L[sub(p)]-General Fractional Opial Inequalities

THEOREM 26.1 Let 7i > 1, v > 2 such that v - 7¿ > 1; i = 1 , . . . ,t and f £ C£{)([a, b})

with f(j)(xo) = 0, j = 0, l , . . . , n - I, n := [i/]. Here x ,x0 € [a, 6] : x > x0Let q\,q2 > 0 continuous functions on [a,b] and r¿ > 0 : 52i=i r i = rLet 5i,Si > 1: 7-+ 7-= 1 and s2,So > l: ¿ + 4 = 1, and p > s2. Furthermore suppose that

(26.1)

and

Call Then it holds

(26.2)

(26.3)

P R O O F Prom (25.15) we have (x0 < w < x)

alH = 1 , . . . , t. Hence it holds

(26.4)

establishing (26.3).