© Hindawi Publishing Corp. OPIAL TYPE L p-INEQUALITIES FOR FRACTIONAL DERIVATIVES

This paper presents a class of Lp-type Opial inequalities for generalized fractional derivatives for integrable functions based on the results obtained earlier by the first author for continuous functions (1998). The novelty of our approach is the use of the index law for fractional derivatives in lieu of Taylor's formula, which enables us to relax restrictions on the orders of fractional derivatives.

1. Introduction and preliminaries. The Opial inequality, which appeared in [8], is of great interest in differential equations and other areas of mathematics, and has attracted a great deal of attention in the recent literature. For classical derivatives it has been generalized in several directions (see, e.g., [1,3,9]), and was a subject of a monograph by Agarwal and Pang [2]. Love [7] gave a generalization for fractional integrals. The present paper takes its inspiration from an earlier paper [4] by Anastassiou. In the present work, we consider Lebesgue integrable functions, whereas [4] dealt with continuous functions using a different definition of fractional derivative.
Our brief survey of basic facts about fractional derivatives is based on the monograph [10] by Samko  For any α ∈ R, we denote by [α] the integral part of α (the integer k satisfying k ≤ α < k+1). If p ∈ R, p > 0, and by L p (0,x), we denote the space of all Lebesgue measurable functions f for which |f | p is Lebesgue integrable on the interval (0,x), and by L ∞ (0,x) the set of all functions measurable and essentially bounded on (0,x). For any For any a ∈ R we write a + = max(a, 0) and a − = (−a) + .
For the sake of completeness, we give a proof of the following known result which provides a basis for the existence of fractional integrals and is needed in another context in the paper.
Then k is measurable on Ω, and Since the repeated integral exists and is finite, the function (s, t) k(s, t)f (t) is integrable over Ω by Tonelli's theorem, and the conclusion follows from Fubini's theorem.
Let α > 0. For any f ∈ L(0,x) the Riemann-Liouville fractional integral of f of order α is defined by (1.5) By Lemma 1.1, the integral on the right-hand side of (1.5) exists for almost all s ∈ [0,x] and where m = [α] + 1, provided that the derivative exists. In addition, we stipulate (1.8) The two fractional derivatives are then defined by an obvious modification of (1.6). All our results stated for the specialized fractional derivative (1.6) have an interpretation for the fractional derivatives with a general anchor point.  We will need the following result on the law of indices for fractional integration and differentiation using the unified notation (1.7).
is valid in the following cases: The following theorem is a powerful analogue of Taylor's formula with vanishing fractional derivatives of lower orders. In this paper, it is used as the main tool for deriving inequalities. Observe that we do not require α ≥ β + 1 but merely α > β. Then (1.14) Proof. Set µ = α−β > 0 and ν =−α < 0. According to Lemma 1.2, f ∈ I −ν (L(0,x)). Then, Lemma 1.3(ii) guarantees that the law of indices holds for this choice of µ, ν, namely this proves the result. Note that, the existence of the integral on the right-hand side of (1.14) is guaranteed by Lemma 1.1.

Main results.
We assume throughout that x, ν are positive real numbers, and that f ∈ L(0,x). The standard assumption on f is that f ∈ I ν (L(0,x)); this is equivalent to f having an integrable fractional derivative D ν f satisfying (1.10). In addition, we require that D ν f is essentially bounded to guarantee that D ν f ∈ L p (0,x) for p > 0. The following notations are used in this section. (The inequalities between ν and µ i are assumed throughout.) l: a positive integer x, ν, r i : positive real numbers, i = 1,...,l r = l i=1 r i µ i : real numbers satisfying 0 ≤ µ i < ν, i = 1,...,l For brevity, we write µ = (µ 1 ,...,µ l ) for a selection of the orders µ i of fractional derivatives, and r = (r 1 ,...,r l ) for a selection of the constants r i .
We derive a very general Opial type inequality involving fractional derivatives of an integrable function f , which is analogous to [9, Theorem 1.3] for ordinary derivatives and to [4, Theorem 2] for fractional derivatives.
Next, we consider the extreme case p = ∞ in analogy with [4, Proposition 1].
Proof. By Theorem 1.4, which implies (2.14) The result then follows when we raise (2.14) to the power r i , take the product from i = 1 to l, multiply by ω(τ), and integrate with respect to τ from 0 to x.
We can now retrace the proof of Theorem 2.1, relying on (2.17) and using the reverse Hölder's inequality in place of Hölder's inequality proper (as 0 < s k < 1 for k = 1, 2 and p < 0).
A possible choice of p in this theorem is p = (s 1 s 2 2 )/(s 1 s 2 − 1). This results in an inequality similar to the one obtained earlier by Anastassiou [4,Theorem 3].
We obtain yet another counterpart of Theorem 2.1 if we assume that s 1 , s 2 , and p lie in the interval (0, 1). In this case, the hypotheses on s 1 , s 2 , and p are of necessity more restrictive.
As in the proof of Theorem 2.3, we have We can now retrace the proof of Theorem 2.1, relying on (2.21) and using the reverse Hölder's inequality in place of Hölder's inequality proper (as 0 < s k < 1 for k = 1, 2 and 0 < p < 1). For the last application of Hölder's inequality, we need τ ρ ∈ L s 1 (0,x). This follows from taking into account the assumption r s 1 ≤ 1.
We present a version of Opial's inequality with l = 2 motivated by Pang and Agarwal's extension [9, Theorem 1.1] of an inequality due to Fink [5] for classical derivatives. This was further extended in [4,Theorem 4] to fractional derivatives. Our proof is similar to the one given in [9]. In view of the auxiliary inequalities used, in particular of (2.26), the theorem does not extend easily to l > 2.
In the following theorem, we address the case when the function |D ν f | is monotonic.
where γ := l i=1 α i and (2.33) Proof. By Theorem 1.4, (2.34) The integrand t (τ − t)  Integrating with respect to τ from 0 to x, we get the result. Condition l i=1 α i p > −1 was needed in order to apply Hölder's inequality to  Proof. As in the proof of Theorem 2.6, we have (2.38) Integrating over [0,x] with respect to τ we obtain (2.36).