Let $k$ be an integer. The following inequality is standard. $$ (a+b)^{k+1} - b^{k+1} \leq (k+1)a(a+b)^k $$ for $a,b > 0$.

However, does the following inequality still hold $$ (a+b)^{k+1} - b^{k+1} \leq (k+1)a\left(a+ \frac{b}{(k+1)^{1/(k+1)}} \right)^k $$ for $a,b > 0$? While $k \rightarrow \infty$, the term $(k+1)^{1/(k+1)} \rightarrow 1$ so that becomes the first inequality. What about if $k$ is large enough?