On the Superstability of Lobačevski's Functional Equations with Involution

Let g be a uniquely 2-divisible commutative group and let f,g:g→c and σ:g→g be an involution. in this paper,generalizing the superstability of lobačevski's functional equation,we consider f(x+σy)/22-g(x)f(y)≤ψ(x) or ψ(y) for all x,y,g,where ψ:g→r+. as a direct consequence,we find a weaker condition for the functions f satisfying the lobačevski functional inequality to be unbounded,which refines the result of gǎvrutǎ and shows the behaviors of bounded functions satisfying the inequality. we also give various examples with explicit involutions on euclidean space. © 2016 jaeyoung chung et al.