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eugene
science forum Guru

Joined: 24 Nov 2005
Posts: 331

Posted: Fri Jun 23, 2006 10:46 pm    Post subject: Re: fractional part of the function at integer points+concave

 Quote: In article 33398815.1151085054168.JavaMail.jakarta@nitrogen.math forum.org>, eugene wrote: In article 15558845.1151055387707.JavaMail.jakarta@nitrogen.math forum.org>, eugene wrote: Let f be a continious concave function from (0, +infty) -> R such that lim_{x -> infty} f(x) = infty and f(x) = o(x) when x -> +infty. How can i prove that sup_{n -naturals} {f(n)} = 1, where {f(n)} is fractional part of f(n). Hint: show that f(n+1) - f(n) tends to 0 as n tends to infinity. Thanks. I had the same idea, surely if we prove it, using the fact that f tends to infitiy everything becomes clear about the problem. But, i'm not sure about my proof of that lim_{n -> infy} (f(n+1)-f(n)) = 0. show f(n+1)-f(n) is a positive decreasing sequence. It thus has a limit. If the limit were positive, then the o(x) condition couldn't hold.

Ah, thanks, it seems i've got it.

First, since f is concave g(x) = (f(x)-f(n))/(x-n) is decsreasing for any fixed natural n , so

g(n+1) >= lim_{x -> infty} g(x) = 0 due to that o(x) condition.
so we have that f(n+1) > f(n).
And to this end, we may notice that since f is concave

1/2(f(n)+ f(n+2)) <= f((n+n+2)/2) = f(n+1), so

f(n+1) - f(n) >= f(n+2) - f(n+1) which means that the sequence f(n+1) - f(n) is decreasing.

Is it ok?
Thanks
science forum Guru

Joined: 24 Mar 2005
Posts: 790

Posted: Fri Jun 23, 2006 6:36 pm    Post subject: Re: fractional part of the function at integer points+concave

In article
<33398815.1151085054168.JavaMail.jakarta@nitrogen.mathforum.org>,
eugene <jane1806@mail.ru> wrote:

 Quote: In article 15558845.1151055387707.JavaMail.jakarta@nitrogen.math forum.org>, eugene wrote: Let f be a continious concave function from (0, +infty) -> R such that lim_{x -> infty} f(x) = infty and f(x) = o(x) when x -> +infty. How can i prove that sup_{n -naturals} {f(n)} = 1, where {f(n)} is fractional part of f(n). Hint: show that f(n+1) - f(n) tends to 0 as n tends to infinity. Thanks. I had the same idea, surely if we prove it, using the fact that f tends to infitiy everything becomes clear about the problem. But, i'm not sure about my proof of that lim_{n -> infy} (f(n+1)-f(n)) = 0.

show f(n+1)-f(n) is a positive decreasing sequence. It thus has a
limit. If the limit were positive, then the o(x) condition
couldn't hold.
eugene
science forum Guru

Joined: 24 Nov 2005
Posts: 331

Posted: Fri Jun 23, 2006 5:50 pm    Post subject: Re: fractional part of the function at integer points+concave

 Quote: In article 15558845.1151055387707.JavaMail.jakarta@nitrogen.math forum.org>, eugene wrote: Let f be a continious concave function from (0, +infty) -> R such that lim_{x -> infty} f(x) = infty and f(x) = o(x) when x -> +infty. How can i prove that sup_{n -naturals} {f(n)} = 1, where {f(n)} is fractional part of f(n). Hint: show that f(n+1) - f(n) tends to 0 as n tends to infinity. Thanks. I had the same idea, surely if we prove it, using the fact that f tends to infitiy everything becomes clear about the problem.

But, i'm not sure about my proof of that
lim_{n -> infy} (f(n+1)-f(n)) = 0.

So,using the properties of f i stated here

f'(x) where it exists is decreasing and thus it may have finite limit L when x -> infty or it may have infinite limit. In both cases using the relation

f(x) = f(a) + int_[a,x] f'(t)dt we would have a

contradiction with that f(x) = o(x), x-> infty unless

L=0, whih means that lim f'(x) = 0 when x -> infty, whihch together with

f(n+1) - f(n) = f'(ksi) , ksi \in (n,n+1) gives the
desired lim_{n ->infty} (f(n+1)-f(n)) = 0.

I think these solutoon has some flaws and i'm not sure about it.
Is it in the whole near to be ok.

Thanks

> Mike Guy
M.J.T. Guy

Joined: 03 May 2005
Posts: 62

Posted: Fri Jun 23, 2006 5:18 pm    Post subject: Re: fractional part of the function at integer points+concave

In article <15558845.1151055387707.JavaMail.jakarta@nitrogen.mathforum.org>,
eugene <jane1806@mail.ru> wrote:
 Quote: Let f be a continious concave function from (0, +infty) -> R such that lim_{x -> infty} f(x) = infty and f(x) = o(x) when x -> +infty. How can i prove that sup_{n -naturals} {f(n)} = 1, where {f(n)} is fractional part of f(n).

Hint: show that f(n+1) - f(n) tends to 0 as n tends to infinity.

Mike Guy
eugene
science forum Guru

Joined: 24 Nov 2005
Posts: 331

 Posted: Fri Jun 23, 2006 9:35 am    Post subject: fractional part of the function at integer points+concave Let f be a continious concave function from (0, +infty) -> R such that lim_{x -> infty} f(x) = infty and f(x) = o(x) when x -> +infty. How can i prove that sup_{n -naturals} {f(n)} = 1, where {f(n)} is fractional part of f(n). Any ideas would be welcome. Thanks.

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