Opial-type inequalities for convex functions and associated results in fractional calculus

This paper is dedicated to Opial-type inequalities for arbitrary kernels using convex functions. These inequalities are further applied to a power function. Applications of the presented results are studied in fractional calculus via fractional integral operators by associating special kernels.


Introduction and preliminary results
Opial obtained the following integral inequality in 1960 [25]. Here h 4 is a best possible constant.
The aim of this paper is to establish some new Opial-type inequalities for convex functions. Therefore, in the subsequent convex function, its properties, characterization, and Opial-type inequalities for convex functions have been summarized as motivation behind the recent work. Definition 1 Let I be an interval in R. Then f : I → R is said to be convex if, for all x, y ∈ I and all α ∈ [0, 1], f αx + (1α)y ≤ αf (x) + (1α)f (y) ( 1 ) holds.
A characterization of convex function is stated in the following lemma.

Lemma 1.2 ([36]) Let f be a differentiable function on (a, b). Then f is convex if and only if f is an increasing function.
The convexity of composition of two functions can be obtained under the conditions stated in the following lemma. In [24] generalized Opial-type inequalities for convex functions have been proved by Mitrinović and Pečarić. Let where v is a continuous function and k is an arbitrary nonnegative kernel such that k(x, t) = 0 for t > x, and v(x) > 0 implies u(x) > 0 for every x ∈ [a, b].
Let U 2 (v, k) denote the class of all the functions u : [a, b] → R having representation where v is a continuous function and k is an arbitrary nonnegative kernel such that k( If the function φ(x 1/q ) is concave, then the reverse inequality holds.
A similar result was obtained for class U 2 (v, k).
In [1] Andrić et al. further extended these results stating the following.
If the function φ(x 1/q ) is concave, then the reverse inequality holds.
A similar result for class U 2 (v, k) is stated in the following theorem.
If the function φ(x 1/q ) is concave, then the reverse inequality holds.
In [11,12] Farid and Pečarić studied these inequalities in a fractional point of view. They considered several fractional integral operators via particular kernels to obtain Riemann-Liouville, Caputo, and Canavati fractional Opial-type inequalities. Extensions of these Opial-type fractional inequalities have been proved in [1] by Andrić et al., Basci and Dumitru in [2] considered some new aspects of these inequalities.
Then the left-sided and right-sided Riemann-Liouville fractional integrals of order α > 0 with a ≥ 0 are defined as follows: where Γ (·) is the gamma function. and In [1] composition identities for the Caputo fractional derivatives are given, they are stated in the following lemmas. .
Then the left-sided Canavati fractional derivative is defined bỹ Composition identity for the left-sided Canavati fractional derivative is given in the following lemma. .
Then the left-sided Riemann-Liouville fractional derivative of order α is defined by The following lemma summarizes conditions in the composition identity for the leftsided Riemann-Liouville fractional derivative.
is valid if one of the following conditions holds: In the following paragraph, the spaces of functions which have been used in definitions and results are summarized.
The space of all continuous functions whose nth time continuous derivative exists on The paper is organized as follows.
In Sect. 2, new Opial-type inequalities for convex functions are established by applying an arbitrary kernel. Moreover, these inequalities are studied for a power function. Furthermore, in Sect. 3 results of Sect. 2 are analyzed for particular kernels, and fractional integral inequalities of Opial-type are produced by using the definitions and composition identities of Riemann-Liouville fractional integral, Caputo fractional derivative, Canavati fractional derivative. The results for fractional inequalities are obtained by using different forms of the weighted functions and kernels.

Opial-type inequalities for convex functions
In this section some new generalized Opial-type inequalities for convex functions are obtained.
Theorem 2.1 Let φ, g : [0, ∞) → R be differentiable convex and increasing functions with φ(g(0)) = 0. Also, let u ∈ U 1 (g • v, k) and |k(x, t)| ≤ K , where K is a constant. Then the following inequalities hold: By applying the function g on both sides and using its monotonicity, we get If we set p(x) := x a |g • v(t)| dt, then p (x) = |g • v(x)|, therefore the last inequality gives g(|u(x)|) ≤ g(K(p(x))). As φ is a differentiable convex function, so by Lemma 1.2, φ is increasing, therefore we get As g is a differentiable convex function, by Lemma 1.2, g is increasing. Using (7), we get From inequalities (8) and (9), the following inequality can be obtained: The right-hand side is computed as follows: Therefore (11) takes the form as follows: As φ and g are convex functions and φ is an increasing function, so by Lemma 1.3, φ • g is convex and the following Jensen's inequality holds: Inequalities (12) and (13) provide the required result.
For power function g(x) = x q , q ≥ 1, the following Opial-type inequality holds.

Theorem 2.2 Let φ : [0, ∞) → R be a differentiable convex and increasing function with
Proof Let g(x) = x q . Then g is convex and increasing for q ≥ 1. Therefore applying Theorem 2.1 for the function g, inequalities in (6) provide inequalities in (14).

Fractional Opial-type inequalities
Here we utilize Riemann-Liouville fractional integrals, Caputo fractional derivatives, composition identities for Caputo fractional derivatives, and composition identities for Canavati fractional derivatives to obtain corresponding fractional Opial-type integral inequalities.
Proof Let us define for x ∈ [a, b] the kernel k(x, t) as follows: Also, if u is defined by Applying Theorem 2.1 for this particular kernel, we get inequalities in (15).
A similar result can be obtained for the class of functions denoted by U 2 (g • v, k) for the right-sided Riemann-Liouville fractional integral which is stated as follows.
Proof The proof is similar to the proof of Theorem 3.1.
Proof Let g(x) = x q . Then g is convex and increasing for q ≥ 1. Therefore, using this power function g in inequalities (15), we get inequalities in (18).
A similar result for right-sided Riemann-Liouville fractional integrals holds.
Proof The proof is similar to the proof of Theorem 3.3.
For composition identity of left-sided Caputo fractional derivative given in Lemma 1.8, the following result holds.
Then, for α < β -1, the following fractional inequalities hold: Proof Let us define the kernel k(x, t) for x ∈ [a, b] as follows: Also let us define the function u by Hence, by applying Theorem 2.1, inequalities in (20) can be obtained.
A similar result can be obtained from composition identity for right-sided Caputo fractional derivative given in Lemma 1.9, for the class of functions denoted by U 2 (g • v, k).
Then, for α < β -1, the following fractional inequalities hold: Proof The proof is similar to the proof of Theorem 3.5.
Proof Let g(x) = x q . Then g is convex and increasing for q ≥ 1. Therefore, using this power function g in inequalities (20), we get inequalities in (23).
A similar result can be obtained for the class of functions denoted by U 2 (g • v, k). and v ∈ AC m [a, b] such that f (i) (b) = 0 for i = n, n + 1, . . . , m -1.
Then, for α < β -1 and q ≥ 1, the following fractional inequalities hold: Proof The proof is similar to the proof of Theorem 3.7.
Next result includes the Canavati fractional derivatives using composition identity given in Lemma 1.10.
Next results are for Riemann-Liouville fractional derivatives using the composition identity given in Lemma 1.11.

Concluding remarks
The aim of this paper is to utilize the classes of functions U i (g • v, k), i = 1, 2, in the establishment of new Opial-type inequalities in general prospect for convex functions. A power function is considered to obtain particular Opial-type inequalities. In application point of view, all these results have been discussed for fractional calculus operators of Riemann-Liouville, Caputo, and Canavati. Fractional inequalities have wide applications in the theory of fractional differential equations and boundary value problems and other modern areas of science like rheology, viscoelasticity, acoustics, optics, chemical and statistical physics, robotics, control theory, electrical and mechanical engineering, bioengineering, etc.