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#### Integration

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- Jan 16, 2013

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Let [TEX]f(x) \in C[a,b] [/TEX] and let [TEX]f(x)>0 [/TEX] on [TEX][a,b][/TEX]. Prove that

[TEX]\exp \Big(\frac{1}{b-a}\int_a^b \ln f(x) dx \Big)\leq \frac{1}{b-a}\int_a^b f(x) dx[/TEX]

I have learnt Gronwall's Inequality and Jensen's Inequality(and inequality deduced from it like Cauchy Schwarz Inequality) but i couldn't use them to fit the condition.

Would you help me please?Thank you.

[TEX]\exp \Big(\frac{1}{b-a}\int_a^b \ln f(x) dx \Big)\leq \frac{1}{b-a}\int_a^b f(x) dx[/TEX]

I have learnt Gronwall's Inequality and Jensen's Inequality(and inequality deduced from it like Cauchy Schwarz Inequality) but i couldn't use them to fit the condition.

Would you help me please?Thank you.

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