When you start any new management job, you will need to spend some time getting to know everyone in your team, understanding their role in the company, appreciating their strengths and weaknesses, and deciding what the company might need from them, as well as what they might from the company and/or their career.
All of that is as true for this person as it is for anyone else on your new team. The only difference is, you already have a clue before you've even met them: they wanted the job you got, which implies they have ambition, and may be frustrated not to have got it.
You will need to find out the extent of their ambition and the depth of their frustration - but this is just as true for anyone and everyone in your team. There is no reason to treat this person differently. Put to the side the fact that you got the job and they didn't - that's just business and nothing to feel guilty about. Simply try to understand, as you would for any of your staff, how this person is performing against what the company wants of them, and what this person wants from their career. Maybe they just weren't ready for a promotion or performing at the right level for it, and you might need to train them to replace you when you move on. Maybe they're not bothered, just saw the opportunity for a pay rise and applied on the off-chance. Maybe this is the final straw and if they can't get a promotion here, they'll go somewhere else. Maybe, maybe, maybe.
All of these 'maybe's are just as likely to be true, and carry just as much weight, for anyone else in your team too. This person deserves your leadership and understanding... just like everyone else in your team does too.