Prove the following refinement of AM-GM inequality

Question

Under what additional condition(s) the following inequality would be holds. In this case, prove your assertion and the inequality. For $x,y \ge 1$ (with $x \ne y$) we have \begin{align} \alpha x + \beta y \ge \alpha x + \beta y - {\rm{e}}^{\frac{{2 - \left| {x - y} \right|}}{{\left| {x - y} \right|}}} \ge \alpha x + \beta y - {\rm{e}}^{\frac{{1 - \beta ^\alpha \alpha ^\beta \left| {x - y} \right|}}{{\beta ^\alpha \alpha ^\beta \left| {x - y} \right|}}} \ge x^\alpha y^\beta \label{eq2.3} \end{align} for all $\alpha,\beta \in [0,1]$ such that $\alpha+\beta=1$.

Answer 1

The first inequality is obvious.

The second inequality is equivalent to $$\alpha^{\beta}\beta^{\alpha}\leq\frac{1}{2},$$ which is true by Jensen and AM-GM: $$\alpha^{\beta}\beta^{\alpha}=e^{\alpha\ln\beta+\beta\ln\alpha}\leq e^{\ln(\alpha\beta+\beta\alpha)}\leq e^{\ln2\left(\frac{\alpha+\beta}{2}\right)^2}=\frac{1}{2}.$$ The last inequality is wrong.

Try $\alpha=\beta=\frac{1}{2}$ and $x-y\rightarrow0.$