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
<br /> \|\dot f(t)\|^2 \leq \int_{t-\tau}^t \|\dot f(\theta)\|^2\,\mathrm{d}\theta<br />
with \tau \neq 0.

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

The inequality does not hold for any \tau &lt; 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 ?