Hadamard's inequality for \(r\)-convex functions (Q1378574)

The upper Hadamard inequality for a convex function \(f\) on \([a,b]\) is \[ (b- a)^{-1} \int^b_a f(t)dt\leq (f(a)+ f(b))/2. \] The power mean \(M_r(x,y; t)\) of order \(r\) for two positive numbers \(x\) and \(y\) is defined by \(M_r(x, y;t)=(tx^r+(1- t)y^r)^{1/r}\) if \(r\neq 0\), and \(M_0(x,y; t)= x^ty^{1- t}\), \(t\in[0,1]\). A positive function \(f\) is called \(r\)-convex on \([a,b]\) if for all \(x,y\in[a, b]\) and \(t\in[0, 1]\), \[ f(tx+ (1-t)y)\leq M_r(f(x), f(y);t). \] The paper contains upper Hadamard-type inequalities for \(r\)-convex functions.