if user logged in wordpress

document.body.classList.contains( 'logged-in' );

<?php
add_action('admin_init', 'wpse74389_check_username');
function wpse74389_check_username()
{
    $user = wp_get_current_user();

    if($user && isset($user->user_login) && 'username_to_check' == $user->user_login) {
        // do stuff
    }
}