In the book Fourier Analysis and Nonlinear PDE, B-C-D establish the following
Proposition 2.22. A constant $C$ exists which satisfies the following properties. If $s_1$ and $s_2$ are real numbers such that $s_1<s_2$ and $\left.\theta \in\right] 0,1[$, then we have, for any $(p, r) \in[1, \infty]^2$ and any $u \in \mathcal{S}_h^{\prime}$, $$\|u\|_{\dot{B}_{p, 1}^{\theta s_1+(1-\theta) s_2}} \leq \frac{C}{s_2-s_1}\left(\frac{1}{\theta}+\frac{1}{1-\theta}\right)\|u\|_{\dot{B}_{p, \infty}^{s_1}}^\theta\|u\|_{\dot{B}_p^{s_2}, \infty}^{1-\theta} . $$
However, I am quite confused in establishing the constant: Following their approach, I was able to reduce down the bound to $$ \lVert u \rVert_{\dot{B}^{\theta s_1 + (1 - \theta) s_2}_{p,1}} \leq C\lVert u \rVert_{\dot{B}^{s_1}_{p,\infty}}^\theta \lVert u \rVert_{\dot{B}^{s_2}_{p,\infty}}^{1 - \theta} \left(\frac{1 }{ 1 - 2^{-(1 - \theta)(s _2 - s_1)} } + \frac{1 }{1 - 2^{ - \theta (s_2 - s_1)} }\right) $$ but I am unable to establish $$ \frac{1 }{ 1 - 2^{-(1 - \theta)x} } + \frac{1 }{1 - 2^{ - \theta x} } \leq \frac{C}{x} \left(\frac{1}{\theta} + \frac{1}{1 - \theta}\right), \quad \theta \in (0,1), x \geq 0 $$ which seems to be incorrect as per plotting for large values of $C$.