Assume $N,M,K$ are positive integers and consider a real convex function, say $r(\cdot)$ with $r(N)=0$. Also $0.092N
Proof of a convex function inequality
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functions
convex-analysis
1 Answers
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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.