# Leave-one-out Harmonic number

I am trying to find a number sequence $$x_k\geq 0$$ such that $$\text{for all integers }n\geq k\geq 1,\quad H_{n-1 - x_k(n-k)}\geq H_n-\frac{1}{k}.$$ I recall that the harmonic numbers $$s\mapsto H_s$$ can be defined for $$s>-1$$ by $$H_s=\int_0^1\frac{1-t^s}{1-t}dt$$. $$H:(-1,\infty)\to\mathbb{R}$$ is a bijection.

In particular, if $$H^{-1}: \mathbb{R}\to (-1,\infty)$$ is the inverse function of $$H$$, how can I prove that $$x_k=-H^{-1}_{-1/k}$$ works ?

Thank you!