Hey there! Sign in to join this conversationNew here? Join for free

Showing inequalities (Context is gamma function converging) Watch

    • Thread Starter
    Offline

    3
    ReputationRep:
    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

    Many thanks in advance.
    Offline

    13
    ReputationRep:
    (Original post by xfootiecrazeesarax)
    Q1)  0\leq x \leq 1 \implies x^{t-1} e^{-x} \leq x^{t-1}
    This is true, as you can (carefully) cancel off the x^{t-1} from both sides and you are left with something nice and easy.

    Q2) that x^{t-1}e^{-x}\leq x^{-x/2} for  x \geq 1
    This is not true in general. Choose x = 2 and t = 4, for example.
    • Thread Starter
    Offline

    3
    ReputationRep:
    (Original post by Gregorius)
    This is true, as you can (carefully) cancel off the x^{t-1} from both sides and you are left with something nice and easy.



    This is not true in general. Choose x = 2 and t = 4, for example.
    ahh okay, thanks. I was trying to follow the proof here:

    http://math.stackexchange.com/questi...onverges-for-z

    the first answer...so this is incorrect? or ok for large  x
    Offline

    17
    ReputationRep:
    (Original post by xfootiecrazeesarax)
    ahh okay, thanks. I was trying to follow the proof here:

    http://math.stackexchange.com/questi...onverges-for-z

    the first answer...so this is incorrect? or ok for large  x
    It's incorrect for general x, but true for large x (i.e. as x->infinity). (If you read the entire thread, you'll see various comments to this effect).
    • Thread Starter
    Offline

    3
    ReputationRep:
    (Original post by DFranklin)
    It's incorrect for general x, but true for large x (i.e. as x->infinity). (If you read the entire thread, you'll see various comments to this effect).
    Okay thanks.

    And yet the integral inequality still holds because the behaviour at x=\infty dominates over the small x behaviour?
    • Thread Starter
    Offline

    3
    ReputationRep:
    anyone on how to prove q2?
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    What newspaper do you read/prefer?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.