FRACTIONAL ORDER HARDY-TYPE INEQUALITY IN FRACTIONAL h-DISCRETE CALCULUS

We investigate the power weights fractional order Hardy-type inequality in the following form: (integral(infinity)(0)integral(infinity)(0)vertical bar f(x) - f(y)vertical bar(p)/vertical bar x- y vertical bar(1+p alpha)dxdy)(p) <= C(integral(infinity)(0)vertical bar f'(x)vertical bar(p) x((1-)(alpha)p)dx)(p) for 0 < alpha < 1 and 1 < p < infinity in fractional h-discrete calculus, where C =2(1/p)alpha(-1)/(p-p alpha)(1/p). For h-fractional function we prove a discrete analogue of above inequality in fractional h-discrete calculus, is proved and discussed. Moreover, we prove that the same constant is sharp also in this case.