How was the series derived(logarithms)

Question

I am revising and my past papers have short answers with no detail. My question is: How was the first series derived, and then why did they put the second series to be smaller than the first one(also how was it derived)?

Answer 1

The function $h(x)=x\log(x)$ with $h(0)=0$ is continuous on $[0,\infty)$ and smooth on $(0,\infty)$. Its first two derivatives are $$ h'(x)=\log(x)+1\\ h''(x)=\frac1x $$ so that the function is convex and monotonically increasing for $x>e^{-1}$.

Use the convexity and the supporting line/tangent at $x=1$ to conclude that $$ h(x)\ge h(1)+h'(1)(x-1)=x-1.\tag1 $$

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To get the claimed inequality into a form that you can apply this above inequality (1), you need to multiply it by $m$ so that $$ m\sum_{i=1}^mp_i\log(mp_i)=\sum_{i=1}^m h(mp_i)\ge\sum_{i=1}^m(mp_i-1)=0 $$

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In total this shows that the entropy is bounded, $$ \sum_{i=1}^m(-p_i\log(p_i))\le\log(m) $$