**Statement**

Assume that $\sigma,R\in (1,+\infty)$, $N\in\mathbb{N}^*$, $p\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that $$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$ Here, the function $B$ is defined as $$ B(n_1):=\sum_{n_2=0}^{N-n_1}\binom{N-n_1}{n_2}p^{n_2}(1-p)^{N-n_1-n_2}\left(n_2+(N-n_1-n_2)R^{\frac{1}{\sigma}-1}\right)^\sigma .$$

**My attempts**

First I tried using the inequality $B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<\left(B(n_1)-B(n_1+1)\right)^\frac{1}{\sigma}$ since $\sigma>1$, but $\left(B(n_1)-B(n_1+1)\right)^\frac{1}{\sigma}$ may be greater than 1 (concluded by computer). So this approach may not work.Then, I tried using $\binom{N-n_1}{n_2}=\binom{N-n_1-1}{n_2}+\binom{N-n_1-1}{n_2-1}$, but I still haven't proven it.

So how to prove or disprove this inequality? Thanks in advance!

**Update**

Thanks to a user who answered earlier (he may have deleted his answer because of a slight error in his argument), I can transform the left side of the inequality to be proved into a form that is more probabilistic.

**Equivalent proposition**: Assume that the random variables $X_1, X_2,\cdots , X_n (n\ge 1)$ are independent and identically distributed, with $\operatorname{P}(X_k=1)=p,\operatorname{P}(X_{k}=t)=1-p$, where $p,t\in (0,1)$. Real number $\sigma > 1$. Prove that
$$ \operatorname{E}^{\frac{1}{\sigma}}\left[\left(\sum_{k=1}^{n}X_k\right)^\sigma\right] - \operatorname{E}^{\frac{1}{\sigma}}\left[\left(\sum_{k=1}^{n-1}X_k\right)^\sigma\right] < 1 .$$
P.S. Actually, $t=R^{\frac{1}{\sigma}-1}$ and $n=N-n_1$.