Inequality for nonegative real numbers
I do not know how to finish with your approach but I give you my proof just using some simple calculations.
Let $x=\frac{a}{b}$ and assume that $x\geq 1$. The inequality is equivalent to
$$x^p-1 \leq p(x^p+1)^{\frac{p-1}{p}}(x-1)$$
Define $f(x)=p(x^p+1)^{\frac{p-1}{p}}(x-1)-x^p+1$ and note that $f(1)=0$.
If my computation is right, then we have
\begin{align} f'(x)&=p(x^p+1)^{\frac{p-1}{p}}+p(p-1)(x-1)(x^p+1)^{-\frac{1}{p}}-px^{p-1}\\ &\geq p(x^p+1)^{\frac{p-1}{p}}-px^{p-1}\\ &>0. \end{align} So $f(x)$ is increasing, then $f(x)\geq f(1)=0$.