A slant immersion is an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger (or Kähler) angle \(\theta\). It is called proper if it is neither holomorphic nor totally real. Let \(\widetilde M^2(4\varepsilon)\) be a complex space form with constant holomorphic sectional curvature \(4\varepsilon\). In a previous paper, the author proved that a proper slant surface \(M\) in \(\widetilde M^2(4\varepsilon)\) satisfies \(H^2\geq 2K- 2(1+3\cos^2(\theta))\varepsilon\), where \(H\) and \(K\) denotes the mean curvature and Gauss curvature of \(M\), respectively. In the present paper, the author classifies all proper slant surfaces in \(\widetilde M^2(4\varepsilon)\) for which this inequality becomes an equality. Further results about these special slant surfaces are obtained.