Redirect logged user depending on user role if he tries to access home page?

Try something like this. Also read about wp_redirect() too.

function redirect_by_role(){
  if (is_user_logged_in()) {
    $user = wp_get_current_user();
    if (is_home()) {
      if ( in_array( 'author', (array) $user->roles ) ) {
         $redirect_url="";
         wp_redirect($redirect_url, 301); exit;
      }
      elseif( in_array( '**another_role_here**', (array) $user->roles ) ) {
         $redirect_url="";
         wp_redirect($redirect_url, 301); exit;
      }
    }

  } else { 
   // Do what you want

  }
}
// Add the action with maximum priority
add_action('init','redirect_by_role',0);