Stability analysis of boundary value problems for Caputo proportional fractional derivative of a function with respect to another function via impulsive Langevin equation

1.   Introduction

In recent years, convexity theory has gained special attention by many researchers because of it engrossing properties and expedient characterizations. It has many applications in fields like biology, numerical analysis and statistics (see ^[1,2,3,4]). Mathematical inequalities are extensively studied with all type of convex functions (see^{[1,3,11,13,14,16]}). One of the fundamental inequality is Hermite-Hadamard inequality. It has been discussed via different types of convexities and became the center of attention for many researchers. Recently, in 2016, Khan et al. have discussed generalizations of Hermite-Hadamard type for $MT$-convex functions ^[26]. In 2017, Khan et al. studied some new inequalities of Hermite-Hadamard types ^[27]. In 2019, Khurshid et al. have utilized conformable fractional integrals via preinvex functions ^[28]. In 2020, Khan et al. have discussed Hermite-Hadamard type inequalities via quantum calculus involving green function ^[29], Mohammed et al. have established a new version of Hermite-Hadamard inequality for Riemann-Liouville fractional integrals ^[30], Han et al. used fractional integral to generalize Hermite-Hadamard inequality for convex functions ^[31], Zhao et al. utilized harmonically convex functions to generalized fractional integral inequalities of Hermite-Hdamrd type ^[32], Awan et al. presented new inequalities of Hermite-Hdamard type for $n$-polynomial harmonically convex functions ^[33]. In 2022, Khan et al. introduced some new versions of Hermite-Hadamard integral inequalities in fuzzy fractional calculus for generalized pre-invex functions via fuzzy-interval-valued settings ^[34]. This reflects the importance of Hermite Hadamard type inequalities among current research.

In ^[9], $s$-convex function is given as,

Definition 1.1. A real valued function $\chi$ is called $s$-convex function on $\mathbb{R}$, if

for each $\rho, \gamma \in \mathbb{R}$ and $\varsigma \in (0, 1)$ where $s\in (0, 1]$.

In ^[10], $m$-convexity is discussed as,

Definition 1.2. A real valued function $\chi$ defined on $[0, b]$ is said to be a $m$-convex function for $m \in [0, 1]$, if

holds for all $\rho, \gamma \in [0, b]$ and $\varsigma \in [0, 1]$.

$(s, m)$-convexity in ^[17] is discussed as,

Definition 1.3. A function $\chi: [0, b] \longrightarrow \mathbb{R}$, $b > 0$ is said to be a $(s, m)$-convex function in the second sense where $s, m \in (0, 1]^{2}$, if

holds provided that all $\rho, \gamma \in [0, b]$ and $\varsigma \in [0, 1]$.

Equivalent definition for $(s, m)$–convex functions:

Let $\rho, \alpha, \gamma \in [0, b]$, $\rho < \alpha < \gamma$

Hölder-İşcan Inequality ^[5]:

Let $p > 1$, $\chi$ and $\psi$ be real valued functions defined on $\left[\rho, \gamma \right]$ and $|\chi|^{p}, |\psi|^{q}$ are integrable functions on interval $\left[\rho, \gamma \right]$

where $\frac{1}{p}+\frac{1}{q} = 1$.

Following lemma is useful to obtain our main results.

Lemma 1.4. ^[8] For $n\in \mathbb{N}$, let $\chi:U\subseteq \mathbb{R} \longrightarrow \mathbb{R}$ be $n$-times differentiable mapping on $U^{\circ}$, where $\rho, \gamma \in U^{\circ}$, $\rho < \gamma$ and $\chi^{n} \in L[\rho, \gamma]$, we have following identity

where an empty set is understood to be nil.

In this paper, Hölder-İşcan inequality is used to modify inequalities involving functions having $s$-convex or $s$-concave derivatives at certain powers. The purpose of this paper is to establish some generalized inequalities for $n$-times differentiable $(s, m)$-convex functions. Applications of these inequalities to means are also discussed. Means are defined as,

Let $0 < \rho < \gamma$,

where $p\neq0, -1$ and $\rho\neq\gamma$.

2.   Main results

2.1.   Modified inequalities for $n$-times differentiable convex functions

Theorem 2.1. For any positive integer $n$, let $\chi:U\subseteq(0, \infty)\rightarrow \mathbb{R}$ be $n$-times differentiable mapping on $U^{\circ}$, where $\rho, \gamma \in U^{\circ}$ with $\rho < \gamma$. If $\chi^{(n)} \in L[\rho, \gamma]$ and $|\chi^{(n)} |^{q}$ for $q > 1$ is $(s, m)$-convex on interval $[\rho, \gamma]$ then

where $\frac{1}{p}+\frac{1}{q} = 1$.

Proof. Since $|\chi^{n}|^q$ is $(s, m)$-convex by using inequality (1.1) for $\rho < \omega < \gamma$, using Lemma 1.4 and Hölder-Işcan inequality (1.2),

Let

Substituting integrals ${I_1}, {I_2}, {I_3}, {I_4}, {I_5}, {I_6}$ in inequality (2.2) we have,

which is required inequality (2.1).

For $n = 1$ inequality (2.1) becomes,

Remark 2.2. For $s = 1$ and $m = 1$ our resulting inequality (2.1) becomes the inequality ($2$) of ^[5].

Theorem 2.3. For $n\in \mathbb{N}$, let $\chi:U\subseteq(0, \infty)\rightarrow \mathbb{R}$ be $n$-times differentiable mapping on $U^{\circ}$, where, $\rho, \gamma \in U^{\circ}$, $\rho < \gamma$, $\chi^{(n)} \in L[\rho, \gamma]$ and $|\chi^{(n)} |^{q}$ for $q > 1$, is $(s, m)$-convex on interval $[\rho, \gamma]$ then following inequality holds

Proof. Since $|\chi^{(n)}|^{q}$ for $q > 1$ is $(s, m)$-convex on $[\rho, \gamma]$, by using Lemma 1.4 and Hölder-İşcan inequality (1.2), since $s \in (0, 1]$, this fact can be used for $\omega, \rho, \gamma \in U\subseteq(0, \infty)$,

Substituting integrals ${I_1}, {I_2}, {I_3}, {I_4}, {I_5}, {I_6}$ in inequality (2.5) we have,

For $n = 1$, Theorem2.3 reduced to the inequality

Remark 2.4. For $s = 1$ and $m = 1$ our resulting inequality (2.4) becomes the inequality $(6)$ of ^[5].

2.2.   Hadamard type inequality via $(s, m)$-convexity

Theorem 2.5. If function $\chi: [0, b] \longrightarrow \mathbb{R}$, $b > 0$ is a (s, m)-convex function in the second sense where $(s, m) \in (0, 1]^{2}$, holds provided that all $\rho, \gamma \in [0, b]$ and $\varsigma \in [0, 1]$, then

Proof. A function $\chi: [0, b] \longrightarrow \mathbb{R}$, $b > 0$ is said to be a $(s, m)$-convex function in the second sense where $s, m \in (0, 1]^{2}$, if

holds provided that all $\rho, \gamma \in [0, b]$ and $\varsigma \in [0, 1]$.

Integrating w.r.t $\varsigma$ on $[0, 1]$,

and

As $\chi$ is $(s, m)$-convex,

Integrating w.r.t $\varsigma$ over $[0, 1]$ and by using (2.8) and (2.9) we get,

Substituting in first integral,

${ { \varsigma \rho + (1 - \varsigma)\gamma m}} = \omega$,

Substituting in the second integral,

$\varsigma \gamma + (1 - \varsigma)\frac{\rho }{m} = l$,

Using (2.11) and (2.12) in (2.10) required inequality (2.7) obtained.

Remark 2.6. For $s, m = 1$ inequality (2.7) becomes classical Hadamard inequality for convex functions.

2.3.   Application of Hadamrd type inequality via $(s, m)$-concavity

Theorem 2.7. For $n\in \mathbb{N}$, let $\chi:U\subseteq(0, \infty)\rightarrow \mathbb{R}$ be n-times differentiable mapping on $U^{\circ}$, where, $\rho, \gamma \in U^{\circ}$, $\rho < \gamma$ and $\chi^{(n)} \in L[\rho, \gamma]$ and $|\chi^{(n)} |^{q}$ for $q > 1$ is (s, m)-concave on interval $[\rho, m\gamma]$, then

Proof. $|\chi^{(n)} |^{q}$ for $q > 1$ is $(s, m)$-concave then by using Theorem 2.5 we have,

Using Lemma 1.4 and Hölder-Îşcan inequality (1.2),

Substituting integrals ${I_1}, {I_2}$ in inequality (2.14) required inequality (2.13) is obtained.

For $n = 1$ inequality (2.13) becomes,

Remark 2.8. For $s = 1$ and $m = 1$ our resulting inequality becomes the inequality obtained in Theorem $4$ of ^[5].

2.4.   Applications of newly obtained inequalities to means

Proposition 2.9. Let $\rho, \gamma \in (0, \infty)$, where $\rho < \gamma$, $q > 1$, $n, i \in \mathbb{N}$ with $i\geq n$,

where

Proof. Let

Let

${g^{''}}(\varsigma) \ge 0$ means g is convex and $g(1) = g(0) = 0$, which omplies $g\leq 0$, hence

By using Theorem 2.1 for $|\chi^{n}(\omega)|^{q}$ which is $(s, m)$–convex for $s, m \in (0, 1]^{2}$ inequality (2.16) obtained.

Remark 2.10. For $s, m = 1$ inequality (2.16) becomes inequality $(3)$ of ^[5].

Example 2.11. Taking $i = 2$, $n = 1$, $p = q = 2$ in Proposition 2.9, the following is valid:

where $A$ and $G$ are classical arithmetic and geometric means, respectively.

Proposition 2.12. Let $\rho, \gamma \in (0, \infty)$, with, $\rho < \gamma$, $q > 1$ and $n \in \mathbb{N}$,

where $L$ is classical logarithmic mean.

Proof.

Let

${g^{''}}(\varsigma) \ge 0$ means $g$ is convex and $g(1) = g(0) = 0$ which implies $g\leq 0$ as

So ${\left| {{\chi ^{(1)}}(\omega)} \right|^q}$ is $(s, m)$-convex. Then by using inequality (2.3) required inequality (2.17) obtained.

Remark 2.13. For $s, m = 1$ inequality (2.17) becomes $(4)$ of ^[5].

Example 2.14. For $n = 1$ and $p = q = 2$, Proposition 2.12 gives:

Proposition 2.15. Let $\rho, \gamma \in (0, \infty)$, $\rho < \gamma$, $q > 1$, $i \in (- \infty, 0] \cup [1, \infty)\backslash\{ - 2q, - q\}$

then

Proof.

Let

$g^{''}(\varsigma) \geq 0$ and $g(1) = g(0)$ so $g\leq0$ and $|\chi^{'} (\omega)|^{q}$ is $(s, m)$-convex, by using inequality (2.3) we have (2.18).

Remark 2.16. For $s, m = 1$ inequality (2.18) becomes $(5)$ of ^[5].

Example 2.17. For $i = 2$ and $p = q = 2$ Proposition 2.15 reduced to

Proposition 2.18. Let $\rho, \gamma \in (0, \infty)$ with $\rho < \gamma$, $q > 1$ and $n \in \mathbb{N}$ then we have

Proof. Let,

As $|\chi^{n}(\omega)|^{q}$ is $(s, m)$-convex on $(0, \infty)$, therefore by using Theorem 2.3 required inequality (2.20) is obtained.

Remark 2.19. For $s, m = 1$ inequality (2.20) becomes inequality obtained in Proposition $4$ of ^[5].

Proposition 2.20. Let $\rho, \gamma \in (0, \infty)$ with $\rho < \gamma$ $q > 1$ and $n \in \mathbb{N}$ then we have,

Proof.

As ${\left| {{\chi ^{(1)}}(\omega)} \right|^q}$ is $(s, m)$–convex, therefore by using inequality (2.6) required (2.21) obtained.

Remark 2.21. For $s, m = 1$ inequality (2.21) becomes inequality obtained in Proposition $5$ of ^[5].

Proposition 2.22. Let $\rho, \gamma \in (0, \infty)$ with $\rho < \gamma$ $q > 1$ and $i \in (-\infty, 0]\backslash\{-2q, q\}$, then

Proof.

$|\chi^{'} (w)|^{q}$ is $(s, m)$-convex by using inequality (2.6) required (2.22) obtained.

For $i = 1$ inequality (2.22) becomes,

Remark 2.23. For $s, m = 1$ inequality (2.22) becomes inequality obtained in Proposition $6$ of ^[5].

Proposition 2.24. Let $\rho, \gamma \in (0, \infty)$ with $\rho < \gamma$, $q > 1$ and $i \in [0, 1]$ we have,

Proof.

As $|\chi^{'} (\omega)|^{q}$ is $(s, m)$-concave by using inequality (2.15) we obtain required inequality (2.24).

Remark 2.25. For $s, m = 1$ inequality (2.24) becomes the inequality obtained in Proposition $9$ of ^[5].

3.   Conclusions

In this paper, Hölder-Isçan inequality is utilized to prove Hermite-Hadamard type inequalities for $n$-times differentiable $(s, m)$-convex functions. The method is adequate and provide many generalizations of existing results as shown in remarks. Moreover, many other inequalities can be generalized for other types of convex functions.

Acknowledgments

This research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation, (grant number B05F650018)

Conflict of interest

The authors declare no conflict of interest.