1. Introduction
The theory of inequality has seen a rise in research activity over the past 20 years in different fields of sciences, both theoretical and applied, including in the study of the qualitative properties of solutions to ordinary, partial, and integral differential equations as well as in numerical analysis, where this tool is essential for estimating quadrature errors, and in a variety of calculation types, including time scale calculus [
1,
2,
3], fractional calculus [
4,
5,
6,
7], quantum calculus [
8,
9], and classical (Newtonian) calculus [
10,
11,
12].
The term multiplicative calculus originates from the classical calculation of Newton and Leibniz, which was introduced by Grossman and Katz when they presented and examined the first non-Newtonian systems [
13].
The multiplicative derivative and integral were presented by Bashirov et al. [
14]. Its relationship to the classical derivative and integral, as well as some of its features, are mentioned below.
The multiplicative derivative of the function with the notation is as follows:
Definition 1 ([14]). For a positive function . The multiplicative derivative is Remark 1. If is positive and differentiable at t, then exists and is related to the standard derivative as follows: The multiplicative integral or integral of the function noted is as follows:
Proposition 1 ([14]). Let . Then, the integral of the function is It is also practical to remember the integration-by-parts formula.
Theorem 1 ([14]). Let , where is a multiplicative differentiable function and χ is a differentiable function. So, the function is a multiplicative integrable function that satisfies Lemma 1 ([15]). Let , where is a differentiable multiplicative function and is a differentiable function. Suppose is a differentiable function, then The analogous multiplicative of the Hermite–Hadamard inequality was provided by Ali et al. in [
16], as follows:
Theorem 2. Let be a positive and multiplicatively convex function on the interval ; then, the following double inequality is true: Since the publication of the aforementioned paper, several works concerning multiplicative inequalities have been published (see, for instance, [
15,
17,
18,
19,
20]).
In [
21], Meftah investigated some Maclaurin-type inequalities for multiplicatively convex functions and established the following results.
Theorem 3. Assume that is a multiplicative differentiable map with multiplicative convex derivative on . Then, we have Theorem 4. Assume that Theorem 3’s whole set of hypotheses is true. Then, we have Theorem 5. Assume that Theorem 3’s whole set of hypotheses is true. Then, we have The multiplicative Riemann–Liouville fractional integrals were first introduced by Abdeljawad and Grossman in [
4] and satisfies the following relations:
Definition 2. The left and right multiplicative Riemann–Liouville fractional integral of order , where , is given as follows:andwhere and are the left and right Riemann–Liouville fractional integrals, respectively, defined as follows:and Budak and Özçelik [
22] proved some multiplicative fractional Hermite–Hadamard-type inequalities by combining the operators (2) and (3) with the definition of multiplicative convex functions. One can also consult [
22,
23,
24,
25,
26,
27,
28] concerning fractional multiplicative inequalities.
Very recently, Peng and Du [
29] established some non-symmetrical fractional Maclaurin-type inequalities as follows:
Theorem 6. is an increasing multiplicative differentiable map. If is multiplicative convex on , then for , the following inequality related to multiplicative RL-fractional integrals holds:wherewith Theorem 7. Under the assumptions of Theorem 6, if with , then we have The goal of the current study is to construct some new symmetrical fractional Maclaurin-type inequalities for multiplicatively convex functions, which are motivated by the previously stated papers. To address this, we provide a novel integral identity, from which the fractional Maclaurin inequality for bounded multiplicative derivatives is derived initially. The situation when the multiplicative derivatives are convex is then covered. Some applications to special means are provided at the end. The remainder of the current paper is organized as follows: Some symmetrical fractional Maclaurin inequalities are presented in
Section 2.
Section 3 provides some applications to special means.
Section 4 draws the conclusion.
2. Main Results
We begin with the auxiliary result that follows.
Lemma 2. Assume that is a multiplicative differentiable mapping with multiplicative integrable derivative on . Then, we have Proof. By using the integration by parts for multiplicative integrals,
yields
Multiplying (5)–(8) yields the desired outcome. □
Theorem 8. Assume that Lemma 2’s hypotheses are all true. If on , then we havewhere is defined by (4). Proof. According to Lemma 2, multiplicative integration, and the hypothesis that
, we have
where we have used
The proof is finished. □
Corollary 1. By assuming that in Theorem 6, we obtain Theorem 9. Assume that Lemma 2’s hypotheses are all true. If is multiplicative convex on , then we havewhere is defined by (4). Proof. According to Lemma 2, multiplicative integration, and the multiplicative convexity of
, we have
The result follows from the calculation of the following integrals:
and
The proof is completed. □
Remark 2. If we put , Theorem 7 may be simplified to Theorem 3.2 from [4]. Corollary 2. Using the multiplicative convexity of , i.e., , Theorem 7 becomes Remark 3. Corollary 2 will be reduced to Corollary 3.3 from [4], if we take . Corollary 3. Using the multiplicative convexity of , i.e., , Theorem 7 becomes Remark 4. Corollary 3 will be reduced to Corollary 3.4 from [4], if we take . 3. Applications to Special Means
Consider the following means of arbitrary real number :
The arithmetic mean: .
The harmonic mean: .
The logarithmic means: , , and .
The k-logarithmic mean: , , and .
Proposition 2. For two positive real numbers , we have Proof. It suffices to apply Corollary 2, taking as a function with where , and . □
Proposition 3. For two positive real numbers and , we have Proof. It suffices to apply Corollary 3 with on the interval to the function , whose and . □
4. Conclusions
The conclusions produced in this work are based on a novel identity. We have constructed certain fractional Maclaurin-type integral inequalities for functions whose multiplicative derivatives are both bounded and multiplicatively convex. We have also discussed some particular cases. A few applications of our findings to special means are given. Our results improve those established in [
29], and they also recover those established in [
21].
Author Contributions
Conceptualization, M.M., B.M., A.M. and M.B.; Methodology, M.M., B.M., A.M. and M.B.; Writing—original draft, M.M., B.M., A.M. and M.B.; Writing—review & editing, M.M., B.M., A.M. and M.B. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by King Khalid University through large research project under grant number R.G.P.2/252/44.
Data Availability Statement
No new data were created or analyzed in this study. Data sharing is not applicable to this article.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work.
Conflicts of Interest
The authors declare no conflict of interest.
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