Abstract
This short note concerns with two inequalities in the geo\-me\-try of gradient Einstein solitons $(g, f, \lambda )$ on a smooth manifold $M$. These inequalities provide some relationships between the curvature of the Riemannian metric $g$ and the behavior of the scalar field $f$ through two quadratic equations satisfied by the scalar $\lambda $. The similarity with gradient Ricci solitons and a slightly generalization involving a $g$-symmetric endomorphism $A$ are provided.
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