def dup_root_lower_bound(f, K):
 bound = dup_root_upper_bound(dup_reverse(f), K)
 if (bound is not None):
  return (1 / bound)
 else:
  return None

def dup_root_lower_bound ( f , K ) : bound = dup_root_upper_bound ( dup_reverse ( f ) , K ) if ( bound is not None ) : return ( 1 / bound ) else : return None
compute the lmq lower bound for the positive roots of f; lmq was developed by akritas-strzebonski-vigklas .