Algebraic independence of certain Mahler functions

A monotonicity-type result for functions (f : mathbb {N}_arightarrow mathbb {R}) satisfying the sequential fractional difference inequality

$$begin{aligned} Delta _{1+a-mu }^{nu }Delta _{a}^{mu }f(t)ge 0, end{aligned}$$

for (tin mathbb {N}_{2+a-mu -nu }), where (0

$$begin{aligned} mu <2(1-nu ). end{aligned}$$

We demonstrate that this result is sharp in the sense that the restriction (mu <2(1-nu )) cannot be improved.