth subharmonic functions considered in Section 2.8 is transferred completely to entire functions. We will mark passing to entire function by changing index u for index f. For example,

$$ M\left( {r,f} \right): = M\left( {r,log\left| f \right|} \right),T\left( {r,f} \right): = T\left( {r,log\left| f \right|} \right). $$

If u(z) ≔ log|f(z)| has order ρ[u] = ρ, then f(z) has order ρ[f] ≔ ρ and so on.