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Why if for a continuous function $f: \mathbb{R}\to\mathbb{R}$ if $|f(t)|\leq a+b |t|, \forall t\in \mathbb{R}$

by integrating

$F(t)\leq a |t|+\frac{b}{2} |t|^2, \forall t\in \mathbb{R}$

i understand this, but i don't understand the following

for a continuous $f: \mathbb{R}\to \mathbb{R}$ and there exists $\sigma\in (0,1)$ and $a,b>0$ such that $|f(t)|\leq a +b |t|^{\sigma}, \, \forall t\in \mathbb{R}$

then $|F(t)|\leq a_1 +b_1 |t|^{\sigma+1}$

why they don't say $|F(t)|\leq a |t| +b_1 |t|^{\sigma+1}$ ?

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  • $\begingroup$ What you're saying is true, but the estimate given is weaker. To see this, think about estimating $|t| \leq 1$ and $|t| \geq 1$ separately here. If $|t|\leq 1$, then $a|t| \leq a$, so $|F(t)| \leq a + b|t|^{\sigma +1}$. On the other hand, if $|t|\geq 1$, then $|t| \leq |t|^{\sigma +1}$, since $\sigma + 1 > 1$, which gives $|F(t)| \leq (a+b)|t|^{\sigma+1}$. Then $$ |F(t)| \leq \max(a + b|t|^{\sigma+1},(a+b)|t|^{\sigma +1}) \leq (a+b|t|^{\sigma+1}) + (a+b)|t|^{\sigma+1} \leq c + d|t|^{\sigma +1}, $$ where $c,d>0$ are some constants depending on $a,b$. $\endgroup$
    – Zach Hunsucker
    Mar 18 at 18:41

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