  In this paper, we deals with isoperimetric-type inequalities for closed
convex curves in the Euclidean plane R^2. We derive a family of parametric
inequalities involving the following geometric functionals associated to a
given convex curve with a simple Fourier series proof: length, area of the
region included by the curve, area of the domain enclosed by the locus of
curvature centers and integral of the radius of curvature. By using our
isoperimetric-type inequalities, we also obtain some new geometric
Bonnesen-type inequalities. Furthermore we investigate stability properties of
such inequalities (near equality implies curve nearly circular).
