  Information-theoretic inequalities play a fundamental role in numerous
scientific and technological areas as they generally express the impossibility
to have a complete description of a system via a finite number of information
measures. In particular, they gave rise to the design of various quantifiers
(statistical complexity measures) of the internal complexity of a (quantum)
system. In this paper, we introduce a three-parametric Fisher-R\'enyi
complexity, named $(p,\beta,\lambda)$-Fisher--R\'enyi complexity. This
complexity measure quantifies the combined balance of the spreading and the
gradient contents of $\rho$, and has the three main properties of a statistical
complexity: the invariance under translation and scaling transformations, and a
universal bounding from below. The latter is proved by generalizing the Stam
inequality, which lowerbounds the product of the Shannon entropy power and the
Fisher information of a probability density function. An extension of this
inequality was already proposed by Bercher and Lutwak, a particular case of the
general one, where the three parameters are linked, allowing to determine the
sharp lower bound and the associated probability density with minimal
complexity. Using the notion of differential-escort deformation, we are able to
determine the sharp bound of the complexity measure even when the three
parameters are decoupled (in a certain range). We determine as well the
distribution that saturates the inequality: the $(p,\beta,\lambda)$-Gaussian
distribution, which involves an inverse incomplete beta function. Finally, the
complexity measure is calculated for various quantum-mechanical states of the
harmonic and hydrogenic systems, which are the two main prototypes of physical
systems subject to a central potential.
