Let $\sigma\in H^{0}(X,E)$ be a holomorphic section of a holomorphic hermitian vector bundle $(E,h)$ and set $S_{\varepsilon}:=\log(|\sigma|^{2}+\varepsilon)$, for some $\varepsilon>0$. We denote by $\{\cdot,\cdot\}$ the exterior product of $E$-valued forms respect to the hermitian metric $h$. We have

$$i\partial S_{\varepsilon}=\frac{i\{\partial_{h}\sigma,\sigma\}}{|\sigma|^{2}+\varepsilon}\,,$$

since $\sigma$ is a holomorphic section. We compute now the complex hessian

	$\displaystyle i\partial\bar{\partial}S_{\varepsilon}$	$\displaystyle=$	$\displaystyle-\bar{\partial}\,\frac{i\{\partial_{h}\sigma,\sigma\}}{|\sigma|^{2}+\varepsilon}$
		$\displaystyle=$	$\displaystyle\frac{-i\{\bar{\partial}\partial_{h}\sigma,\sigma\}+i\{\partial_{h}\sigma,\partial_{h}\sigma\}}{|\sigma|^{2}+\varepsilon}\,+\,i\{\partial_{h}\sigma,\sigma\}\wedge\bar{\partial}\left(\frac{1}{|\sigma|^{2}+\varepsilon}\right)$
		$\displaystyle=$	$\displaystyle\frac{i\{\partial_{h}\sigma,\partial_{h}\sigma\}\,-\,\{i{\cal C}_{h}(E)\sigma,\sigma\}}{|\sigma|^{2}+\varepsilon}\,-\,\frac{i\{\partial_{h}\sigma,\sigma\}\wedge\{\sigma,\partial_{h}\sigma\}}{(|\sigma|^{2}+\varepsilon)^{2}}$
		$\displaystyle=$	$\displaystyle\underbrace{\frac{(|\sigma|^{2}+\varepsilon)i\{\partial_{h}\sigma,\partial_{h}\sigma\}-i\{\partial_{h}\sigma,\sigma\}\wedge\{\sigma,\partial_{h}\sigma\}}{(|\sigma|^{2}+\varepsilon)^{2}}}_{iT(S_{\varepsilon})}\,-\,\frac{\{i{\cal C}_{h}(E)\sigma,\sigma\}}{|\sigma|^{2}+\varepsilon}\,.$

We show that the $(1,1)$-form $iT(S_{\varepsilon})$ is nonnegative. In fact by using twice the Lagrange inequality

$$i\{\partial_{h}\sigma,\sigma\}\wedge\{\sigma,\partial_{h}\sigma\}\leq|\sigma|^{2}\,i\{\partial_{h}\sigma,\partial_{h}\sigma\}\,,$$

(which is an equality in the case of line bundles) we get

$\displaystyle iT(S_{\varepsilon})\geq\frac{\varepsilon i\{\partial_{h}\sigma,\partial_{h}\sigma\}}{(|\sigma|^{2}+\varepsilon)^{2}}\geq\frac{\varepsilon i\{\partial_{h}\sigma,\sigma\}\wedge\{\sigma,\partial_{h}\sigma\}}{|\sigma|^{2}(|\sigma|^{2}+\varepsilon)^{2}}=\frac{\varepsilon}{|\sigma|^{2}}\,i\partial S_{\varepsilon}\wedge\bar{\partial}S_{\varepsilon}\geq 0\,.$

Observe that the last form is smooth. Consequently, we find the inequalities

	$\displaystyle i\partial\bar{\partial}S_{\varepsilon}$	$\displaystyle\geq$	$\displaystyle\frac{\varepsilon}{|\sigma|^{2}}\,i\partial S_{\varepsilon}\wedge\bar{\partial}S_{\varepsilon}\,-\,\frac{\{i{\cal C}_{h}(E)\sigma,\sigma\}}{|\sigma|^{2}+\varepsilon}$
		$\displaystyle\geq$	$\displaystyle\frac{\varepsilon}{|\sigma|^{2}}\,i\partial S_{\varepsilon}\wedge\bar{\partial}S_{\varepsilon}\,-\,\|{\cal C}_{h}(E)\|_{h,\omega}\,\frac{|\sigma|^{2}}{|\sigma|^{2}+\varepsilon}\,\omega$

where $\omega$ is a positive $(1,1)$-form.

Remark. Let $(X,\omega_{X})$ be a polarised compact Kähler manifold of complex dimension $n$, let $(Y,\omega_{Y})$ be a compact irreducible Kähler space of complex dimension $m\leq n$, let $\pi:X\rightarrow Y$ be a surjective holomorphic map and let $0\leq f\in L\log^{n+\varepsilon}L(X,\omega^{n}_{X})$, for some $\varepsilon>0$ such that $1=\int_{X}f\omega^{n}_{X}$. Set $K_{t}:=\{\pi^{*}\omega_{Y}+t\omega_{X}\}^{n}>0$ for $t\in(0,1)$. Consider the complex Monge-Ampère equations $(\pi^{*}\omega_{Y}+t\omega_{X}+i\partial\bar{\partial}\psi_{t})^{n}=K_{t}\,f\,\omega^{n}_{X}$. The hypothesis $(C1)$ of statement $(C)$ in theorem 3 is obviously satisfied. The hypothesis $(C2b)$ is also satisfied since

$$\lim_{t\rightarrow 0}\,\frac{(\pi^{*}\omega_{Y}+t\omega_{X})^{n}}{K_{t}\,\omega^{n}_{X}}=\left(\;\int\limits_{y\in Y}\omega^{m}_{Y}(y)\cdot\int\limits_{z\in\pi^{-1}(y)}\omega^{n-m}_{X}\right)^{-1}\frac{\pi^{*}\omega_{Y}\wedge\omega^{n-m}_{X}}{\omega^{n}_{X}}<+\infty.$$

We deduce $\operatorname{Osc}(\psi_{t})\leq C<+\infty$ for all $t\in(0,1)$ by the statements $(C)$ and $(A)$ of theorem 3. This solves in full generality a Tian’s conjecture [Ti-Ko].

Acknowledgments. The second named author is grateful to Professor Gang Tian for bringing this type of problems to his attention. He expresses also his gratitude to the members of Institut Fourier for providing an excellent research environment. In particular he thanks Adrien Dubouloz, Hervé Pajot, Olivier Lablée and Eric Dumas for useful conversations.