Let $(\mathbb{R}^{+},\frac{dt}{t})$ be the multiplicative group of positive real numbers with the usual topology and Haar measure $\frac{dt}{t}$. Define functions $g:\mathbb{R}^{+}\to [0,\infty)$, $h:\mathbb{R}^{+}\to [0,\infty)$ by $g(x)=\left|f(x)\right|x^{1-\frac{b}{p}}$ and $h(x)=x^{-\frac{b}{p}}\chi_{[1,\infty)}(x)$. We will apply Minkowski's inequality to the convolution $F=g\star h$. Note that:
\begin{align*} F(x)= &\int_{0}^{\infty} \left|f(t)\right|t^{1-\frac{b}{p}}\;\;\frac{t^{\frac{b}{p}}}{x^{\frac{b}{p}}}\;\;\chi_{[1,\infty)}\left(\frac{x}{t}\right)\frac{dt}{t} \ =& \frac{1}{x^{\frac{b}{p}}}\int_{0}^{x} \left|f(t)\right| dt \ \end{align*}
if $x\in\mathbb{R}^{+}$. Furthermore,
$ \left\|h\right\|_{L^1(\mathbb{R}^{+},\frac{dt}{t})}= \int_{1}^{\infty} t^{-\frac{b}{p}-1}dt=\frac{p}{b} $
and
$\left\|g\right\|_{L^p(\mathbb{R}^{+},\frac{dt}{t})}=\left(\int_{0}^{\infty} \left|f(t)\right|^p t^{p-b-1} dt \right)^{\frac{1}{p}} $
Minkowski's inequality thus implies that
\begin{align*} \left(\int_{0}^{\infty} \left(\int_{0}^{x} \left|f(t)\right| dt\right)^p x^{-b-1} dx\right)^{\frac{1}{p}}\leq \frac{p}{b}\left(\int_{0}^{\infty} \left|f(t)\right|^p t^{p-b-1} dt\right)^{\frac{1}{p}} \end{align*}
Let us now redefine the functions $g:\mathbb{R}^{+}\to [0,\infty), h:\mathbb{R}^{+}\to [0,\infty)$ by $g(x)=\left|f(x)\right|x^{1+\frac{b}{p}}$ and $h(x)=x^{\frac{b}{p}}\chi_{(0,1]}(x)$. We will apply Minkowski's inequality to the convolution $F=g\star h$. Note that,
$ F(x)= \int_{0}^{\infty} \left|f(t)\right|t^{1+\frac{b}{p}}\;\;\frac{x^{\frac{b}{p}}}{t^{\frac{b}{p}}}\chi_{(0,1]}\left(\frac{x}{t}\right)\frac{dt}{t} = x^{\frac{b}{p}}\int_{x}^{\infty} \left|f(t)\right| dt $
if $x\in\mathbb{R}^{+}$. Furthermore,
$ \left\|h\right\|_{L^1(\mathbb{R}^{+},\frac{dt}{t})}=\int_{0}^{1} t^{\frac{b}{p}-1} dt = \frac{p}{b} $
and
$\left\|g\right\|_{L^p(\mathbb{R}^{+},\frac{dt}{t})}=\left(\int_{0}^{\infty} \left|f(t)\right|^p t^{p+b-1} dt \right)^{\frac{1}{p}} $
Minkowski's inequality thus implies that,
\begin{align*} \left(\int_{0}^{\infty} \left(\int_{x}^{\infty} \left|f(t)\right| dt\right)^p x^{b-1} dx\right)^{\frac{1}{p}}\leq \frac{p}{b} \left(\int_{0}^{\infty} \left|f(t)\right|^p t^{p+b-1} dt\right)^{\frac{1}{p}} \end{align*}