Prompt for sudo password and programmatically elevate privilege in bash script?

I run sudo directly from the script:

if [ $EUID != 0 ]; then
    sudo "$0" "$@"
    exit $?
fi

Add this as the first line of the script:

[ "$UID" -eq 0 ] || exec sudo bash "$0" "$@"

Change sudo to gksu or gksudo if you prefer a graphical prompt.

I suggest:

#!/bin/bash

if (($EUID != 0)); then
  if [[ -t 1 ]]; then
    sudo "$0" "$@"
  else
    exec 1>output_file
    gksu "$0 $@"
  fi
  exit
fi

# some example stuff
ls -l /root
echo "app: $0"
for f; do
  echo ">$f<"
done