  In this paper, we show several bounds for the numerical radius of a Hilbert
space operator in terms of the Euclidean operator norm. The obtained forms will
enable us to find interesting refinements of celebrated results in the
literature.
  Then, the $f$-operator radius, recently defined as a generalization of the
Euclidean operator radius, will be studied. Many upper bounds will be found and
matched with existing results that treat the numerical radius. Special cases of
this discussion will lead to some refinements and generalizations of some
well-established results in the field.
  Further, numerical examples are given to support our findings, and a simple
optimization application will be presented.
