q,h-Opial-Type Inequalities via Hahn Operators

In this paper, the well-known H¨older’s inequality is proved via Hahn differential and integral operators, which is a helping tool to establish some Opial-type inequalities via Hahn’s calculus. The weight functions involved in these Opial-type inequalities are positive and monotone. In search of applications, some new as well as some existing inequalities in the literature are obtained by applying suitable limits.


Introduction
In 1960, rst time Opial's inequality was founded by Opial [1]. He established the following important integral inequality.
where v is an absolutely continuous function on [ where a is a constant. Furthermore, the proof of Opial's inequality is simpli ed by Olech [2], Beescak [3], Levison [4], Pederson [5], Mallows [6]. Levison [7] proved that if v is an absolutely continuous function on (0, d) with v(0) 0, in (1), then, For the extension of (1), Beescak [3] demonstrated that if u is an absolutely continuous function on [c, x 1 ] with u(c) 0, then, where p(t) is a continuous and positive function with Keng [8] generalized the inequality (3) in the following form: If u is an absolutely continuous function with u(c) 0, then, where j is a positive integer. Extensions of Beescak's inequalities (4) and (5) are proved by Yang [9]. He assumed that if u is absolutely continuous on [c, x 1 ] with u(c) 0, then, where r is continuous and positive function with c (1/r(t))dt < ∞ and m is positive, bounded and nonincreasing function on [c, where m is positive, bounded and nondecreasing func- where K is defined by K � Also, another extension of Opial's inequality is given by Yang [9] is generalization of (6).
where j, k ≥ 1. e following inequality is presented by Lee [10], which is a generalization of the inequality (7). It demonstrates that if u is an absolutely continuous function on [c, x 1 ] with given condition u(c) � 0, then, where j ≥ 0, k ≥ 1 and r is positive with ) < ∞ and m is nonincreasing and positive on [c, x 1 ]. He also proved a generalization of inequality (8) and suppose that if u is an absolutely continuous function on [x 1 , d] and u(d) � 0, then, where k ≥ 1 and j ≥ 0 and r(·) is positive function with (dt/r k (t)) < ∞ and m(·) is nondecreasing and positive on [x 1 , d]. Lee [10] has combined (11) and (12) with u(c) � u(d) � 0, to find extension of (9) in the following form: where C � ( Further discussion on Opial-type inequalities may include the work: some Opial inequalities in q-calculus [11] by Mirković, q-Opial-type inequality by Alp,et al.,in [12], refinements of Opial-type inequalities in two variables [13], Opial-type inequalities for conformable fractional integrals [14,15], dynamic Opial inequalities on time scales [16][17][18][19], Opialtype inequalities in (p, q)-calculus by Li et al.,in [20], interval valued Opial-type inequalities by Zhao et al., in [21]. e history of quantum calculus is 300 years old. It is considered the most difficult subject to engage in mathematics by Bernoulli and Euler. e quantum has been derived from "quantus" a Latin word meaning "how much", commonly quantum deals with the measurement to its smallest unit. Quantum calculus treats the sets of nondifferentiable functions without using limits.
e h-calculus (h > 0), deals with calculus of finite differences (Boole [22]). For the study of h − calculus, readers are suggested to Milne omson [23]. Hahn's calculus unifies h-calculus and q-calculus, initiated by Hahn [24]. It is utilized to construct families of orthogonal polynomials and to deal with some approximation problems [25].
In 2015, Saker et al. ( [18], eorems 3.3 and 3.4) initiated the study of dynamic versions of (11) and (12) on time scales (a time scale is a closed subset of real line). In 2019, Fatma et al., [16] have also studied (11)-(13) with the help of time scales calculus. In this paper, we present q, h− analogues of Opial-type inequalities proved in [10,16,18] with the help of Hahn's calculus. e paper is arranged as follows: In Section 2, some basic concepts of Hahn's calculus and useful lemmas are presented. In Section 3, Hölder inequality and some Opial-type inequalities for monotonic functions via Hahn integrals (also called Jackson Nörlund integrals) are established. Section 4 consists of the conclusion of the paper.  [24].

Some Essentials of Hahn's Calculus
where When h � 0, we get q-derivative of u(·) at t 1 .
e arithmetical properties of Hahn-differential are simply concluded in the following theorem, which is given in [26].
en, there must exists c between t 1 and qt 1 + h, such that, The right inverse of Hahn-differential operator ( [28], Chapter 6) is as follows: Definition 2. Let I be any closed interval of real numbers, which contains h°. Suppose that g: where [k] q � (1 − q k /1 − q) and the series on the right hand side is convergent at x � c, x � d.

Lemma 2.
Assume u, v: I ⟶ R are Hahn − integrable and y ∈ I.
In particular, (31) leads to the following inequality. If The following q, h integration by parts formula can be found in ( [28], Lemma 6.2.8).
Discrete Dynamics in Nature and Society 3

Hölder's Inequality via Hahn's Calculus.
e Hölder inequality plays a fundamental role in the field of mathematics. Different variants can be found in [29,30]. All through this section, j and i are conjugate to each other's, as (1/j) + (1/i) � 1. In order to extend Opial-type inequalities by using Hahn calculus, we first prove the Hölder's inequality involving Hahn calculus.
Proof. For nonnegative α, β real numbers, the basic inequality holds.
In (35), choose, Use (35) and integrating the resultant inequality from a to b to obtain Hence, we get (34). e proof is complete. Next, consider the notations y m � yq m + h[m] q , y � (y + mh) and � y � yq m , where y ∈ a, x 1 , b .

Remark 1. In form of sums (34) can be written as
When h � 0 in (39), it recaptures the Hölder's inequality in q-calculus [31]. 4 Discrete Dynamics in Nature and Society

Opial-Type Inequalities on
Proof. Let us consider the following integral Since T(a) � 0, one has that, Use of Hölder inequality (34) with indices k and (k/(k − 1)), provides, By taking power j on both sides, we have Since s(·) is nonincreasing and positive on [a, x 1 ], one has that, Discrete Dynamics in Nature and Society Multiply (48) by l k(k− 1) (z)s (k 2 /(j+k)) (z)|D q,h u(z)| k ≥ 0, to get Multiply (49) by (j + k), integrate it from a to x 1 and use monotonicity of ( z a l − k (t)d q,h t) to obtain Now from Chain rule (23), we have Since D q,h T(z) ≥ 0 and z ≤ c, one gets From (50) and (52), it is noted that T(a) � 0 and (by using (43)), Denote, By applying Hölder inequality (34) with indices (j + k/j) and (j + k/k) on L, one gets By combining (54) and (56), we get where (42), it is converted to the following Opial inequality in q-calculus. where Case 3. When q ⟶ 1 and h � 0 in (42). We get (11), which is ([10], eorem 1.1), ( [16], Corollary 3.2).

Conclusions
In this article, Hölder's inequality is proved with the help of Hahn calculus whose special cases contain Hölder in qcalculus [31], in h-discrete calculus [32], and in usual calculus [33]. Furthermore, the obtained Hölder's inequality is utilized to find some Opial-type inequalities. Limiting cases of newly proved Opial-type inequalities coincide with some Opial-type inequalities of [9,10,16]. Some extensions of Opial-type inequalities in q-calculus (q > 1) and in h-discrete calculus (h > 0) as well [28].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.