camp exam problem-10

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nafistiham
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camp exam problem-10

Unread post by nafistiham » Fri Nov 04, 2011 10:56 pm

Let $f$ be a function from the set of real numbers to itself such that for all real numbers $x,y$ ,

\[\frac{f(x) + f(y)}{2} - f(\frac{x+y}{2}) \geq |x-y|\]


Prove that,

\[\frac{f(x) + f(y)}{2} - f(\frac{x+y}{2}) \geq 2^n|x-y|\]


for all real numbers $x,y$ and all non-negative integers $n$ .
Also, prove that, no such function can exist.
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
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nafistiham
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Re: camp exam problem-10

Unread post by nafistiham » Sat Nov 05, 2011 1:04 am

i couldn't even decide if it was a convex function or not :oops: .would anyone clear me out ?
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
Introduction:
Nafis Tiham
CSE Dept. SUST -HSC 14'
http://www.facebook.com/nafistiham
nafistiham@gmail

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