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- Homework Statement
- Let a>0, b>0, and 0<λ<1 begiven. Show that ln(λa + (1 − λ)b) ≥ λ ln a + (1 − λ) ln b.

- Relevant Equations
- We were given a few exercises on a worksheet in order understand a proof in a Topology book.

The listed HW problem was 4.

Now, For problem 5, we have:

(AM-GMInequality)Let a > 0,b > 0,and 0 < λ <1 be given. Show that (a^λ)(b^(1−λ)) ≤ λa+(1−λ)b.

Note that if we prove problem 4, the proof for problem 5 follows directly. We use properties of logarithms to combine the right hand side of ln into a single logarithm. Then we raise both side of the inequality to a power of e. Which leads us to the desired inequality.

But, when I try to be prove 4 using 5, it leads to circular reasoning. Since, If I prove 5 first without using problem 4, I can do a proof by contradiction.

Assume instead that ln(λa + (1 − λ)b) < λ ln a + (1 − λ) ln b. Which leads to (a^λ)(b^(1−λ)) > λa+(1−λ)b. But by problem 5, we know that (a^λ)(b^(1−λ)) ≤ λa+(1−λ)b.

QED

I want to avoid circular reasoning and complete the problems in order. Was wondering if anyone can give me a hint and point me in the right direction on how to complete 4? I tried contraction, but I do not think this is the way to go.

Thanks.

But, when I try to be prove 4 using 5, it leads to circular reasoning. Since, If I prove 5 first without using problem 4, I can do a proof by contradiction.

Assume instead that ln(λa + (1 − λ)b) < λ ln a + (1 − λ) ln b. Which leads to (a^λ)(b^(1−λ)) > λa+(1−λ)b. But by problem 5, we know that (a^λ)(b^(1−λ)) ≤ λa+(1−λ)b.

QED

I want to avoid circular reasoning and complete the problems in order. Was wondering if anyone can give me a hint and point me in the right direction on how to complete 4? I tried contraction, but I do not think this is the way to go.

Thanks.