The following version of the weighted Hermite–Hadamard inequalities for set-valued functions is presented: Let Y be a Banach space and F : a , b → cl ( Y ) be a continuous set-valued function. If F is convex, then F(xμ)⊃1μ(a,b)∫abF(x) dμ(x)⊃b−xμb−aF(a)+xμ−ab−aF(b),$$F(x_mu ) supset {1 over {mu (a,b)}}intlimits_a^b {F(x);dmu (x) supset {{b - x_mu } over {b - a}}} F(a) + {{x_mu - a} over {b - a}}F(b),$$ where μ is a Borel measure on a , b and x μ is the barycenter of μ on a , b . The converse result is also given.
展开▼
