The complex Hessian of a $C^2$ function dominates that of a minorant at a common minimum

Given: An integer $m\ge 1$, an open set $U\subseteq {\mathbb{C}}^m$, real-valued functions $u,v\in C^2(U)$, a point $a\in U$, a neighbourhood $V\subseteq U$ of $a$ on which $u\ge v$, and $u(a)=v(a)$.

The identification of ${\mathbb{C}}^m$ with ${\mathbb{R}}^{2m}$ transports open sets and real coordinate regularity, and $C^2$ means all ordered real coordinate derivatives through order two exist and are continuous (Complex $m$-space and its real coordinate dictionary, $C^k$ maps and multi-index derivative notation in Euclidean space).

For a real $C^2$ function on a real open set, its real Hessian quadratic form is nonpositive at an interior local maximum (The Hessian is negative semidefinite at an interior local maximum).

The Wirtinger operators are ${\mathrm{\partial }}_{z_k}=\frac{1}{2}({\mathrm{\partial }}_{x_k}-i{\mathrm{\partial }}_{y_k})$ and ${\mathrm{\partial }}_{{\bar{z}}_k}=\frac{1}{2}({\mathrm{\partial }}_{x_k}+i{\mathrm{\partial }}_{y_k})$ (Wirtinger operators in ${\mathbb{C}}^m$).

If a map has continuous coordinate partial derivatives on a neighbourhood, it is totally differentiable there, and the ordinary chain rule for total derivatives applies (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$). Applied to an affine complex line and to the first coordinate derivatives of a real $C^2$ function, it gives the line-composition derivative formulas used below.

The real coordinate mixed partial derivatives of a $C^2$ function commute (Clairaut--Schwarz theorem for continuous second partial derivatives).

The Levi form is ${\mathcal{L}}_u(a;X)={\sum }_{j,k=1}^m{u}_{z_j{\bar{z}}_k}(a)X_j\overline{X_k}$, with the one-based indices of The Levi form and strict plurisubharmonicity.