Let $s\in(1,2)$ and $p\geq 2$, then there exists a constant $c(p)$ large depending on $p$ such that $$ s^p-1-(s-1)^p\leq c(p)(s-1). $$ For $p=2,3$ I have verified it but for other values of $p$, I am unable to verify it. Can somebody kindly help.
Thanks in advance.