  In this article, we prove an inner product inequality for Hilbert space
operators. This inequality, then, is utilized to present a general numerical
radius inequality using convex functions. Applications of the new results
include obtaining new forms that generalize and extend some well known results
in the literature, with an application to the newly defined generalized
numerical radius.
  We emphasize that the approach followed in this article is different from the
approaches used in the literature to obtain the refined versions.
