How can we show that if the function $y=f(x)$ is convex in $0\leq x<\infty$, then the points $a_{n}=f(n)$ form a convex sequence.
To show if function f(x), 0\leq x<\infty is convex then $a_{n}=f(n)$ is convex sequence.
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real-analysis
convex-analysis
1 Answers
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Is it true that $f(n+1)\leq\frac{1}{2}\left(f(n)+f(n+2)\right)\,\,?$Hint: take $\,\,x_1:=n\,\,,\,x_2:=n+2\,$ and apply convexity of $f$:$f\left(\frac{x_1+x_2}{2}\right)\leq\frac{1}{2}\left(f(x_1)+f(x_2)\right)$
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0Thanku so much! And it means to write the proof of given result i have to show first$ f(n+1)\leq \frac{1}{2}(f(n)+f(n+2)).$ – 2012-06-05