Given polynomials \(P_1,\dots,P_n\) of degree \(k_1,\dots, k_n\), respectively, on a complex Banach space \(E\), C. Benítez et al. [Math. Proc. Camb. Philos. Soc. 124, No. 3, 395–408 (1998; Zbl 0937.46044)] showed that \[ \|P_1\|\cdots\|P_n\|\leq {(k_1+\cdots +k_n)^{k_1+\cdots +k_n}\over k_1^{k_1} \cdots k_n^{k_n}}\cdots \|P_1\dots P_n\|, \] with this inequality being sharp on \(\ell_1^n\). When \(E\) is a Hilbert space, D. Pinasco [Trans. Am. Math. Soc. 364, No. 8, 3993–4010 (2012; Zbl 1283.30004)] showed that the constant \({(k_1+\cdots +k_n)^ {k_1+\cdots +k_n}\over k_1^{k_1}\cdots k_n^{k_n}}\) may be replaced with \(\sqrt{{(k_1+\cdots +k_n)^{k_1+\cdots +k_n}\over k_1^{k_1}\cdots k_n^{k_n}}}\). In this paper, the authors show that, if \(E\) is a \(d\)-dimensional Banach space, then \[ \|P_1\|\cdots\|P_n\|\leq {(C_K4ed)^{k_1+\cdots +k_n}\over 2^{{n\over C_K}}}\cdots \|P_1\dots P_n\| \] where \(C_K\) is \(1\) if \(E\) is real and \(2\) if \(E\) is complex. Moreover, when \(P_1,\dots, P_n\) are homogeneous polynomials on a Hilbert space, \(\displaystyle{\left(C_K4ed\right)^{k_1+\cdots +k_n}\over 2^{{n\over C_K}}}\) may be replaced with \(\displaystyle{\left({e^{H_dC_K}\over 4}\right)^{k_1+\cdots +k_n}\over 2^{{n\over C_K}}}\) where \(H_d=\sum_{n=1}^d{1\over k}\). For large values of \(n\), this gives an improvement of the above inequality of Benítez et al. [loc. cit.]. Using this result, the authors show that, if \(P_1,\dots,P_n\) are polynomials of degree \(k_1,\dots , k_n\), respectively, on a complex Banach space and \(a_1,\dots, a_n\) are strictly positive real numbers with \(\sum_{k=1}^na_k<{1\over n^{n-1}}\), then there is \(z_0\) in the unit ball of \(E\) with \(|P_i(z_0)|\geq a_i^{k_i}\) for \(1\leq i\leq n\). In the case where \(E\) is \(L_p(\mu)\) or \(S_p\) for \(1\leq p\leq 2\) and \(P_1,\dots,P_n\) are homogeneous, the condition \(\sum_{k=1}^na_k<{1\over n^{n-1}}\) can be replaced with the condition \(\sum_{k=1}^na_k^p<{1\over n^{n-1}}\).