Conditional convexity (Q1840560)

We know that a real-valued function \(f: D\to\mathbb{R}\) defined on a nonempty convex set \(D\) is called convex (on \(D\)) if the inequality \[ f(\lambda x+ (1-\lambda) y)\leq \lambda f(x)+ (1-\lambda) f(y)\quad (x,y\in D,\;\lambda\in (0,1)) \] holds. Here, the author studies the conditional version of the above inequality in the cases when (i) \(y= \omega\) is a fixed element of \(D\), (ii) \(\lambda x+(1- \lambda)y= \omega\) is a fixed element of \(D\); for details we refer the reader to the paper.