Let $(a_k)$ be a log-concave positive decreasing sequence. Is $\sum\limits_{k=1}^n a_k(1-e^x)^{k-1}$ log-concave in $x<0$, for each natural $n$?
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6
Yes, and under weaker conditions.
Set $s = (1-e^x) > 0$, and multiply your partial sums by $s$; this has no effect on log concavity. Then the resulting sequence is the Hadamard product of the coefficients of $(\sum_{i=1}^n a_i)$ with $(s^n)$, each of which is log concave (the first is the convolution of $(1,1,1,\dots,)$ with $(a_k)$), so the outcome is.
This argument requires only that $(a_k)$ be log concave (no monotonicity, etc). [There is a quick reduction to $(a_k)$ being a finite sequence if you don't like the possibly infinite convolution, since the $n$th partial sum only involve the first $n$ terms of $(a_k)$, and log concavity is checked just using the $n-1,n+1,n$th terms for each $n$.]
Edit: The original question has been edited (and not by the original proposer) to be quite different. The answer here is to what I thought was the original question, which was whether $(\sum_{i=1}^n a_i (1-e^x)^{n-1})$ is a log concave sequence in $n$.
answered Jan 25, 2016 at 21:21
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4$\begingroup$ Perhaps you understood the question to be about the log-concavity in $n$. However, I believe it is about the log-concavity in $x$; otherwise, indeed, why not use $s$ instead of $1-e^x$? I am now going to edit the question for clarity. $\endgroup$ Commented Jan 26, 2016 at 5:23
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$\begingroup$ I think it would be better for the OP to edit it in this case, just to explain what he meant. And now (after editting), the question is ambiguous, because it could mean either (1) as a function of $x$, the function is log concave, or (2) the Maclaurin series coefficients (perhaps with absolute values) is a log concave sequence. In any event, is it usual to talk about log concavity with a negative parameter? $\endgroup$ Commented Jan 27, 2016 at 2:30
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$\begingroup$ And log concave (no hyphen) is preferred to log-concave. $\endgroup$ Commented Jan 27, 2016 at 2:34
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1$\begingroup$ To answer David's questions: For two reasons, I was and am pretty sure that the OP meant the log-concavity in $x$, and not in $n$: (i) if the OP had meant the log-concavity in $n$, then there would have been no reason for him to use $1-e^x$ rather than a simple symbol (say $s$) and (ii) because I had seen a related question by the same OP at mathoverflow.net/questions/229028/… and, especially, his comments there. $\endgroup$ Commented Jan 27, 2016 at 16:44
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$\begingroup$ Continued: I don't see any ambiguity in the edited question. In particular, I don't see any substantial difference between "log-concave in $x$, for each natural $n$" there and your "(1) as a function of $x$, the function is log concave". I also don't see how the question about the log-concavity in $x$ could be possibly interpreted as a question about the log-concavity of the sequence of the Maclaurin series coefficients. $\endgroup$ Commented Jan 27, 2016 at 16:44