Here's yet another example:
VARTEST="value of VARTEST"
#export VARTEST="value of VARTEST"
sudo env | grep -i vartest
sudo echo ${SUDO_USER} ${SUDO_UID}:${SUDO_GID} "${VARTEST}"
sudo bash -c 'echo ${SUDO_USER} ${SUDO_UID}:${SUDO_GID} "${VARTEST}"'
Only by using export VARTEST the value of VARTEST is available in sudo bash -c '...'!
For further examples see:
bash-hackers.org/wiki/doku.php/scripting/processtree