The problem is this: I have an abstract class that does some work in its constructor, and a set of child classes that implement the abstract class:
class Abs
Any instance of CustomizedClass1 is also an instance of ConcreteClass1, by definition, so it must be constructed as a valid ConcreteClass1 instance before the CustomizedClass1 constructor can run. Otherwise, what would happen if you called ConcreteClass1 methods on it? They'd be trying to operate on variables that haven't been initialized yet.
If you think you need to do this, chances are your design needs re-thinking. If you only want some of the functionality from ConcreteClass1, for example, that functionality could be factored out into a superclass of ConcreteClass1, and CustomizedClass1 could extend that to get just the functionality that it needs.
Please provide more information about the relationship between these classes.