  The paper gives a precise asymptotic relation between higher order
logarithmic difference and logarithmic derivatives for meromorphic functions
with order strictly less then one. This allows us to formulate a useful
Wiman-Valiron type estimate for logarithmic difference of meromorphic functions
of small order. We then apply this estimate to prove a classical analogue of
Valiron about entire solutions to linear differential equations with
polynomials coefficients for linear difference equations.
