For any model function $\varphi$ we set

(6.1)

$$E_{\omega}(\varphi)=\frac{1}{n+1}\sum_{j=0}^{n}\int\varphi(\omega+dd^{c}\varphi)^{j}\wedge\omega^{n-j}$$

and call $E_{\omega}(\varphi)$ the energy of $\varphi$. It follows formally from an integration by parts argument, see Proposition 2.20 and [Tia00, Lemma 6.2] that if $\varphi,\psi$ are any two model functions, then

(6.2)

$$E_{\omega}(\psi)-E_{\omega}(\varphi)=\frac{1}{n+1}\sum_{j=0}^{n}\int(\psi-\varphi)(\omega+dd^{c}\varphi)^{j}\wedge(\omega+dd^{c}\psi)^{n-j}.$$

Writing $\varphi_{t}=(1-t)\varphi+t\psi$, and expanding $E_{\omega}(\varphi_{t})-E_{\omega}(\varphi)$ in $t$ leads to the following formulas for first and second derivatives of $E_{\omega}$:

(6.3)		$\displaystyle E^{\prime}_{\omega}(\varphi)\cdot(\psi-\varphi)$	$\displaystyle=\frac{d}{dt}\bigg|_{t=0+}E_{\omega}(\varphi_{t})=\int(\psi-\varphi)\MA(\varphi);$
(6.4)		$\displaystyle E^{\prime\prime}_{\omega}(\varphi)\cdot(\psi-\varphi)$	$\displaystyle=\frac{d^{2}}{dt^{2}}\bigg|_{t=0+}E_{\omega}(\varphi_{t})=n\,\int(\psi-\varphi)dd^{c}(\psi-\varphi)\MA(\varphi).$