On analytic solutions of a differential equation of Shah type in the unit disk. Boundedness of l-index

For an analytic in the unit disk ${\mathbb D}$ solution of the form $f(z)=F(1/(1-z))$,where $F$ is an entire transcendental function, of the differential equation $(1-z)^nw''+a(1-z)^mw'+bw=0$(with $n>m\ge 0,\, a\in {\mathbb R},\,b\in {\mathbb R}$) the boundedness of $l$-index of $f$ in ${\mathbb D}$ and $l$-index of $F$ in ${\mathbb C}$ are investigated.