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Let $F:[0,\infty) \to [0,\infty)$ be a $C^1$ strictly convex function. Let $\lambda_n \in [0,1],a_n\le c<b_n \in [0,\infty)$ satisfy $$ \lambda_n a_n +(1-\lambda_n)b_n=c_n \tag{1}$$ and assume that ...

I am not sure if this is exactly research-level, but I am struggling to find a proof for the following claim: Let $F:[0,\infty) \to [0,\infty)$ be a $C^2$ strictly convex function. Let $\lambda_n \in [...