  Quantum channel estimation and discrimination are fundamentally related
information processing tasks of interest in quantum information science. In
this paper, we analyze these tasks by employing the right logarithmic
derivative Fisher information and the geometric R\'enyi relative entropy,
respectively, and we also identify connections between these distinguishability
measures. A key result of our paper is that a chain-rule property holds for the
right logarithmic derivative Fisher information and the geometric R\'enyi
relative entropy for the interval $\alpha\in(0,1) $ of the R\'enyi parameter
$\alpha$. In channel estimation, these results imply a condition for the
unattainability of Heisenberg scaling, while in channel discrimination, they
lead to improved bounds on error rates in the Chernoff and Hoeffding error
exponent settings. More generally, we introduce the amortized quantum Fisher
information as a conceptual framework for analyzing general sequential
protocols that estimate a parameter encoded in a quantum channel, and we use
this framework, beyond the aforementioned application, to show that Heisenberg
scaling is not possible when a parameter is encoded in a classical-quantum
channel. We then identify a number of other conceptual and technical
connections between the tasks of estimation and discrimination and the
distinguishability measures involved in analyzing each. As part of this work,
we present a detailed overview of the geometric R\'enyi relative entropy of
quantum states and channels, as well as its properties, which may be of
independent interest.
