Suppose we have a differentiable function $g(x)$. Let's define a function $h(x)=axg(x)$ where $a$ is a constant (this is just as an example). I want to check the convexity/concavity of function $h(x)$. Is there any interpretation for the ration of the second derivative of a function to its first derivative, that is, $\frac{g^{\prime \prime}(x)}{g^{\prime}(x)}$? $h^{\prime \prime}(x)=a\left[x g^{\prime \prime}(x) + 2g^{\prime}(x) \right]$
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As stated this seems to be two different questions mixed together. To answer the question that's explicitly asked, one interpretation of $\frac{g''(x)}{g'(x)}$ is the derivative of $\log g'(x)$. – Greg Martin Apr 10 '22 at 18:56
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@GregMartin Yes, you are right. I am wondering if $log(g^{\prime}(x)) $ has any meaningful interpretation? Based on the formula, can we say anything about the convexity of function $h(x)$? – Amin Apr 10 '22 at 19:31