Constructing transcendental entire functions

In this talk we use a lemma of Clunie and Kovari to construct some examples of transcendental entire functions with specific properties of their maximum modulus.  It is a very useful tool that allows us to obtain transcendental entire functions with certain properties by first constructing real, increasing, convex functions. In particular, we give an example of a transcendental entire function whose lower order is different from its order.