The relationship between these two characters is somewhat complex. Oliver Bacon is in awe of the Duchess. So in that way, he is subordinate to her. The Duchess looks down on Oliver and people like him yet, at the same time, she needs him badly. So each is superior in a way to the other.
Oliver comes from a poor, lower-class background. Because of this, he wants badly to be accepted by people of the Duchess's class and status. This is why he allows himself to be fooled by the Duchess's pearls. Even though he is materially much better off than the Duchess and even though she needs him, he still feels inferior to her.
The Duchess is an aristocrat and so she thinks she is better than everyone else. But she has no money and so she needs Oliver. She is trying to ingratiate herself with him, trying to buy him off by giving him access to her social class. In this way, she is begging from him even though she looks down on him.
So there is something of a complicated relationship between the two with each being both superior to and subordinate to the other.