Proving one function is greater than other?

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Fatima Majeed 2024-6-9

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2126841-proving-one-function-is-greater-than-other

评论： Matt J 2024-6-10

采纳的回答： Matt J

I want to see where is this inequality true

where x in (e^e,∞).

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Matt J 2024-6-9

The notation is ambiguous. We must know whether

is to be interpreted as

or as

Fatima Majeed 2024-6-9

@Matt J It is as (log(x))^y

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Matt J 2024-6-10

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https://ww2.mathworks.cn/matlabcentral/answers/2126841-proving-one-function-is-greater-than-other#answer_1469321

编辑：Matt J 2024-6-10

Make the change of variables

and rearrange the inequality as,

Since

is a convex function on

such that

>0 and

, it readily follows that

for all

. QED.

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Fatima Majeed 2024-6-10

@Sam Chak that is stunning

Matt J 2024-6-10

I want to thank both of you for your efforts.

You're welcome, but please Accept-click the answer if it has resolved your question.

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Torsten 2024-6-9

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编辑：Torsten 2024-6-10

在 MATLAB Online 中打开

syms x y
f = 53.989/21.233 * (log(log(x))).^(4/3)./(log(x)).^(1/3);
%x is solution where f starts getting greater than 1
xstart = vpasolve(f==1,x,5)
xstart = 5.8933180867157239497727768797558
log(log(xstart))/(21.233*log(xstart)) 
ans = 0.01521725041175722892398806828762
1/(53.989*log(xstart)^(2/3)*log(log(xstart))^(1/3))
ans = 0.01521725041175722892398806828762
ftrans = subs(f,x,exp(exp(y)));
%exp(exp(y)) is solution where f ends being greater than 1
yend = vpasolve(ftrans==1,y,13)
yend = 13.085773977624348668463856655909
log(log(exp(exp(yend))))/(21.233*log(exp(exp(yend))))
ans = 0.0000012785232351771168083804508742678
1/(53.989*log(exp(exp(yend)))^(2/3)*log(log(exp(exp(yend))))^(1/3))
ans = 0.0000012785232351771168083804508742678