Let $Z\_1,\ldots, Z\_n$ be independent standard gaussian random variables. Is it true that $X=\max\{Z\_1,\ldots,Z\_n\}$ has a log-concave distribution function?



Log-concavity of the maximum of gaussians
From the context, it appears that by "distribution function" the OP meant the pdf (usually, the distribution function is understood as the cdf). Let $F$ and $G$ denote the cdf's of $X$ and $Z\_1$, respectively, so that $F=G^n$. (Note that $1-(1-G)^n$ is the cdf of the minimum, not the maximum, of $Z\_1,\ldots,Z\_n$. Thus, ofer zeitouni actually showed that the pdf of $\min\{Z\_1,\ldots,Z\_n\}$ is log concave.) 


Let us show that the pdf $f=F'$ of $\max\{Z\_1,\ldots,Z\_n\}$ is log concave. 
We have $f=F'=nG^{n-1}g$, where $g:=G'$, the standard normal pdf. 
Next,  

\begin{equation}
 G^2(\ln f)''=-(n-1)gh-G^2,\quad\text{where } h(x):=xG(x)+g(x). 
\end{equation}
Moreover, $h''=g>0$ and $h(-\infty+)=0=h'(-\infty+)$. So, $h>0$ and hence $(\ln f)''<0$, on the entire real line. Thus, the pdf $f$ is indeed log concave. 


**Remark.** For any $j=1,\dots,n$, let $f\_j$ denote the pdf of the $j$th order statistic $Z\_{n:j}$ for $Z\_1,\ldots,Z\_n$, so that $Z\_{n:1}=\min\{Z\_1,\ldots,Z\_n\}$ and $Z\_{n:n}=\max\{Z\_1,\ldots,Z\_n\}$. It follows from a known formula for the pdf of an order statistic (see formula (2.1.6) on page 10 in [David--Nagaraja](https://books.google.com/books?id=bdhzFXg6xFkC&printsec=frontcover&dq=Order%20Statistics&hl=en&sa=X&ved=0ahUKEwi1tvfSuK_ZAhUCxmMKHbVdAgQQ6AEIJzAA#v=onepage&q=single%20order&f=false)) that 
\begin{equation}
 f\_j=\binom{n-1}{j-1}f\_1^{\frac{n-j}{n-1}}f\_n^{\frac{j-1}{n-1}}. 
\end{equation}
So, combining the log-concavity of $f\_1$ proved by ofer zeitouni with the log-concavity of $f\_n$ proved in this answer, we obtain the more general fact that the pdf $f\_j$ of any order statistic $Z\_{n:j}$ for $Z\_1,\ldots,Z\_n$ is log concave. 
Moreover, by shifting and rescaling, we now see that the pdf of any order statistic for any iid normal sample is log concave. 


