What is the best policy to deal with students using their cellphones inside the class?

Is it OK to prevent students from using their cellphones during the lecture by stating that in the syllabus?

and how to deal with students who insist on using their cellphones during the lecture?

(Most of what they do is either to text message or to browse the internet. Often I see those who are used to use the cellphones during the lecture smiling or showing something on the phone to the person next to them which might start a side chat. Is that OK to ignore?)

What is the best policy to deal with students using their cellphones inside the class?

In my opinion: Just ignore them. As long as they are not disturbing the class I don't see a point in doing anything.

There are many different reasons for using a mobile phone during class, maybe they have to check something you said in the course material, maybe they just want to check their time or maybe their learning speed is faster than your teaching speed.

What you are trying to fight here seem to be two things:

1. The students are distracted by their phones.
2. The students are distracting someone else by talking to them.

The first thing is, in my opinion, not your problem. They are old enough to decide for themselves if they want to pay attention or not. You are providing them with the opportunity to learn something and if they take advantage of it is up to them.

The second point is (again, only my opinion) a valid point. However, I think you have the wrong approach here. You can't possibly know if banning mobile phones achieves that goal. The reason for using mobile phones is, most of the time, boredom (for whatever reason) and if not allowed to distract them with their phone they might try to distract them by talking with their neighbor. Therefore, a ban of mobile phones might be the cause for more side chats. If you want to reduce these chats maybe you should try it more directly.

Is it OK to prevent students from using their cellphones during the lecture by stating that in the syllabus?

The question should be: Is it OK to try to prevent students from using their cellphones. Here you have (as already said by @Nicholas) a problem of enforceability. Students are still going to use mobile phones (most of them (including me ;-)) don't even have a watch anymore) and then you have the problem that it will be hard to enforce your rule.

Again, I think you are trying to fight two things with your mobile ban: Distraction for themselves (which is, in my opinion) non of your business (and most students would just switch to something else to distract them) and distraction of others by talking to their neighbors.

If you want to fight the second point you should, in my opinion, try to enforce this directly, e.g. by asking them if they have a question (or, if it happened more often, to tell them to shut up or, in the worst case, just kicking them out) or, as @Nicholas suggested, ask them an easy question. Personally, I think it is impossible to distinguish if they are discussing something relevant to the course (which, in my opinion, is great and should not be banned) and irrelevant chats which are a (minor) distraction and should be banned, therefore I tend to ask if there is a question.