  We construct nonlinear coherent states or f-deformed coherent states for a
nonpolynomial nonlinear oscillator which can be considered as placed in the
middle between the harmonic oscillator and the isotonic oscillator (Cari\~nena
J F et al, J. Phys. A: Math. Theor. 41, 085301 (2008)). The deformed
annihilation and creation operators which are required to construct the
nonlinear coherent states in the number basis are obtained from the solution of
the Schr\"odinger equation. Using these operators, we construct generalized
intelligent states, nonlinear coherent states, Gazeau-Klauder coherent states
and the even and odd nonlinear coherent states for this newly solvable system.
We also report certain nonclassical properties exhibited by these nonlinear
coherent states. In addition to the above, we consider position dependent mass
Schr\"odinger equation associated with this solvable nonlinear oscillator and
construct nonlinear coherent states, Gazeau-Klauder coherent states and the
even and odd nonlinear coherent states for it. We also give explicit
expressions of all these nonlinear coherent states by considering a mass
profile which is often used for studying transport properties in
semiconductors.
