K}}_n^\theta$ in for a fixed $\theta$: $$\log {{\cal K}}_n^\theta = {\theta{'}}\sum_{i=1}^n c_i x_i  - \sum_{i=1}^n \log(1+p_i (e^{{\theta{'}}c_i}-1)).$$ For notational simplicity we write $${u}(\theta)=\log {{\cal K}}_n^\theta.$$ We investigate the behavior of ${u}(\theta)$ for $\theta$ close the origin. Fix $\theta \in {\mathbb R}^d$ with unit length (i.e. $\Vert \theta\Vert=1$) and consider ${u}(s\theta)$, $0\le s\le \epsilon$. Note that ${u}(0)=0$. We will choose $\epsilon$ appropriately later in .

The derivative of ${u}(s\theta)$ with respect to $s$ is written as follows. $$\begin{aligned}
\frac{\partial}{\partial s} {u}(s\theta) &=  {\theta{'}}\sum_{i=1}^n c_i x_i  
- \sum_{i=1}^n \frac{{\theta{'}}c_i p_i e^{s{\theta{'}}c_i}}{1+p_i (e^{s{\theta{'}}c_i}-1)}
\nonumber \\
&={\theta{'}}\sum_{i=1}^n c_i (x_i - p_i) - \sum_{i=1}^n \frac{{\theta{'}}c_i p_i e^{s{\theta{'}}c_i} - {\theta{'}}c_i p_i (1+p_i (e^{s{\theta{'}}c_i}-1))
}{1+p_i (e^{s{\theta{'}}c_i}-1)}\nonumber \\
&={\theta{'}}{{\cal S}}_n  - \sum_{i=1}^n {\theta{'}}c_i p_i (1-p_i) 
\frac{e^{s{\theta{'}}c_i}-1} {1+p_i (e^{s{\theta{'}}c_i}-1)} .
\label{eq:fixed-theta-capital}\end{aligned}$$ Note that ${\theta{'}}c_i$ and $e^{s{\theta{'}}c_i}-1$ have the same sign and hence each summand in the second term on the right-hand side of is non-negative.

Let $$\gamma_p(y)= \frac{e^y-1}{1+p(e^y-1)}$$ be a function of $y\in {\mathbb R}$ depending on the parameter $p\in [0,1]$. Note $\gamma_p(0)=0$. Its derivative is computed as $$\label{eq:gamma-deriv}
\gamma_p'(y)= \frac{e^y}{(1+p(e^y-1))^2} > 0.$$ Hence $$\gamma_p(y)=\int_0^y \gamma_p'(z)dz = \int_0^y  \frac{e^z}{(1+p(e^z-1))^2} dz,$$ where for negative $y<0$ we interpret $\int_0^y (\cdots) dz $ as $-\int_y^0 (\cdots) dz$. Now $\gamma_p'(z)$ in is monotone 