In this paper, we prove that the Watanabe-Yoshida conjecture holds up to dimension 7. Our primary new tool is a function, $\varphi_J(R;z^t),$ that interpolates between the Hilbert-Kunz multiplicities of a base ring, $R$, and various radical extensions, $R_n$. We prove that this function is concave and show that its rate of growth is related to the size of $e_{\textrm{HK}}(R)$. We combine techniques from Celikbas et al. (Nagoya Math. J. 205, 149–165, 2012) and Aberbach and Enescu (Nagoya Math. J. 212, 59–85, 2013) to get effective lower bounds for $\varphi ,$ which translate to improved bounds on the size of Hilbert-Kunz multiplicities of singular rings. The improved inequalities are powerful enough to show that the conjecture of Watanabe and Yoshida holds in dimension 7.