Order is Maximum of Exponent of Convergence and Degree

Theorem
Let $f: \C \to \C$ be an entire function.

Let $\omega$ be its order.

Let $\tau$ be its exponent of convergence.

Let $h$ be the degree of the polynomial in its canonical factorization.

Then:
 * $\omega = \map \max {\tau, h}$

Proof
By Exponent of Convergence is Less Than Order:
 * $\tau \le \omega$

By Hadamard Factorization Theorem:
 * $h \le \omega$

Thus:
 * $\map \max {\tau, h} \le \omega$

By Order is Less Than Maximum of Exponent of Convergence and Degree:
 * $\omega \le \map \max {\tau, h}$