In the present paper, the authors study standard graded algebras over Artinian rings, for example, an associated graded ring of an \(\mathfrak m\)-primary ideal in a Noetherian local ring \((A, \mathfrak m)\). The Poincaré-Hilbert series of such a graded ring \(G\) is a formal power series \(P_G(t) = \sum_{n \geq 0} \ell(G_n) t^n\). For two formal power series, we define an order between them by \[ \sum a_n t^n \geq \sum b_n t^n \iff a_n \geq b_n \quad \text{for all \(n\)}. \] The purpose of this paper is to give a sharp upper bound of Poincaré-Hilbert series by using the dimension, Hilbert coefficients and other extended multiplicities [cf. \textit{L. Rodrigues Doering, T. Gunston} and \textit{W. V. Vasconcelos} [Am. J. Math. 120, 493-504 (1998; Zbl 0924.13011)]. By using this bound, the authors show that there are only finitely many Poincaré-Hilbert series of standard graded algebras with given dimension and given extended multiplicity.