You could also ask this question on Workplace.SE.
Here I'll try to give an answer that relevant to Buddhism .SE.
As I speak kindly with my subordinates, they are taking advantage of me by commanding me to do their work and are trying to dominate.
It has happened, when I was working somewhere, that a friend/colleague told me that he'd decided to leave and go and work elsewhere. Instead of being annoyed (that they were leaving), I wanted to be happy for them (i.e. mudita): that they were free to find another job that they preferred.
An opposite (a negative), to be avoided, is to have a "mind of ill-will". I think you can (should) "speak kindly to your subordinates" and avoid having ill-will towards them.
That needn't mean they "dominate you", though, for a few reasons:
"Compassion" means, "may they be free from suffering". You seem to be treating this as a zero-sum game (they win if you lose, and compassion means that you should let them win). Instead of "zero-sum" I prefer to see work as a cooperative endeavour, i.e. "we win if we cooperate". You can want them to be free from suffering, without wanting them to dominate you. You might also do this with children or animals: if you know what's good and bad for them, you tell them and insist on that.
Also, "equanimity": part of the Buddhist teaching on equanimity is to reflect that everyone is the owner of their kamma. For example once, when I was working as a software developer, I had one colleague, who couldn't do the work, and who smelled of alcohol at work. I told my boss, and he was fired. I didn't have ill-will towards this man, and it wasn't my fault he was drinking: but this wasn't the right job for him.
Buddhism has a concept of "right livelihood", i.e. that in order to be moral you shouldn't have immoral employment. Buddhism doesn't have much to say about it for lay people, there are very few rules describing right livelihood for lay-people. Nevertheless I think that one of the rules is "not cheating": for example if your employer hires you to be a manager, then you owe it (to your employer) to be a good manager. This needn't mean being a cruel manager (if it does then maybe you need a different employer), but IMO your job as manager is to balance the needs of the employer with the needs of the employee (and the rest of society too, i.e. the employer's owners and customers etc.).
So for example you define work, assign it to employees, supervise the quality of the work, integrate their output with other people's work, etc. The fact that you're trying to do right livelihood means that it's inappropriate as a manager to be a push-over, to be too dominated. You might find it help to read this collection of definitions of right speech ... which even permits some things said which other people don't like, which they find "unendearing and disagreeable":
In the case of words that the Tathagata knows to be factual, true, beneficial, but unendearing & disagreeable to others, he has a sense of the proper time for saying them.
That quote is taken from the Abhaya Sutta, which also includes this line:
What do you think, prince: If this young boy, through your own negligence or that of the nurse, were to take a stick or a piece of gravel into its mouth, what would you do?"
I would take it out, lord. If I couldn't get it out right away, then holding its head in my left hand and crooking a finger of my right, I would take it out, even if it meant drawing blood. Why is that? Because I have sympathy for the young boy.
Another concept more-or-less related to Buddhism might be being impersonal. For example when I was the "chief software developer" at a company, I didn't think of it as my job to supervise the people: I saw it as my job to supervise the software that was being created. I'd talk to people about the software (e.g. "this software which you wrote is good and/or bad"), not try to dominate the people (e.g. "you are good and/or bad").
When reviewing their software I tried to have three (not just two) categories of comment:
- This is good: acceptable, well done.
- This is, in my opinion, not (yet) good enough, for this and that reason: please change it in this and that way before I accept it.
- Well, OK; I wouldn't have done it that way. FYI I would have done it so and so. If you like/agree/prefer you could change it to include my suggestion, or we can keep it the way you've done it: it's acceptable.
Giving my opinion on the software was something I was relatively competent at. Because I was focused on the software, people couldn't easily "dominate" me except my giving me high-quality software, which is what I (and my employer) wanted.
Similarly you find it better to be shoulder-to-shoulder with your subordinates, assessing the work, working together, instead of face-to-face and trying to dominate each other.
Also I remember, just once, it happened that someone wasn't willing to let me review their changes to the software. Well, responsibility and authority tend to go together. In this case I was (part) responsible for the quality of the software and I had the corresponding authority (e.g. I had the Administrator password and authority to use it). Instead of arguing with the person, I used Administrator functionality to remove their write-access to the software, so that they couldn't (so that only I could) write their changes. That was kind of forceful but it was the really least amount of force, harmless, an appropriate response to the risk/threat of unreviewed software being written.
They have started speaking in rude ways to me as they have started feeling that I am not harmful for them and cannot do anything.
It might be worth telling them: "Don't be rude to me. I'm trying to do a useful job here, and trying to help you do a good job."
If that doesn't work, you might talk to your boss about it. Note that "X is rude to me" and "X is rude to others" and "X isn't doing their job" and "X doesn't accept my advice" are various different statements you could make to your boss.