How should a hegemonic state deal with the rise of another power?
This depends very much upon which theory of international relations one believes in.
To realists, a hegemonic state should react to the rise of another power with great wariness. The hegemonic state should take steps to ensure that its own power will remain greater than that of the rising power. If the rising country continues to build its capabilities to the point where it might threaten the hegemon, war might even become necessary. This might be seen as how WWI got started, with France and England as hegemonic countries trying to prevent Germany from becoming too strong.
To idealists, the correct approach is to entangle the rising power in various international entities. A rising power should not be directly resisted. Instead, the hegemonic country should attempt to get the rising power to participate in things such as the United Nations. It should attempt to create multiple levels of contact with the rising nation to allow for open communications and to prevent misunderstandings. This might be seen as the approach that the US is taking with China as it tries to encourage Chinese participation in international affairs (such as North Korea) and to engage with China through such things as military contacts.
These are very different approaches, but each would be advocated by a particular group of scholars of international relations.