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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 $Z_{n:n}=\max\{Z_1,\ldots,Z_n\}$. Note 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.

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) 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.

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 $Z_{n:n}=\max\{Z_1,\ldots,Z_n\}$. Note 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.

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 $Z_{n:n}=\max\{Z_1,\ldots,Z_n\}$. Note 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.

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.

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 $Z_{n:n}=\max\{Z_1,\ldots,Z_n\}$. Note 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.