SOME INEQUALITIES FOR WEIGHTED AND INTEGRAL MEANS OF CONVEX FUNCTIONS ON LINEAR SPACES

Let f be a convex function on a convex subset C of a linear space and x, y is an element of C, with x not equal y. If p : [0, 1] -> R is a Lebesgue integrable and symmetric function, namely p (1 - t) = p (t) for all t is an element of [0, 1] and such that the condition 0 <= integral(tau)(0) p(s) ds <= integral(1)(0) p(s) ds for all tau is an element of [0, 1] holds, the we have vertical bar 1/integral(1)(0) p(tau)d tau integral(1)(0) p(tau) f ((1 - tau)x + tau y)d tau - integral(1)(0)f((1 - tau)x + tau y)d tau vertical bar <= 1/integral(1)(0)p(tau) d tau integral(1)(0)(integral(tau)(0)p(s) ds) (1- tau)d tau [del( -) f(y) (y - x) -del (+) f(x) (y - x)] <= 1/2 [del (-)f(y) (y - x) - del(+) f(x) (y - x)]. Some applications for norms and semi-inner products are also provided.