= Band-edge effective mass in a weak periodic potential
In a one-dimensional two-wave <nearly-free electron model>, set $\varepsilon_B=\hbar^2G^2/(8m)$ and let the coupling magnitude be $U>0$. Near the Bragg plane,
$$
E_\pm(q)=\varepsilon_B+\frac{\hbar^2q^2}{2m}\pm\sqrt{U^2+\left(\frac{\hbar^2Gq}{2m}\right)^2}.
$$
Expansion at $q=0$ gives $E_\pm=\varepsilon_B\pm U+[\hbar^2/(2m)](1\pm2\varepsilon_B/U)q^2+O(q^4)$, hence
$$
\frac{m_\pm^*}{m}=\frac1{1\pm2\varepsilon_B/U}\simeq\pm\frac{U}{2\varepsilon_B}\qquad(U\ll\varepsilon_B).
$$
The lower edge has negative electronic curvature, and the upper edge positive curvature. The last expression is a leading weak-potential approximation, not the exact curvature of the two-wave dispersion.
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