Symbol for link to my site on mobile device? Eg iPhone.

Hello SI Team

Over the weekend I was looking at my Site on a friends iPhone.
He showed me how he could save the site to his iPhone as an icon.
This icon resembled an App icon and is a link to my site for fast fast access.

Probably similar to how the icons work in my Browser. please see screen shot.

The thing is this icon needs some kind of information that makes it distinctive = Branding/Logo what ever.
In my case my site showed up as a symbol with a white background and tiny illegible text = lines.
I do not have an iPhone so can not demonstrate this!?!Sorry.

I am wondering is there any way to influence the appearance of my site if and when saved as a link in this manner on a mobile device?

I hope this makes sense?

thanks

Paul