How to show negative entropy function \(\displaystyle{f{{\left({x}\right)}}}={x}{\log{{\left({x}\right)}}}\)

Quinn Moses

Quinn Moses

Answered question

2022-04-06

How to show negative entropy function f(x)=xlog(x) is strongly convex?

Answer & Explanation

betazpvaf4

betazpvaf4

Beginner2022-04-07Added 9 answers

We have f(x)=1+log(x). Strict convexity therefore means that there exists a strictly positive α such that
[log(y)log(x)](yx)α(yx)2
holds. Without loss of generality I assume y>x. Then the above inequality requires that
log(y)log(x)α(yx).
Although you don't state it, I assume that the variables x and y live on (0,1) (because they are probabilities). Since the slope of the log function on the interval (0,1) is larger than or equal to 1, you can choose any positive α smaller than 1.
If x and y live on a general bounded interval (0,M), then the argument goes through with α<1M.

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