The authors introduce and study Hessian quartic forms for an arbitrary Hermitian metric, patterned after the Hessian curvature for the Carathéodory metric on bounded domains in \({\mathbb{C}}^ n\) [see the reviewer, Rocky Mt. J. Math. 8, 555-559 (1978; Zbl 0347.32014)]. These Hessian quartic forms provide another proof of a result of \textit{H. Wu} [Indiana Univ. Math. J. 22, 1103-1108 (1973; Zbl 0265.53055)] stating that the holomorphic sectional curvature coincides with the maximum of the Gaussian curvatures to all local one-dimensional submanifolds that contact at the point in the direction under consideration. Furthermore, as another application, the authors consider the Hessian quartic form of the so called ''m-th order Bergman metric'' [see the reviewer, Proc. Am. Math. Soc. 67, 50-54 (1977; Zbl 0346.32030)] and prove several distortion theorems.