Hadamard and Fejér-type inequalities
Let \(f: [a,b]\to \mathbb{R}\), \(g: I\to\mathbb{R}\) be a bijective, continuous mapping defined on an interval \(I\), containing \(\text{range}(f)\). Then \(f\) is named \(g\)-convex if \[ f(ux+ (1- u)y)\leq g^{-1}[u(g\circ f)(x)+ (1- u)(g\circ f)(y)] \] holds true for all \(x,y\in[a, b]\); \(u\in [0,1]\). The generalized logarithmic means of two arguments are defined by \[ L_g(x,y)= (g(y)- g(x))^{- 1} \int^{g(y)}_{g(x)} g^{-1}(t) dt\quad\text{for }x\neq y,\quad L_g(x, x)= x. \] One of the main results of this paper says that if \(f\) is \(g\)-convex, then \[ {1\over b-a} \int^b_a f(x) dx\leq L_g(f(a), f(b)). \] Other results are related to a generalization of the Fejér-type inequalities.
scientific article; zbMATH DE number 1437115