When studying the $\text{UMD}$ constants of spaces like $L_{p_1}(L_{p_2}(\cdots (L_{p_n})\cdots))$, I encounter the following question: Let $\alpha > 0$, define $$C(\alpha) : = \sup_{a > 0, b> 0, 0< \delta<1} \frac{(1 + a)^{1+ \alpha} + (1 + b)^{1 + \alpha}}{(1 + \delta^2 + 2 \delta + a)^{1+\alpha} + (1 + \delta^2 - 2 \delta + b)^{1 + \alpha}}.$$
My QUESTION is:
Does there exist $M> 0$ such that for $\alpha$ sufficiently small, we have $$C(\alpha) \ge 1 + M \alpha?$$ By taking $a = 0$ in the supremum, I can show that there exists $m > 1$ such that $$C(\alpha) \ge (1 + m^{-\frac{1}{\alpha}})^\alpha,$$ this of course does not answer the preceding question.
