How Can Norm Integration Address Inequality?

[nalkapo]

I am struggling with this question. I need a different perspective. Any recommendation is appreciated.
Please click on the attached Thumbnail.

 

[pasmith]

What exactly is the question? I see
$$\|\dot f(t)\|^2 \leq \int_{t-\tau}^t \|\dot f(\theta)\|^2\,\mathrm{d}\theta$$
with $\tau \neq 0$.

The inequality does not hold for all $\tau > 0$ unless $\|\dot f(t)\| = 0$, since the right hand side can be made arbitrarily small by taking $\tau > 0$ sufficiently small.

The inequality does not hold for any $\tau < 0$ unless $\|\dot f(\theta)\|$ vanishes identically on $(t,t+|\tau|)$ and $\|\dot f(t)\| = 0$, since otherwise the right hand side is non-positive ($\int_{t+|\tau|}^t = -\int_{t}^{t + |\tau|}$) and the left hand side is non-negative.

 

[nalkapo]

Follow up question

Thanks pasmith,

Yeah, by using definition of Riemann integration rule I already proved that the inequality is wrong. I tried to find a domain in which the inequality holds; however, there is no such domain.

How about we multiply only left-hand side with τ (tau)? Will this inequality be hold in some domain? What do you think?

 

[sobuz]

what if 0<tau<1 ?