You can use below code to check homepage
$routeName = Mage::app()->getRequest()->getRouteName();
$identifier = Mage::getSingleton('cms/page')->getIdentifier();
if($routeName == 'cms' && $identifier == 'home') {
echo 'This is Magento Homepage.';
} else {
echo 'This is not a Magento Homepage.';
}
From your comment i suggest you use above code $identifier == 'home' change cms page 'home' identifier with your homepage identifier.
also alternative way to check is this home page
if(Mage::getBlockSingleton('page/html_header')->getIsHomePage()) {
echo 'This is Magento Homepage.';
} else {
echo 'This is not a Magento Homepage.';
}