Proof of a convex function inequality

Question

Assume $N,M,K$ are positive integers and consider a real convex function, say $r(\cdot)$ with $r(N)=0$. Also $0.092N<M\leq N$ and $\tilde{M}$ the largest multiple of $N/K$ less than or equal to $0.092N$ so that $$0\leq0.092N-N/K\leq \tilde{M}\leq0.092N$$ Then I cannot reach the following result $$r(M)\leq\frac{r(\tilde{M})}{1-\tilde{M}/N}(1-M/N)$$ trying various triplets of $a,x_1,x_2$ for the property $$r(ax_1+(1-a)x_2)\leq ar(x_1)+(1-a)r(x_2)$$ with $a\in[0,1]$ of course.

Answer 1

Observe that because $\tilde{M}\le M\le N$, then $M$ is expressed as a convex combination of $\tilde{M}$ and $N$ as follows: $$M=\frac{N-M}{N-\tilde{M}}\cdot\tilde{M}+\frac{M-\tilde{M}}{N-\tilde{M}}\cdot N.$$ By convexity (you quote a desired inequality) we have $$r(M)\le \frac{N-M}{N-\tilde{M}}\cdot r(\tilde{M})+\frac{M-\tilde{M}}{N-\tilde{M}}\cdot r(N)=\frac{N-M}{N-\tilde{M}}\cdot r(\tilde{M})$$ by $r(N)=0$. This is the inequality in question.