On Opial's type inequalities for an integral operator with homogeneous kernel

In this article, we establish some new Opial's type integral inequalities for an integral operator with homogeneous kernel. The results in special cases yield some of the interrelated results and provide new estimates on inequalities of this type. MS (2000) Subject Classification: 26D15.


Statement of results
Here, we shall extend some of the previous results for the singular integral operator T which have an integral representation. For this, we say that the integral operator T (f)y belongs to the class U(f, K) if it can be represented in the following form Remark 2.2 Let the singular integral operator T(f)y change to a function x(t), so we say that the function x(t) belongs to the classŪ(y, K) if it can be represented in the following form where y(t) is a continuous function on [a, τ], and K(t, s) is a non-negative kernel defined on [a, τ] × [a, τ]; such that x(t) > 0 if y(t) > 0, t [a, τ]. Taking these in Theorem 2.1 and with suitable modifications, then (2.2) becomes the following established by Mitrinović and Pečarić [17].

Remark 2.3
In particular, if l >0 we have then y(t) is the derivative of order l of x(t) in the sense of Riemann-Liouville. Thus, if x(t) is differentiable, then for l = 1, it follows that y(t) = x'(t).
Further, taking for f(x, y) = f(x) and K(t, s) = K l (t, s) in (2.3), (2.3) reduces to a result of Godunova and Levin [18]. Now, let T(f)y U(f, K), where K(y, x) = 0 for x > y. Such the singular integral operator we shall say belong to the class U 1 (f, K). It is clear that in the case, we have (2:5) Remark 2.5 Let the singular integral operator T(f)y change to a function x(t), so we say that the function x(t) belongs to the classŪ(y, K). Taking these in Theorem 2.4 and with suitable modifications, then (2.5) becomes the following result.
This is just a new result established by Mitrinović and Pečarić [17].

Proofs of results
Proof of Theorem 2.1 From the hypotheses of Theorem 2.1, it turn out that By using the Jensen integral inequality, we have This completes the proof. Proof of Theorem 2.4 From the hypotheses of Theorem 2.4 and in view of Hölder's inequality, we obtain Hence z (y) = | f (y) | v .
Further, from the convexity of f(t 1/v ) it follows that the function t 1-v f'(t) is nondecreasing, thus we have This completes the proof.