Quantum Hermite-Hadamard and quantum Ostrowski type inequalities for $s$-convex functions in the second sense with applications

1.   Introduction

In convex functions theory, Hermite-Hadamard (H-H) inequality is very important and was discovered by C. Hermite and J. Hadamard independently (see, also ^[23], [41,p.137]),

where $f$ is a convex function. In the case of concave mappings, the above inequality is satisfied in reverse order. For more recent developments, one can consult ^{[17,18,42,48,49]}.

Hudzik and Maligranda defined $s$-convex functions in the second sense in ^[28], which may be expressed as: a mapping $f: \mathbb{R} ^{+}\rightarrow \mathbb{R}$, where $\mathbb{R} ^{+} = [0, \infty),$ is called $s$-convex in the second sense if

for all $x, y\in \mathbb{R} ^{+}$, $t\in \left[0, 1\right]$, $s\in \left(0, 1\right]$ and these functions are denoted by $f\in k_{s}^{2}.$ After that, Dragomir and Fitzpatrick ^[22] used this newly class of functions and proved the following H-H inequality:

For more recent integral inequalities related to the class of $s$-convex functions and its generalizations via different integral operators, one can consult ^{[11,19,20,24,25,34,36,40]}.

On the other hand, several studies have been carried out in the domain of $q$ -analysis, beginning with Euler, in order to achieve proficiency in mathematics that constructs quantum computing $q$-calculus, which is considered a relationship between physics and mathematics. It has a wide range of applications in mathematics, including combinatorics, simple hypergeometric functions, number theory, orthogonal polynomials, and other sciences, as well as mechanics, relativity theory, and quantum theory ^[27,32]. Euler is thought to be the inventor of this significant branch of mathematics. He used the $q$-parameter in Newton's work on infinite series. Later, Jackson presented the $q$-calculus, which knew no limits calculus, in a methodical manner ^[26,30]. In 1966, Al-Salam ^[10] introduced a $q$-analogue of the $q$-fractional integral and $q$ -Riemann-Liouville fractional. Since then, the related research has gradually increased. In particular, in 2013, Tariboon and Ntouyas introduced $_{a}D_{q}$-difference operator and $q_{a}$-integral in ^[46]. In 2020, Bermudo et al. introduced the notion of $^{b}D_{q}$ derivative and $q^{b}$-integral in ^[12].

Many integral inequalities have been studied using quantum integrals for various types of functions. For example, in ^{[3,6,8,9,12,13,14,31,37,43,44,45]}, the authors used $_{a}D_{q}, \; ^{b}D_{q}$-derivatives and $q_{a}, q^{b}$-integrals to prove H-H integral inequalities and their left-right estimates for convex and coordinated convex functions. In ^[38], Noor et al. presented a generalized version of quantum H-H integral inequalities. For generalized quasi-convex functions, Nwaeze et al. proved certain parameterized quantum integral inequalities in ^[39]. Khan et al. proved quantum H-H inequality using the green function in ^[35]. Budak et al. ^[15], Ali et al. ^[2,4], Chu et al. ^[21] and Vivas-Cortez et al. ^[47] developed new quantum Simpson's and quantum Newton's type inequalities for convex and coordinated convex functions. For quantum Ostrowski's inequalities for convex and co-ordinated convex functions, one can consult ^[5,7,16].

Inspired by these ongoing studies, we offer a variant of quantum H-H inequality (1.2) and Ostrowski type inequalities for $s$-convex functions in the second sense. Since the $s$-convexity is the generalization of convexity, therefore the inequalities proved in this paper using the $s$ -convexity are the generalization of already proved inequalities for convexity that is the main motivation of this paper.

The following is the structure of this paper: A brief overview of the concepts of $q$-calculus, as well as some related works, is given in Section 2. In Section 3, we show the relationship between the results presented here and comparable results in the literature by proving quantum H-H inequalities for $s$-convex functions in the second sense. Quantum Ostrowski type inequalities for $s$-convex functions in the second sense are presented in Section 4. Some applications to special means are given in Section 5. Section 6 concludes with some recommendations for future studies.

2.   Preliminaries of $q$-calculus and some inequalities

In this section, we recollect some formerly regarded concepts. Also, here and further we use $q\in \left(0, 1\right)$ and the following notation (see ^[32]):

In ^[30], Jackson gave the $q$-Jackson integral from $0$ to $b$ as follows:

provided the sum converges absolutely.

Definition 2.1. ^[46] The $q_{a}$-derivative of a mapping $f:\left[a, b \right] \rightarrow \mathbb{R}$ at $x\in \left[a, b\right]$ is defined as:

If $x = a$, we define $_{a}D_{q}f\left(a\right) = \lim_{x\rightarrow a} \begin{array}{l} _{a}D_{q}f\left(x\right) \end{array}$ if it exists and it is finite.

Definition 2.2. ^[12] The $q^{b}$-derivative of a mapping $f:\left[a, b \right] \rightarrow \mathbb{R}$ at $x\in \left[a, b\right]$ is defined as:

If $x = b$, we define $^{b}D_{q}f\left(b\right) = \lim_{x\rightarrow b} \begin{array}{l} ^{b}D_{q}f\left(x\right) \end{array}$ if it exists and it is finite.

Definition 2.3. ^[46] The $q_{a}$-integral of a mapping $f:\left[a, b\right] \rightarrow \mathbb{R}$ is defined as:

where $x\in \left[a, b\right].$

Definition 2.4. ^[12] The $q^{b}$-integral of a mapping $f:\left[a, b\right] \rightarrow \mathbb{R}$ is defined as:

where $x\in \left[a, b\right].$

In ^[12], Bermudo et al. established the following quantum H-H type inequality:

Theorem 2.1. For the convex mapping $f:\left[a, b\right] \rightarrow \mathbb{R}$, the following inequality holds

In ^[16], Budak et al. proved the following Ostrowski inequality by using the concepts of quantum derivatives and integrals:

Theorem 2.2. Let $f:\left[a, b\right] \subset \mathbb{R} \rightarrow \mathbb{R}$ be a function and $^{b}D_{q}f$ and $_{a}D_{q}f$ be two continuous and integrable functions on $\left[a, b\right].$ If $|^{b}D_{q}f\left(t\right) |, |_{a}D_{q}f\left(t\right) |\leq M$ for all $t\in \left[a, b \right],$ then we have the following quantum Ostrowski inequality

3.   Hermite-Hadamard inequalities

In this section, we prove H-H inequalities for $s$-convex functions in the second sense using the quantum integrals.

Theorem 3.1. Assume that the mapping $f: \mathbb{R} ^{+}\rightarrow \mathbb{R}$ is $s$-convex in the second sense and $a, b\in \mathbb{R} ^{+}$ with $a < b.$ If $f\in L_{1}([a, b])$, then the following inequality holds for $s\in (0, 1]$:

Proof. As $f$ is $s$-convex in the second sense on $\mathbb{R} ^{+}$ we have

for all $x, y\in \mathbb{R}^{+}$ and $t\in \lbrack 0, 1].$

Obverse that

We get the following, by putting $x = tb+\left(1-t\right) a$ and $y = ta+\left(1-t\right) b$ in (3.2)

From Definitions 2.3 and 2.4, we have

and the first inequality in (3.1) is proved.

To proved the second inequality, we use the $s$-convexity and we have

and

By adding (3.3) and (3.4), from Definition 2.3 and 2.4, we have

and the proof is completed.

Remark 3.1. If we set $s = 1$ in Theorem 3.1, then we recapture the inequality (2.3).

Remark 3.2. In Theorem 3.1, if we take the limit as $q\rightarrow 1^{-}$, then inequality (3.1) becomes the inequality (1.2).

4.   Ostrowski's inequalities

In this section, we prove Ostrowski's type inequalities for $s$-convex functions in the second sense. We use the following lemma to prove the new results.

Lemma 4.1. ^[16] Let $f:\left[a, b\right] \subset \mathbb{R} \rightarrow \mathbb{R}$ be a function. If $^{b}D_{q}f$ and $_{a}D_{q}f$ are two continuous and integrable functions on $\left[a, b\right]$, then for all $x\in \left[a, b \right]$ we have

Theorem 4.1. Assume that the mapping $f:I\subset \mathbb{R} ^{+}\rightarrow \mathbb{R}$ is differentiable and $a, b\in I$ with $a < b.$ If $|_{a}D_{q}f|$ and $|^{b}D_{q}f|$ are $s$-convex mappings in the second sense, then the following inequality holds:

where

Proof. From Lemma 4.1 and properties of the modulus, we have

Since the mappings $|_{a}D_{q}f|$ and $|^{b}D_{q}f|$ are $s$-convex in the second sense, therefore

and

We obtain the resultant inequality (4.2) by putting (4.4) and (4.5) in (4.3).

Remark 4.1. If we set $s = 1$ in Theorem 4.1, then we obtain the following inequality

which is given by Budak et al. in ^[16].

Corollary 4.1. If we assume $|_{a}D_{q}f\left(x\right) |$, $|^{b}D_{q}f\left(x\right) |\leq M$ in Theorem 4.1, then we have following quantum Ostrowski's type inequality for $s$-convex functions in the second sense:

Remark 4.2. If we set $s = 1$ in Corollary 4.1, then we recapture inequality (2.4).

Remark 4.3. In Corollary 4.1, if we take the limit as $q\rightarrow 1^{-}$, then Corollary 4.1 reduces to [1,Theorem 2].

Theorem 4.2. Assume that the mapping $f:I\subset \mathbb{R} ^{+}\rightarrow \mathbb{R}$ is differentiable and $a, b\in I$ with $a < b.$ If $|_{a}D_{q}f|^{p_{1}}$ and $|^{b}D_{q}f|^{p_{1}}$, $p_{1}\geq 1$ are $s$-convex mappings in the second sense, then the following inequality holds:

Proof. From Lemma 4.1, using properties of the modulus and power mean inequality, we have

Since the mappings $|_{a}D_{q}f|^{p_{1}}$ and $|^{b}D_{q}f|^{p_{1}}$ are $s$ -convex in the second sense, therefore

and

We obtain the resultant inequality (4.7) by putting (4.9) and (4.10) in (4.8).

Remark 4.4. If we set $s = 1$ in Theorem 4.2, then we obtain the following inequality

which is proved by Budak et al. in ^[16].

Corollary 4.2. If we assume $|_{a}D_{q}f\left(x\right)$, $|_{a}D_{q}f\left(a\right) |\leq M$ in Theorem 4.2, then we have following quantum Ostrowski's type inequality for $s$-convex functions in the second sense:

Remark 4.5. In Corollary 4.2, if we take the limit as $q\rightarrow 1^{-}$, then Corollary 4.2 reduces to [1,Theorem 4].

Theorem 4.3. Assume that the mapping $f:I\subset \mathbb{R} ^{+}\rightarrow \mathbb{R}$ is differentiable and $a, b\in I$ with $a < b.$ If $|_{a}D_{q}f|^{p_{1}}$ and $|^{b}D_{q}f|^{p_{1}}$, $p_{1} > 1$ are $s$-convex mappings in the second sense, then the following inequality holds:

where $\frac{1}{r_{1}}+\frac{1}{p_{1}} = 1.$

Proof. From Lemma 4.1, using properties of the modulus and Hölder's inequality, we have

Since the mappings $|_{a}D_{q}f|^{p_{1}}$ and $|^{b}D_{q}f|^{p_{1}}$ are $s$ -convex in the second sense, therefore

and

We obtain the resultant inequality (4.11) by putting (4.13) and (4.14) in (4.12).

Remark 4.6. If we set $s = 1$ in Theorem 4.3, then we obtain the following inequality

which is proved by Budak et al. in ^[16].

Corollary 4.3. If we assume $|_{a}D_{q}f\left(x\right) |$, $|^{b}D_{q}f\left(a\right) |\leq M$ in Theorem 4.3, then we have the following quantum Ostrowski's type inequality for $s$-convex functions in the second sense:

Remark 4.7. In Corollary 4.3, if we set $s = 1$, then we recapture the following inequality

which is obtained by Budak et al. in ^[16].

Remark 4.8. In Corollary 4.3, if we take the limit as $q\rightarrow 1^{-}$, then Corollary 4.3 reduces to [1,Theorem 3].

Theorem 4.4. Assume that the mapping $f:I\subset \mathbb{R} ^{+}\rightarrow \mathbb{R}$ is differentiable and $a, b\in I$ with $a < b.$ If $|_{a}D_{q}f|^{p_{1}}$ and $|^{b}D_{q}f|^{p_{1}}$, $p_{1}\geq 1$ are $s$-convex mappings in the second sense, then the following inequality holds:

where

Proof. From Lemma 4.1, using properties of the modulus and improved power mean inequality (see ^[33]), we have

Since the mappings $|_{a}D_{q}f|^{p_{1}}$ and $|^{b}D_{q}f|^{p_{1}}$ are $s$ -convex in the second sense, therefore

and

We obtain the resultant inequality (4.16) by putting (4.18), (4.19), (4.20) and (4.21) in (4.17).

Corollary 4.4. If we assume $|_{a}D_{q}f\left(x\right) |$, $|^{b}D_{q}f\left(x\right) |\leq M$ in Theorem 4.4, then we have the following quantum Ostrowski's type inequality for $s$-convex functions in the second sense:

Corollary 5. In Corollary 4.4, if we set $s = 1$, then we obtain the following new Ostrowski type inequality:

Theorem 4.5. Assume that the mapping $f:I\subset \mathbb{R} ^{+}\rightarrow \mathbb{R}$ is differentiable and $a, b\in I$ with $a < b.$ If $|_{a}D_{q}f|^{p_{1}}$ and $|^{b}D_{q}f|^{p_{1}}$, $p_{1} > 1$ are $s$-convex mappings in the second sense, then the following inequality holds:

where $\frac{1}{r_{1}}+\frac{1}{p_{1}} = 1.$

Proof. From Lemma 4.1, using properties of the modulus and Hölder's İşcan inequality (see ^[29]), we have

Since the mappings $|_{a}D_{q}f|^{p_{1}}$ and $|^{b}D_{q}f|^{p_{1}}$ are $s$ -convex in the second sense, therefore

and

We obtain the resultant inequality (4.23) by putting (4.25), (4.26), (4.27) and (4.28) in (4.24).

Corollary 4.6. If we assume $|_{a}D_{q}f\left(x\right) |$, $|^{b}D_{q}f\left(x\right) |\leq M$ in Theorem 4.5, then we have the following quantum Ostrowski's type inequality for $s$-convex functions in the second sense:

Corollary 4.7. In Corollary 4.6, if we use $s = 1$, then we obtain the following new Ostrowski type inequality:

5.   Applications to special means

For arbitrary positive numbers $\mathcal{\kappa }_{1}, \mathcal{\kappa }_{2}$ $(\mathcal{\kappa }_{1}\neq \mathcal{\kappa }_{2})$, we consider the means as follows:

$1.$The arithmetic mean

$2.$The logarithmic mean

Proposition 5.1. For $0 < a < b$ and $0 < q < 1$, the following inequality is true:

where

Proof. The inequality (4.2) in Theorem 4.1 with $x = \frac{a+b}{2}$ for $f\left(x\right) = \frac{x^{s+1}}{s+1}$, where $x > 0$ and $s\in \left(0, 1\right)$ leads to this conclusion.

Proposition 5.2. For $0 < a < b$ and $0 < q < 1$, the following inequality is true:

Proof. The inequality (4.6) in Corollary 4.1 with $x = \frac{a+b}{2}$ for $f\left(x\right) = \frac{x^{s+1}}{s+1}$, where $x > 0$ and $s\in \left(0, 1\right)$ leads to this conclusion.

Proposition 5.3. For $0 < a < b$ and $0 < q < 1$, the following inequality is true:

Proof. The inequality (4.7) in Theorem 4.2 with $x = \frac{a+b}{2}$ for $f\left(x\right) = \frac{x^{s+1}}{s+1}$, where $x > 0$ and $s\in \left(0, 1\right)$ leads to this conclusion.

Proposition 5.4. For $0 < a < b$ and $0 < q < 1$, the following inequality is true:

Proof. The inequality (4.11) in Theorem 4.3 with $x = \frac{a+b}{2}$ for $f\left(x\right) = \frac{x^{s+1}}{s+1}$, where $x > 0$ and $s\in \left(0, 1\right)$ leads to this conclusion.

Proposition 5.5. For $0 < a < b$ and $0 < q < 1$, the following inequality is true:

Proof. The inequality (4.15) in Corollary 4.3 with $x = \frac{a+b}{2}$ for $f\left(x\right) = \frac{x^{s+1}}{s+1}$, where $x > 0$ and $s\in \left(0, 1\right)$ leads to this conclusion.

6.   Conclusions

In this investigation, Hermite-Hadamard and Ostrowski type inequalities for $s$-convex mappings in the second sense are derived, by applying quantum integrals. It is also showed that the results established in this paper are potential generalization of the existing comparable results in the literature. As future directions, one can find similar inequalities for co-ordinated $s$-convex functions in the second sense.

Acknowledgments

This research was funded by King Mongkut's University of Technology North Bangkok. Contract no. KMUTNB-61-KNOW-015. The work of second author is partially supported by the National Natural Science Foundation of China (Grant No. 11971241).

Conflict of interest

The authors declare that they have no competing interest.