Show two inequalities - (context gamma function converges)

In summary, the author is trying to solve a problem in a proof from a textbook, but is having difficulty understanding why the inequality holds.
  • #1
binbagsss
1,277
11

Homework Statement



I'm not after another proof.
I've just got a couple of inequalities I don't know how to show when following a given proof in my book.

These are:

Q1) ## 0\leq x \leq 1 \implies x^{t-1} e^{-x} \leq x^{t-1} ##

So this is obvioulsy true, however I think I'm being dumb because surely this is true for all ##x##

(the integral over the gamma function is split into ##\int ^{\infty} _1 ## and ## \int^1_0 ##) and so this inequality is used in the latter, but I don't see why it can't be used in the former, e.g ##2^{t-1}/e^{2} \leq 2^{t-1} ## is also true isn't it? and as ## x \to \infty ## this is also true, because the LHS ## \to 0 ##..

Q2) that ##x^{t-1}e^{-x}\leq x^{-x/2}##, for ##x \geq 1##
I am unsure how to show this, or understand why it holds,

and then secondly I need to show that ##x^{t-1}e^{-x}\leq x^{-x/2} \iff x^{t-1}\leq e^{x/2}##

I can write ## x^{-x/2} = e^{(-x/2) ln (x) } ##
I am unsure what to do next or whether this helps

Homework Equations



see above

The Attempt at a Solution



see above
 
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  • #2
binbagsss said:

Homework Statement



I'm not after another proof.
I've just got a couple of inequalities I don't know how to show when following a given proof in my book.

These are:

Q1) ## 0\leq x \leq 1 \implies x^{t-1} e^{-x} \leq x^{t-1} ##

So this is obvioulsy true, however I think I'm being dumb because surely this is true for all ##x##

(the integral over the gamma function is split into ##\int ^{\infty} _1 ## and ## \int^1_0 ##) and so this inequality is used in the latter, but I don't see why it can't be used in the former, e.g ##2^{t-1}/e^{2} \leq 2^{t-1} ## is also true isn't it? and as ## x \to \infty ## this is also true, because the LHS ## \to 0 ##..

[tex]\int_1^R x^{t-1}\,dx = \begin{cases} \ln R & t = 0, \\
\frac{R^t - 1}{t} & t \neq 0\end{cases}.[/tex] For [itex]t \geq 0[/itex] these do not converge as [itex]R \to \infty[/itex]. You're trying to prove convergence so this bound is of no assistance.
 
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Likes binbagsss
  • #3
pasmith said:
[tex]\int_1^R x^{t-1}\,dx = \begin{cases} \ln R & t = 0, \\
\frac{R^t - 1}{t} & t \neq 0\end{cases}.[/tex] For [itex]t \geq 0[/itex] these do not converge as [itex]R \to \infty[/itex]. You're trying to prove convergence so this bound is of no assistance.

thank you, makes sense !
Q2 anyone?
 
  • #4
binbagsss said:
thank you, makes sense !
Q2 anyone?
bump
 

FAQ: Show two inequalities - (context gamma function converges)

1. What is the gamma function?

The gamma function is a mathematical function that extends the concept of factorial to real and complex numbers. It is denoted by the Greek letter "𝛤" and is defined as the integral from 0 to infinity of x^(𝛤-1)e^(-x)dx.

2. How is the gamma function related to inequalities?

The gamma function is often used to prove and solve inequalities involving real and complex numbers. It has many properties that make it a useful tool in inequality proofs.

3. What does it mean for the gamma function to converge?

A function converges when its values approach a certain limit as its input approaches a specific value. In the case of the gamma function, it converges if the integral from 0 to infinity of x^(𝛤-1)e^(-x)dx exists and is finite.

4. How can we show two inequalities using the gamma function?

To show two inequalities using the gamma function, we can use its properties such as monotonicity, convexity, and log-convexity to manipulate the inequalities and prove them using the definition and properties of the gamma function.

5. Can the gamma function be used to solve real-world problems?

Yes, the gamma function has numerous applications in various fields such as physics, biology, and economics. It is used to model and solve problems involving continuous quantities and can also be used in statistical analysis and probability calculations.

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