  In this paper we introduce homological and homotopical Poincar\'e polynomials
$P_f(t)$ and $P^{\pi}_f(t)$ of a continuous map $f:X \to Y$ such that if $f:X
\to Y$ is a constant map, or more generally, if $Y$ is contractible, then these
Poincar\'e polynomials are respectively equal to the usual homological and
homotopical Poincar\'e polynomials $P_X(t)$ and $P^{\pi}_X(t)$ of the source
space $X$. Our relative Hilali conjecture $P^{\pi}_f(1) \leqq P_f(1)$ is a map
version of the the well-known Hilali conjecture $P^{\pi}_X(1) \leqq P_X(1)$ of
a rationally elliptic space X. In this paper we show that under the condition
that $H_i(f;\mathbb Q):H_i(X;\mathbb Q) \to H_i(Y;\mathbb Q)$ is not injective
for some $i>0$, the relative Hilali conjecture of product of maps holds,
namely, there exists a positive integer $n_0$ such that for $\forall n \geqq
n_0$ the \emph{strict inequality $P^{\pi}_{f^n}(1) < P_{f^n}(1)$} holds, where
$f^n:X^n \to Y^n$. In the final section we pose a question whether a
"Hilali"-type inequality $HP^{\pi}_X(r_X) \leqq P_X(r_X)$ holds for a
rationally hyperbolic space $X$, provided the the homotopical Hilbert--Poincare
series $HP^{\pi}_X(r_X)$ converges at the radius $r_X$ of convergence.
