Is $g(x) = \log f(x)$ locally strongly concave if $f$ is a linear function? What if $f$ is a homogeneous function of degree 1?
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2Let $f(x)=x$ which is linear. Then $g(x)$ is concave. Are you sure this is what you wanted to ask? – CyclotomicField May 22 '23 at 01:43
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$\log(x)$ is concave, but it's not strongly concave. The question is: "is it locally strongly concave?" – smz May 22 '23 at 02:07
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1Look at the Hessian. – copper.hat May 22 '23 at 03:41
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If we assume that $f$ is differentiable and maps to the positives ($f:\mathbb{R}\to ]0,\infty[$), we can study its derivatives. Notice that:
$$g''(x)=\frac{f''(x)}{f(x)}+\left(\frac{f'(x)}{f(x)}\right)^2.$$
Concavity is then met whenever $g''(x)<0$, equivalently:
$$f''(x)f(x)<\left(f'(x)\right)^2.$$
Weierstraß Ramirez
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