How is this inequality obtained?

In summary: This inequality means that for every real number $x$, the second derivative of the function $f$ with respect to the real variable $x$ is smaller than the first derivative of $f$ with respect to the real variable $x$.
  • #1
Boromir
38
0
Introduction to Operator Theory and Invariant Subspaces - B. Beauzamy - Google BooksIn page 144 of this preview I don't know how they obtain the inequality in (1). It looked like cauchy schwarz but I don't think it is.

I also don't know how they connect the norm of the integral to the supremum of an inner product.

Finally, is it necessary to show the integral is bounded? Is that the motivation?
 
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  • #2
Re: integral inequality

Boromir said:
Introduction to Operator Theory and Invariant Subspaces - B. Beauzamy - Google BooksIn page 144 of this preview I don't know how they obtain the inequality in (1). It looked like cauchy schwarz but I don't think it is.

I also don't know how they connect the norm of the integral to the supremum of an inner product.

Finally, is it necessary to show the integral is bounded? Is that the motivation?
On my computer, the Google Books preview of this text only goes up to p.139. Please state the problem and its context more explicitly.
 
  • #3
Re: integral inequality

Opalg said:
On my computer, the Google Books preview of this text only goes up to p.139. Please state the problem and its context more explicitly.

How can it be different for different computers?
 
  • #4
Re: integral inequality

Boromir said:
How can it be different for different computers?

On my computer pages 144-151 are not shown in the preview. It would really be best if you took the time to state the problem yourself.
 
  • #5
Re: integral inequality

Boromir said:
Opalg said:
On my computer, the Google Books preview of this text only goes up to p.139. Please state the problem and its context more explicitly.

How can it be different for different computers?
It depends entirely on what Google's server chooses to send to different users.
 
  • #6
Here is what the inequality says. It seems like a bunch of notation is required to be explained to make sense of this. But it seems like Opalg said just CS inequality.

$$
\left| \left< (\int |f| ~ dE)x,y\right> \right|^2 \leq \left< (\int |f| ~ dE)x,x \right> \left< \int (|f| ~ dE)y,y\right> $$
 

FAQ: How is this inequality obtained?

What is an integral inequality?

An integral inequality is a mathematical statement that describes the relationship between two integrals. It typically involves one or more functions and their integrals over a given interval. The inequality can be used to determine the relative size of the integrals and their corresponding functions.

What is the significance of integral inequalities?

Integral inequalities are important in mathematics because they allow us to make comparisons between different functions and their integrals. They also have various applications in fields such as physics, economics, and engineering.

How are integral inequalities different from regular inequalities?

While regular inequalities involve only numbers or variables, integral inequalities involve integrals of functions. This means that the functions' behavior over a certain interval is taken into account, rather than just their values at specific points. In addition, solving integral inequalities often requires different techniques and methods compared to regular inequalities.

What are some common types of integral inequalities?

Some commonly used integral inequalities include the Cauchy-Schwarz inequality, the Hölder inequality, and the Minkowski inequality. Each of these has its own specific conditions and applications, but they all involve comparing integrals of different functions.

Are there any special considerations when solving integral inequalities?

Yes, there are a few things to keep in mind when solving integral inequalities. First, it is important to make sure that the functions involved are well-behaved and continuous over the given interval. Additionally, some integral inequalities may have specific conditions or restrictions that need to be taken into account when solving them. It is also helpful to have a good understanding of integration techniques and properties in order to solve integral inequalities efficiently.

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