Can you help me? I understand that it has to do with areas. I can draw pictures and unsteratsnd thet the inequality holds. But how to prove it correctly?
$f$ is increasing and concave up, determine definite integral
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definite-integrals
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0The first inequality follows from the fact that $f$ is increasing, hence $f(a) < f(x)$ if $a < x \leq b$. The second inequality follows from the inequality $f(x) < f(a) + (f(b)-f(a))(x-a)$ for $a < x < b$, which follows from strict convexity of $f$. – 2017-01-21
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0This is a chord I mentioned in my answer. – 2017-01-21
1 Answers
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Because $f$ is increasing, we have $f(a)\le f\left(\frac{a+b}{2}\right)$. Next see the Hermite-Hadamard inequality.
The proof of this inequality goes by the basic properties of convexity (i.e. up-concavity). The left part of HH inequality: integrate the affine support at the midpoint. The right part: integrate the chord through $a,b$.
The HH inequality reverts in the case of down-concavity, because $f$ is down concave $\iff -f$ is up concave. Use this variant of HH to prove your inequality. Monotonocity also reverts. The inequalities are strict because of (strict) monotonicity properties.