I've encountered the following problem that involves alternating sums of multinomial coefficients. Let $$f(k)=\sum_{i=0}^{n-k}(-1)^i\binom{n}{k,i,n-k-i}(k+i)^\alpha$$ where $\binom{n}{k,i,n-k-i}=\frac{n!}{k!i!(n-k-i)!}$ is the multinomial coefficient, and $\alpha\in\mathbb{R}$ is a constant that satisfies $-1\leqslant\alpha<0$

Problem: I'm trying to prove that $$\frac{f(k)}{f(k+1)}\leqslant\frac{k+1}{k}$$, and equality happens when $\alpha=-1$.

Background: this problem stems from my study of expected order statistics of Weibull distribution. When a Weibull distribution has shape parameter larger than 1 (assuming scale parameter equals 1), the expected difference between the $k$-th largest order statistic and the $k+1$-th is exactly $f(k)$. Numerical computations showed the results hold, but I cannot prove it analytically. Any suggestions or pointers are much appreciated!