iTunes APP download does not show up on iPhone (model 4S)

I use iTunes on iMAC to manage iPhone APPs. The APP appears to download and install on the iMac image of the iPhone, but the APP is not visible on the iPhone. If I use the iPhone APP store I can see information about the app but the download icon is grayed out - as though the app has been installed.

I found the app in settings/general/usage/manage storage. I deleted it and started over but had the same result.

Confused!