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Let $X$ and $Y$ be two continuous random variables with support $\mathbb{R}^{+}$ and with PDF $f(x)$ and $g(y)$. If the Bhattacharyya distance of $f$ and $g$ is less than $\epsilon$, then is there any (lower or upper) bound on $\int_{0}^{\infty}\log(1+x)(f(x)-g(x))\,\mathrm{d}x$?

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    $\begingroup$ No. The Bhattacharyya distance is invariant under shifting both distributions to larger $x$, but your integral increases in magnitude unboundedly under such a transformation. $\endgroup$
    – Yoav Kallus
    Commented Dec 2, 2015 at 15:56

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