Inequality: Integration of Norm

[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 ?

 

[blue_raver22]

I approach this question by looking at the evidence and data surrounding inequality and the integration of norms. Inequality, in terms of social and economic disparities, has been a long-standing issue in societies around the world. It has been linked to a variety of negative outcomes, including lower levels of well-being and health, decreased social cohesion, and hindered economic growth.

One way to address inequality is through the integration of norms, or shared values and beliefs that guide behavior and decision-making. This can involve promoting values such as equality, fairness, and justice, and ensuring that they are reflected in policies and practices. For example, implementing policies that promote equal access to education and healthcare can help to reduce disparities and promote greater social and economic equality.

However, the success of integrating norms in addressing inequality also depends on other factors, such as the willingness of individuals and institutions to adopt and uphold these values. This can be influenced by a variety of factors, including cultural, political, and economic contexts. Therefore, it is important to consider these factors when implementing policies aimed at promoting greater equality and integration of norms.

In addition, as a scientist, I also recognize the importance of continuously evaluating and adjusting strategies to address inequality. This can involve gathering and analyzing data on the impact of policies and initiatives, and using this information to inform future actions. It is also important to involve diverse perspectives and voices in these discussions and decision-making processes to ensure that all members of society are represented and their needs are addressed.

Overall, while the integration of norms can be a valuable approach to addressing inequality, it is important to consider the complexity of this issue and to continuously monitor and adapt strategies to achieve meaningful and lasting change.