Mean Value Majorization

Suppose $f$ is differentiable in an open set $\mathcal{X}$ containing both $x$ and $y$ and the line connecting them. We can use the mean value theorem in the inequality form $$f(x)\leq f(y)+\sup_{0\leq\lambda\leq 1} (x-y)\mathcal{D}f(y+\lambda(x-y))$$ Define, for fixed $x,y$ $$h(\lambda)\mathop{=}\limits^{\Delta}(x-y)\mathcal{D}f(y+\lambda(x-y))$$ The maximum of $h$ is attained at either $\lambda=0$, or $\lambda=1$, or at a point in the interior of the unit interval where the derivative with respect to $\lambda$ vanishes. Now $$\mathcal{D}h(\lambda)=(x-y)\mathcal{D}^2f(y+\lambda(x-y))(x-y).$$ Thus for concave $f$ we see that $h$ is decreasing, and we recover our previous result $$f(x)\leq f(y)+(x-y)\mathcal{D}f(y).$$ For convex $f$, for which $h$ is increasing, we find $$f(x)\leq f(y)+(x-y)\mathcal{D}f(x).$$

$$cz+\frac12 dz(x+y)>0$$