On Opial-Type Inequalities for Superquadratic Functions and Applications in Fractional Calculus

Many authors have been working continuously on Opial’s inequality and succeeded to establish very interesting results. For its numerous generalizations and extensions, we refer the readers to [2–14] and references therein. Opial-type inequalities are useful in the study of difference and differential equations, for example, in the uniqueness of initial value problems, in the existence and uniqueness of boundary value problems, and setting upper bounds of their solutions. For a historical survey of the inequalities given after the publication of inequality (1), one can study the book by Agarwal and Pang; see [2]. Mitrinović and Pečarić, in 1988, proved Opial-type inequalities for convex functions with respect to the power function by defining two classes of functions involving general kernels; see [15, 16]. Motivated by these inequalities, recently, we have studied such inequalities for convex functions by considering a generalized class of functions with arbitrary kernel [17]. Also, for special kernels, fractional Opial-type inequalities are proved for different fractional integral and derivative operators [18, 19]. Our aim in this paper is to study Opial-type inequalities for superquadratic functions which are also connected with inequalities that hold for convex functions. )e established results estimate the Opial-type inequalities with the help of some suitably defined functions. )e definitions and results needful for the presentation of results of this paper are given as follows.


Introduction and Preliminary Results
Opial's inequality is very important and useful in the study of differential and difference equations. It was introduced by Opial in 1960. It is stated in the following theorem.
Many authors have been working continuously on Opial's inequality and succeeded to establish very interesting results. For its numerous generalizations and extensions, we refer the readers to [2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein. Opial-type inequalities are useful in the study of difference and differential equations, for example, in the uniqueness of initial value problems, in the existence and uniqueness of boundary value problems, and setting upper bounds of their solutions. For a historical survey of the inequalities given after the publication of inequality (1), one can study the book by Agarwal and Pang; see [2]. Mitrinović and Pečarić, in 1988, proved Opial-type inequalities for convex functions with respect to the power function by defining two classes of functions involving general kernels; see [15,16]. Motivated by these inequalities, recently, we have studied such inequalities for convex functions by considering a generalized class of functions with arbitrary kernel [17]. Also, for special kernels, fractional Opial-type inequalities are proved for different fractional integral and derivative operators [18,19].
Our aim in this paper is to study Opial-type inequalities for superquadratic functions which are also connected with inequalities that hold for convex functions. e established results estimate the Opial-type inequalities with the help of some suitably defined functions. e definitions and results needful for the presentation of results of this paper are given as follows.
Definition 2 (see [20]). A function ψ: [0, ∞) ⟶ R is called a superquadratic function provided for all x ≥ 0, there exists a real constant C x such that (3) e following lemma explores convexity from the superquadratic function.
Lemma 1 (see [20]). Let ψ be a superquadratic function with C x as in Definition 2. en, we have where u is a continuous function and k is an arbitrary nonnegative kernel such that k(x, t) � 0 for t < x and where u is a continuous function and k is an arbitrary nonnegative kernel such that k(x, t) � 0 for t < x and Theorem 2 (see [15,16]). Let ψ: [0, ∞) ⟶ R be a differentiable function such that, for p 2 > 1, the function e reverse of the above inequality holds when ψ(x (1/p 2 ) ) is concave.
A similar result for the class U 2 (u, k) is given as follows.
Recently, the following Opial-type inequalities for superquadratic functions are established.
Definition 3 (see [24] en, the left-sided and right-sided Riemann-Liouville fractional integrals of order c > 0 are defined by where Γ(.) is the gamma function.
en, the left-sided and right-sided Caputo fractional derivatives of order c are defined by In [25], Andrić et al. presented the composition identities for the left-sided and right-sided Caputo fractional derivatives; these are stated in the following two lemmas.
In the upcoming section, we prove the mean value theorems for the estimation of the nonnegative differences of the inequalities given in eorems 4-7. In Section 3, we prove the fractional versions of the results of Section 2 for Riemann-Liouville integrals and Caputo fractional derivatives.

Main Results
First, we define the linear functional Φ ψ j (w, u; K), j � 1, 2, from nonnegative differences of Opial-type inequalities for superquadratic functions given in eorems 4 and 5 as follows: Remark 1. Under the assumptions of eorems 4 and 5, we Theorem 8. With the same assumptions as given in eorem 4, furthermore, let I⊆(0, ∞) be a compact interval and ψ ∈ C 2 (I).
en, there exists η 1 ∈ I such that the following result holds: Proof. By replacing ψ with ψ 1 (defined in Lemma 3) in eorem 4, one can have the following inequality: From inequality (21), after simplification, one can obtain Similarly, if we take ψ 2 from Lemma 3 instead of ψ in eorem 4, then the following inequality holds: e above two inequalities lead to the following inequality: erefore, there exists η 1 ∈ I such that the following equation is obtained: which gives equation (20). □ Theorem 9. With the same assumptions on ψ 1 and ψ 2 as given for ψ in eorem 4, furthermore, if I⊆(0, ∞) is a compact interval and ψ 1 , ψ 2 ∈ C 2 (I) where Φ x 3 1 (w, u; K) ≠ 0, then there exists η 1 ∈ I such that the following result holds: where denominators should not be zero.
where λ 1 and λ 2 are given by en, f ∈ C 2 (I); by applying eorem 8 for f, it follows that there exists η 1 such that we have From this, one can get the required equation.
Mathematical Problems in Engineering Theorem 10. With the same assumptions as given in eorem 5, furthermore, if I⊆(0, ∞) is a compact interval and ψ ∈ C 2 (I), then there exists η 2 ∈ I such that the following result holds: Proof. By replacing ψ with ψ 1 from Lemma 2 in eorem 5, we get the following inequality: Similarly, adopting the method for functional Φ ψ 2 (w, u; K) as we did for Φ where denominators should not be zero.
Proof. Let us define the function g by g � λ 1 ψ 1 − λ 2 ψ 2 , where λ 1 and λ 2 are given by (32) en, g ∈ C 2 (I); by applying eorem 10 for g, it follows that there exists η 2 such that we have From this, one can get the required equation.

Theorem 12.
With the same assumptions of eorem 6 on ψ, furthermore, if I⊆(0, ∞) is a compact interval and ψ ∈ C 2 (I), then there exists η 3 ∈ I such that we have Proof. By replacing ψ with ψ 1 from Lemma 3 in eorem 6, we get the following inequality: From inequality (36), after simplification, one can obtain Similarly, if we apply ψ 2 from Lemma 3 instead of ψ in eorem 6, then the following inequality holds: e above two inequalities lead to the following inequality: erefore, there exists η 3 ∈ I such that the following equation is obtained: which gives the required equation.

Proof. Let us define the function h by
where ω 1 and ω 2 are given by (42) en, h ∈ C 2 (I); by applying eorem 12 for h, it follows that there exists η 3 such that we have From this, one can get the required equation.
Proof. By replacing ψ with ψ 1 from Lemma 3 in eorem 4, we get the following inequality: Rest of the proof can be followed from the proof of eorem 12.
where denominators should not be zero.
Proof. e proof can be followed from the proof of eorem 13.

Fractional Versions of Mean Value Theorems
In this section, we give applications of results proved in Section 2 by choosing particular kernels. We get fractional versions of mean value theorems by applying definitions of fractional integral/derivative operators. First, we give results for Riemann-Liouville fractional integrals.

Theorem 16.
With same assumptions of eorem 4 on ψ, furthermore, let I⊆(0, ∞) be a compact interval and ψ ∈ C 2 (I). Also, let u ∈ L 1 [a, b] be a Riemann-Liouville fractional integral of order c, where c ≥ 1; then, there exists η 1 ∈ I such that the following result holds: Proof. Let us define the kernel k(x, t) as follows: Furthermore, we take w as follows: where denominators should not be zero.
Proof. It is easy to prove by applying eorem 9 for the kernel defined in (48) and using the function w given by (49). □ Theorem 18. With the same assumptions of eorem 5 on ψ, furthermore, let I⊆(0, ∞) be a compact interval and ψ ∈ C 2 (I). Also, let u ∈ L 1 [a, b] be a Riemann-Liouville fractional integral of order c where c ≥ 1; then, there exists η 2 ∈ I such that the following result holds: Proof.
e proof is similar to the proof of eorem 16. □ Theorem 19. With the same assumptions of eorem 5 on ψ 1 and ψ 2 , furthermore, let I⊆(0, ∞) be a compact interval and ψ 1 , ψ 2 ∈ C 2 (I). Also, let u ∈ L 1 [a, b] be a Riemann-Liouville fractional integral of order c where c ≥ 1; then, there exists η 2 such that we have where denominators should not be zero.
Proof. It is easy to prove by applying eorem 11 for the kernel defined in (48) and using the function w given by (49).
□ Theorem 20. With the same assumptions of eorem 6 on ψ, furthermore, let I⊆(0, ∞) be a compact interval and ψ ∈ C 2 (I). Let u ∈ L 1 [a, b] be a Riemann-Liouville fractional integral of order c where c ≥ (1/p 2 ); then, there exists η 3 ∈ I such that the following result holds: Proof. Let us consider the kernel k(x, t) as defined in (48) and w given in (49).
By applying eorem 12, we get the required result. □ Theorem 21. With the same assumptions of eorem 6 on ψ 1 and ψ 2 , furthermore, let I⊆(0, ∞) be a compact interval and ψ 1 , ψ 2 ∈ C 2 (I). Furthermore, let u ∈ L 1 [a, b] be a Riemann-Liouville fractional integral of order c. If c > (1/p 2 ) and Π x 3 2 (I c a+ u, u) ≠ 0, then there exists η 3 ∈ I such that the following equality holds: where denominators should not be zero.
Proof. It is easy to prove by applying eorem 13 for the kernel defined in (48) and using the function w given by (49). Proof.
where denominators should not be zero.
Proof. It is easy to prove by applying eorem 15 for the kernel defined in (48) and using the function w given by (49).
Next, we give the results for Caputo fractional derivatives using their composition identities.  1 [a, b]. en, for c ≤ μ − 1, the following result holds: Proof. Let us consider the kernel k(x, t) as follows: Furthermore, we take w as follows:  c)) and applying eorem 8 for the kernel given in (59), one can get the required result.
where denominators should not be zero.
Proof. It is easy to prove by applying eorem 9 for the kernel defined in (58) and using function w given by (59).
where denominators should not be zero.
Proof. It is easy to prove by applying eorem 11 for the kernel defined in (58) and using function w given by (59).
Proof. Let us consider the kernel k(x, t) as defined in (58) and w given in (59). If we set Q(x) � (