  The Cauchy-Schwarz, Buzano and Kre\u{\i}n inequalities are three inequalities
about inner product. The main goal of this article is to present refinements of
Buzano and Cauchy-Schwarz inequalities, and to present a new proof of a refined
version of a Kre\u{\i}n-type inequality. Applications that include Buzano-type
inequalities for certain operators, operator norm and numerical radius
inequalities of Hilbert space operators will be presented.
