b f#x$1 0 aIa b g#x$1 . Proof. By using the local fractional integrals for mappings f,g ( Ia x 0a, b1, we have. ' ' #' % a$ b. :a b. :a. 0f#x$ f#y$1 0g#x$ g#y$1 #dy$ a.
ON GENERALIZED SOME INTEGRAL INEQUALITIES FOR LOCAL FRACTIONAL INTEGRALS MEHMET ZEKI SARIKAYA, TUBA TUNC, AND HÜSEYIN BUDAK Abstract. In this study, we establish generaized Grüss type inequality and some generaized Cebysev type inequalities for local fractional integrals on fractal sets R (0 < 1) of real line numbers.
1. Introduction In 1935, G. Grüss [8] proved the following inequality: (1.1) 1 b
a
Zb
1
f (x)g(x)dx
b
a
a
Zb
f (x)dx
1 b
a
a
Zb
g(x)dx
1 (M 4
m)(N
n);
a
provided that f and g are two integrable function on [a; b] satisfying the condition (1.2)
m
f (x)
M and n
g(x)
1 4
N for all x 2 [a; b]:
The constant is best possible. µ In 1882, P. L. Cebyš ev [3] gave the following inequality: 1 (b a)2 kf 0 k1 kg 0 k1 ; 12 where f; g : [a; b] ! R are absolutely continuous function, whose …rst derivatives f 0 and g 0 are bounded, 0 10 1 Zb Zb Zb 1 1 1 (1.4) T (f; g) = f (x)g(x)dx @ f (x)dxA @ g(x)dxA b a b a b a
(1.3)
jT (f; g)j
a
a
a
and k:k1 denotes the norm in L1 [a; b] de…ned as kpk1 = ess sup jp(t)j : t2[a;b]
The following result of Grüss type was proved by Dragomir and Fedotov [5]: Theorem 1. Let f; u : [a; b] ! R be such that u is L-Lipshitzian on [a; b]; i.e, (1.5)
ju(x)
u(y)j
L jx
yj for all x 2 [a; b];
f is Riemann integrable on [a; b] and there exist the real numbers m; M so that (1.6)
m
f (x)
M
for all x 2 [a; b]:
2000 Mathematics Subject Classi…cation. 26D07, 26D10, 26D15, 26A33. Key words and phrases. Grüss inequality, Cebysev inequality, local fractional integrals. 1
2
M EHM ET ZEKI SARIKAYA, TUBA TUNC, AND HÜSEYIN BUDAK
Then we have the inequality; Zb
f (x)du(x)
u(b) b
u(a) a
a
Zb
f (x)dx
1 L(M 2
m)(b
a):
a
From [10], if f : [a; b] ! R is di¤erentiable on (a; b) with the …rst derivative f 0 integrable on [a; b]; then Montgomery identity holds: (1.7)
f (x) =
1 b
a
Zb
f (t)dt +
a
Zb
P (x; t)f 0 (t)dt;
a
where P (x; t) is the Peano kernel de…ned by P (x; t) =
