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GH from MO
GH from MO
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A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect to $x$ if $a_k\ge 0$ for all $k$ and $x\ge0$. I have tested some log-concave sequences. The results provesupport my guess. If this proposition is true, how to prove it rigorously?

A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect to $x$ if $a_k\ge 0$ for all $k$ and $x\ge0$. I have tested some log-concave sequences. The results prove my guess. If this proposition is true, how to prove it rigorously?

A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect to $x$ if $a_k\ge 0$ for all $k$ and $x\ge0$. I have tested some log-concave sequences. The results support my guess. If this proposition is true, how to prove it rigorously?

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Johnny Yin
Johnny Yin
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Log-concave polynomial is a log-concave function?

A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect to $x$ if $a_k\ge 0$ for all $k$ and $x\ge0$. I have tested some log-concave sequences. The results prove my guess. If this proposition is true, how to prove it rigorously?