For a convex function $f$, I am interested in bounds of the form $$\frac{f(y)-f(x)}{y-x}\geq (1-\lambda)f'(y)+\lambda f'(x),\quad x<y.$$
For $\lambda=1$, this becomes true of any convex function $f$. On the other hand, for $\lambda= 0$, this only holds for a line. So I'm wondering which functions satisfy the above for small (but positive) values $\lambda$. For the parabola $f(x)=x^2$ this holds with $\lambda=1/2$. However, this also holds for functions like $f(x)=ax^2$ with $\lambda=1/2$ for any $a>0$, which tells me this $\lambda$ isn't a measure of curvature.
Does this quantity $\lambda$ have a name? Does this come up in the literature? (By $\lambda$ I mean the best possible $\lambda$.)