On restricted functional inequalities associated with quadratic functional equations

In this paper it is proved that, for a function f : X → E mapping from a normed linear space X into an inner product space E , the functional inequality ‖ 2 f ( x ) + 2 f ( y ) - f ( x - y ) ‖ ⩽ ‖ f ( x + y ) ‖ , ‖ x ‖ + ‖ y ‖ ⩾ d for some d > 0 , implies f is quadratic. Other types of functional inequalities related to the quadratic functional equation have also been investigated. Besides we establish the Hyers–Ulam stability on restricted domains, and we improve the bounds and thus the stability results obtained in Jung (J Math Anal Appl 222:126–137, 1998) and Rassias (J Math Anal Appl 276: 747–762, 2002). Finally we apply our recent results to the asymptotic behavior of quadratic functional equations of different types.