Bound for non-integer power of sum

Question

Let $x > 1$, $y \in (0,1)$ and $z \in (0,1)$. I need to bound $$(x+y)^z - x^z \leq B_z(x)$$ where I guess something like $B_z(x) \approx x^{z-1}$.

Is there anything known on these non-integer powers of sums?

Thanks..

Answer 1

Define $h(x) = x^z$. Then: $$ (x+y)^z - x^z \leq (x+1)^z - x^z = h(x+1)-h(x)$$ Note that $h'(x) = zx^{z-1}$ is nonincreasing in $x$. We then have: \begin{align} h(x+1) &= h(x) + \int_{x}^{x+1}h'(t)dt \\ &\leq h(x) + \int_x^{x+1}h'(x)dt \: \: \: \: \mbox{[since $h'(t)$ is nonincreasing]}\\ &= h(x) + h'(x) \\ &= h(x) + zx^{z-1} \end{align} Thus, we can define $B_z(x) = zx^{z-1}$.