Remarks on strongly convex functions
Let \(I\subseteq \mathbb{R}\) be an interval and \(c\) be a positive number. A function \(f:I\rightarrow \mathbb{R}\) is called strongly convex with modulus \( c\) if \[ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-ct(1-t)(x-y)^{2}, \] for all \(x,y\in I\) and \(t\in [0,1]\). Using the property that a function \(f:I\rightarrow \mathbb{R}\) is strongly convex with modulus \(c\) if and only if the function \(g:I\rightarrow \mathbb{R}\) defined by \( g(x)=f(x)-cx^{2}\) is convex, the authors prove the following properties related to strong convex functions: a sandwich theorem, a Hyers-Ulam stability result, an integral Jensen-type inequality, a Hermite-Hadamard-type inequality. A relationship between strong convexity and the generalized convexity in the sense of Beckenbach is also shown.
scientific article; zbMATH DE number 5851467