Abstract

In this paper, the well-known Hölder’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.

1. Introduction

In 1960, first time Opial’s inequality was founded by Opial [1]. He established the following important integral inequality.where is an absolutely continuous function on and , the constant is the most suitable. Equality (1) holds, if and only ifwhere is a constant.

Furthermore, the proof of Opial’s inequality is simplified by Olech [2], Beescak [3], Levison [4], Pederson [5], Mallows [6]. Levison [7] proved that if is an absolutely continuous function on with , in (1), then,

For the extension of (1), Beescak [3] demonstrated that if is an absolutely continuous function on with , then,where is a continuous and positive function with , and if , then,

Keng [8] generalized the inequality (3) in the following form: If is an absolutely continuous function with , then,where is a positive integer. Extensions of Beescak’s inequalities (4) and (5) are proved by Yang [9]. He assumed that if is absolutely continuous on with , then,where is continuous and positive function with and is positive, bounded and nonincreasing function on . If , then,where is positive, bounded and nondecreasing function on . Further he assumed that if is an absolutely continuous function on with , then,where is defined by and is a positive bounded function on interval . Further is nonincreasing on and is nondecreasing on . Also, another extension of Opial’s inequality is given by Yang [9] is generalization of (6). He proved that, if , then,where . The following inequality is presented by Lee [10], which is a generalization of the inequality (7). It demonstrates that if is an absolutely continuous function on with given condition , then,where and is positive with and is nonincreasing and positive on . He also proved a generalization of inequality (8) and suppose that if is an absolutely continuous function on and , then,where and and is positive function with and is nondecreasing and positive on . Lee [10] has combined (11) and (12) with , to find extension of (9) in the following form:where . 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 [1619], Opial-type inequalities in (p, q)-calculus by Li et al., in [20], interval valued Opial-type inequalities by Zhao et al., in [21].

The history of quantum calculus is 300 years old. It is considered the most difficult subject to engage in mathematics by Bernoulli and Euler. The 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.

The -calculus , deals with calculus of finite differences (Boole [22]). For the study of , readers are suggested to Milne Thomson [23]. Hahn’s calculus unifies -calculus and -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], Theorems 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 analogues of Opial-type inequalities proved in [10,16,18] with the help of Hahn’s calculus.

The 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.

2. Some Essentials of Hahn’s Calculus

The generalization of both quantum calculus and -calculus is another kind of quantum calculus called Hahn’s quantum calculus. Hahn’s difference operator is defined by Wolfgang Hahn in 1945 [24].

Definition 1. Letand. Defineand letbe any interval incontaining. Suppose. Theofis described bywhere , if , then, . is the of at and is over , if exists for all . Generally, when , we get h-derivative of at .When , we get q-derivative of at .If and , on condition that exists. It is noted that, by making both restrictions simultaneously we get usual derivative of at .The arithmetical properties of Hahn-differential are simply concluded in the following theorem, which is given in [26].

Theorem 1. Assumeboth are Hahn-differentiable (q,h-differentiable) at, then,where is a constant.Provided by .

Theorem 2. (Chain Rule involving Hahn-differential operator) (see [27]).
Consider is and continuous. Let is continuously differentiable. Then, there must exists between and , such that,The right inverse of Hahn-differential operator ([28], Chapter 6) is as follows:

Definition 2. Letbe any closed interval of real numbers, which contains. Suppose thatand, such that,. Theoffromtois defined by,where and the series on the right hand side is convergent at , .
The following properties of Hahn’s integral are given in ([28], Lemma 6.2.2).

Lemma 1. Assumearefunctions on interval,and, whereis a constant. Then,Next result can be found in ([28], Lemma 6.2.8).

Lemma 2. Assumeareand. Denote. If, then,In particular, (31) leads to the following inequality. If is , , then , one has that,The following integration by parts formula can be found in ([28], Lemma 6.2.8).

Lemma 3. Ifare continuous at. Then,

3. Opial Inequalities for Monotone Functions

3.1. Hölder’s Inequality via Hahn’s Calculus

The Hölder inequality plays a fundamental role in the field of mathematics. Different variants can be found in [29,30]. All through this section, and are conjugate to each other’s, as . In order to extend Opial-type inequalities by using Hahn calculus, we first prove the Hölder’s inequality involving Hahn calculus.

Theorem 3. Ifandare continuous functions, then,where and .

Proof. For nonnegative real numbers, the basic inequality holds.Now consider,In (35), choose,Use (35) and integrating the resultant inequality from to to obtainHence, we get (34). The proof is complete.
Next, consider the notations , and , where .

Remark 1. In form of sums (34) can be written asWhen in (39), it recaptures the Hölder’s inequality in q-calculus [31].By using in (39), we get Hölder’s inequality in -discrete calculus, which is ([32], Theorem 3.1).When , in (34), we obtain Hölder’s inequality in classical calculus, which can be found in [33].

3.2. Opial-Type Inequalities on

Theorem 4. Letbe any interval,and. Assumeare positive continuous functions on.is non-increasing function onand. Furthermore, supposeis a continuous function onwith. Then,,, and, we have,

Proof. Let us consider the following integralSince , one has that,Equation (32) implies,Use of Hölder inequality (34) with indices and , provides,By taking power on both sides, we haveSince is nonincreasing and positive on , one has that,Multiply (48) by , to getMultiply (49) by , integrate it from to and use monotonicity of to obtainNow from Chain rule (23), we haveSince and , one getsFrom (50) and (52), it is noted that and(by using (43)),Denote,By applying Hölder inequality (34) with indices and on , one getsBy combining (54) and (56), we getHence, (42) is obtained.

Corollary 1. Case 1. When, then (42) reduces to the following Opial inequality incalculus.where .

Case 2. Whenin (42), it is converted to the following Opial inequality in q-calculus.where

Case 3. Whenandin (42). We get (11), which is ([10], Theorem 1.1), ([16], Corollary 3.2).

Case 4. When,and. We obtain (7) given by Yang in ([9], Theorem 3), ([16], Remark 3.3).

Remark 2. Ifin (26), then, it recaptures Opial-type discrete inequality given in ([16], Corollary 1).

3.3. Opial-Type Inequalities on

Theorem 5. Letbe any interval,and. Assumeare positive continuous functions on.is nondecreasing onand. Furthermorer, assumeis a continuous function onwith, then,,,, we have,

Proof. Let us consider the following integral:By using , we getUse of (32) givesUse of (34) with indices and impliesBy taking power on both sides, one getsSince is nondecreasing and positive on , therefore,((63)implies),Multiply both sides of (69) by , to find,(70) can be written in the following subsequent form,Multiply (71) by , integrate from to and use monotonicity of to find,Now, from Chain rule (23), we have