Most common name is "Lofty Peak":
http://listsofjohn.com/peak/18150
Earliest source I know of calling it Lofty Peak is the book below from 1955:
http://www.amazon.com/Scenery-Counties- ... B0007EU5G8
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Usually known as West Notch Mountain:
http://listsofjohn.com/peak/18138
Published in this guidebook if you want to trust the author:
http://www.amazon.com/Western-Uinta-Bac ... 1890828173
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http://listsofjohn.com/peak/18282
Known as both Duke Mountain and Flat Iron Mountain. Flat Iron is the name used by the Forest Service and Utah Natural Resources-Wildlife Resources.
For example, the name is on the map in the book Lakes of the High Uintas Provo and Weber River Drainages, Revision 1983, Publication No. 83-6, page 36, calls it Flat Iron Mountain.
Still, it is possible that Duke Mountain may be used more frequently.