I believe I would go with Capt, I would assume you get a written contract to protect you from law suits and to protect the peoples rights also.
When you send out the contract, merely have the new price in place.
People would expect price increases in all things. Now of course expect to lose a few people, My mower for example has increaed his prices so I shoped my mowing out and got it cheaper than his old price by a firm that does a lot of my other neighbors already.
So not only does a new year mean that someone else may move in and advertise in the paper, by leaving door hangers and just beat you out of business. But some people just ever so often bid out a lot of services, internet service, telephone, car insurance and so on are all bidded out at times, I do most at least every two years.
And of course a price increase may cause some to shop around that would not have anyway.
If you don't use a contract, do you carry liabity insurance, ( i.e. mower throws rock though house window or car window, you knock over and break the concrete bird bath or yard item. If you do be sure and sell your professional service. If you use high speed mowers, or always include triming, besure and always sell your good points.
My mower a few years back, put a rock right though my very large three pain gas sealed picture window. About 800 to replace. I ended up sueing him in court when he did not want to pay for the damage, I don't think he stopped mowing after that.
I also would not do it in a letter, but before the season was ready, I would go by and visit each of them and merely disucss their mowing needs, ask them what they want done this year, try to sell more service, like mulching, some land scaping, maybe chemicals added. Then a small increase in mowing will never be noticed in the new contract.
Best way to increase price is to increase all of the service for more money in general