Should we ignore our old customers to get new ones?

I have seen many businesses offer a special discount or a free gift to their new customers. If they have introduced this gift or discount from the first day of their business, then it sounds really good. Because all their existing customers already have taken this opportunity, so when the new customers take this offer, those existing customers feel good. But when an existing customers came know about such an offer, which is not for them, but for new customers, they feel cheated. And if they are your loyal customers, then you may loose them in the process of attracting new customers.