On Hardy type inequality in variable exponent Lebesgue space Lp(.)(0, l)

We study a variable exponent Hardy type inequality parallel to x(-1) H f parallel to(Lp(.)(0, l)) <= C parallel to f parallel to(Lp(.)(0, l)); f >= 0. in the norms of variable exponent Lebesgue spaces L-p(.)(0, l). In terms of regularity conditions for p : (0, l) -> (1, infinity), we derive necessary and sufficient conditions for this inequality to hold for all f is an element of L-p(.)(0, l).