Salary/Compensation

In my new supervisory position, I will be managing two people who may already resent me. "Grace" worked with my predecessor for twenty years and says they are best friends. "Becky" applied for the job, but was not selected. Fortunately, the third employee is a recent hire who seems perfectly happy. How can I get off to a good start with this group?

As the manager of a newly-formed team, I am concerned about my most senior team member. "Stephanie" also applied for the manager position, but I was chosen instead.  I have avoided meeting with her one-on-one because I don't believe I owe her an explanation for my promotion.  Now I'm afraid Stephanie may be trying to undermine my authority and turn the team against me.  How should I handle this?