\(a(x)\)-convex functions and their inequalities
The authors define \(a(x)\)-convex functions as follows: Let \(f\) and \(a\) be real functions defined on interval \(I\subseteq\mathbb{R}\) such that \(f\) is differentiable and \(a\,f^{\prime}\) integrable. The function \(f\) is called \(a(x)\)-convex on the interval \(I\) if for every \(x,\,y\in I\) \[ (y-x)\,(f^{\prime}(y)-f^{\prime}(x))\geq (y-x)\,\int_{x}^{y}a(t)\,f^{\prime}(t)\,dt \] holds. For \(a(x)=0\), \(a(x)\)-convexity reduces to ordinary convexity. The authors give properties and characterizations of \(a(x)\)-convex functions. They also prove analogues of several inequalities for ordinary convex functions, such as Jensen, Hermite-Hadamard, Lah-Ribarić inequalities, in the case of \(a(x)\)-convex functions.
scientific article; zbMATH DE number 6052953