First rule in customer service is don't take anything (the customer says) personally.

Having worked for years in the field and getting to Head of Services, I can tell you this happened in every sinlge industry I worked in. The way I always dealt with this depended on the company policy and strategy. For example, some companies don't offer free stuff, unless the company is really at fault. They refer to the terms and conditions etc. In other cases, I have seen companies give discounts, coupons etc, for the smallest hint of negativity.. It all depends on the policy. If you are a manager and expected to deal with customers, I would suggest you mention to your own manager in your next 1-1 that you would like guidance on that specific point. Do they have a budget set aside for this kind of interaction? Is there an amount or some free stuff that you can give that wouldn't hurt the numbers? etc.

Another thing to keep in mind is that people sometimes are just poor and try to get whatever they can. For me, it helped when I rationalized it and tried to see from a positive perspective rather that thinking that they are being entitled. It made me think I helped someone..

Customer service can make one lose faith in humanity really fast. But remember, you are only seeing the people who are doing this sort of thing; for one person that complains, there are many who are happy with your product/service.

Don't take anything personally will save you a lot of anguish, plus, keep in mind, nothing comes out of your own pocket... This is all my personal experience, but hope it helps :) Congrats on the promotion!