  We prove an extended convexity for quantum Fisher information of a mixed
state with a given convex decomposition. This convexity introduces a bound
which has two parts: i. classical part associated to the Fisher information of
the probability distribution of the states contributing to the decomposition,
and ii. quantum part given by the average quantum Fisher information of the
states in this decomposition. Next we use a non-Hermitian extension of
symmetric logarithmic derivative in order to obtain another upper bound on
quantum Fisher information, which enables to derive a closed form for a fairly
general class of system dynamics given by a dynamical semigroup. We combine our
two upper bounds together in a general (open system) metrology framework where
the dynamics is described by a quantum channel, and derive the ultimate
precision limit for quantum metrology. We illustrate our results and their
applications through two examples, where we also demonstrate that how the
extended convexity allows to track transition between quantum and classical
behaviors for an estimation precision.
