$p^{r+s+t}+(1-p)^{r+s+t} \geqslant [p^{r+s}+(1-p)^{r+s}]\times[p^{s+t}+(1-p)^{s+t}]$ How to prove this? and is there any generalization based on some algebraic inequalities?
It seems that it's a special case of Jensen or Inequality of arithmetic and geometric means.
Set $0 \leqslant  p \leqslant 1 $ and  $r,s,t \in \mathbb{N}$
prove that:
$$p^{r+s+t}+(1-p)^{r+s+t} \geqslant [p^{r+s}+(1-p)^{r+s}]\times[p^{s+t}+(1-p)^{s+t}]$$
 A: Here is an algebraic/geometric proof.
We have to assume $s \geqslant 1$ (otherwise, the inequality is false).
Let $q=1-p$. We can assume WLOG that 
$$\tag{0}p \geqslant q.$$
As all quantities involved in the inequality to be proven are $>0$, it is equivalent to establish that 
$$\tag{1} p^{s}p^{t}+q^{s}q^{t} \leqslant \dfrac{p^{r+s}p^t+q^{r+s}q^t}{p^{r+s}+q^{r+s}}$$
It is sufficient to prove that 
$$\tag{2} \dfrac{p^{s}p^{t}+q^{s}q^{t}}{p^s+q^s} \leqslant \dfrac{p^{r+s}p^t+q^{r+s}q^t}{p^{r+s}+q^{r+s}}$$
(because for $s \geqslant 1$, $ \ p^s+q^s \leqslant 1$. Thus $(2) \implies (1)$ ).
Setting 
$$\tag{3} w_1=p^s, \ \ \ w_2=q^s, \ \ \ w'_1=p^{r+s}, \ \ \ w'_2=q^{r+s},$$
(2) can be interpreted as a relationship between barycenters (weighted means) of the same numbers $p^t$ and $q^t$:
$$\tag{4} \dfrac{w_1 p^{t}+w_2q^{t}}{w_1  +w_2} \leqslant  \dfrac{w'_1p^t+w'_2q^t}{w'_1+w'_2}$$
Let us recall that hypothesis (0) implies that 
$$\tag{5} 0 \leqslant q^t \leqslant p^t \leqslant 1$$ 
Thus (4) will be established if the weights' ratios are like this:
$$\tag{6} \dfrac{w_1}{w_2} \leqslant \dfrac{w'_1}{w'_2} $$
(meaning that the barycenter is more attracted by $p^t$ on the RHS than on the LHS). 
(6) is easy to check because:
$$ p \geqslant q \implies (\tfrac{p}{q})^s \leqslant (\tfrac{p}{q})^{r+s} $$
In plain terms, the RHS of (4) interpreted as a barycenter is on the right of the barycenter on the LHS of (4).
Remarks: 
1) Relationship (2) is a stronger inequality than the proposed inequality.
2) The hypothesis that $r,s,t$ are integers can be relaxed: it suffices that $r>0, s \geqslant 1, t>0$ with real values. 
