Natural Parameter Space of an Exponential Family is a convex set (Proof)? A natural parameter family is defined as follows
$$p(x|\eta) = h(x) \exp(\eta T(x) + A(\eta))$$
where 
T: sufficient statistics 
A: log partition function.

We want to prove that the natural parameter space $\mathcal{N}$ given by $$\mathcal{N} = \left\{ \eta: \int(\exp(A(\eta))) &lt; \infty \right\}$$ 
is convex.

The proof rests on holder inequality and is given here. 
I am attaching a picture for a quick reference
I have looked at the definition of holder inequality. 
I am not really sure how the $1/\lambda$ and $1/1-\lambda$ are written in the denominator in eq 8.35 when applying the holder inequality in the proof given. 

Also in Eq 8.36 how is the $e^{\lambda n^T T(x)}$ is discared for the integral. 

Please help in explaining these things?
That looks like a mistake in print. Ideally, the $1/\lambda$ should be in the power for $\exp(\lambda\eta T(x))^{1/\lambda}$ where $p=1/\lambda$ in the Holder's inequality.

Also, it's not $\eta^T$ ($\eta$ raised to the power T), but $\eta^\mathrm{'}T$ which is just the multiplication of 2 matrices.
That looks like a mistake in print. Ideally, the $1/\lambda$ should be in the power for $\exp(\lambda\eta T(x))^{1/\lambda}$ where $p=1/\lambda$ in the Holder's inequality.

Also, it's not $\eta^T$ ($\eta$ raised to the power T), but $\eta^\mathrm{'}T$ which is just the multiplication of 2 matrices.