Let L ( X ) = exp ⁡ ( log ⁡ X log ⁡ log...

Emma Hobbs

Emma Hobbs

Answered

2022-11-22

Let L ( X ) = exp ( log X log log X )
Prove that if c>0, Y = L ( X ) c , and u = log X / log Y , then
u u = L ( X ) ( 1 / 2 c ) ( 1 + o ( 1 ) )
I've tried to write u u = ( log X / log Y ) log X / log Y
But that doesn't seem to get me anywhere.

Answer & Explanation

kuthiwenihca

kuthiwenihca

Expert

2022-11-23Added 23 answers

First, we have ln u u = u ln u = 1 ln Y ( ln X ln ln X ln Y ) = 1 ln Y ( ln X ( ln ln X ln ln Y ) ) , and ln ln Y = ln ( c ln X ln ln X ) = 1 2 ln ln X + o ( ln ln X ) ,   a s   X + . Therefore, ln u u = 1 2 c ln X ln ln X ln X ln ln X ( 1 + o ( 1 ) ) . On the other hand, we have ln L ( X ) = ln X ln ln X , and so
ln u u ln L ( x ) = 1 2 c ( 1 + o ( 1 ) ) .

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