1. Introduction
Jensen’s inequality is one of the most studied results in the literature. In the last few decades, quite a few researchers have been interested in refining and generalizing this inequality (see, e.g., [
1,
2,
3,
4,
5,
6]).
Let
and let
(
) be positive weights associated with these
and let their sum demonstrate unity. Then, Jensen’s inequality
holds (see [
7]).
Mercer investigated a generalized form of Jensen’s inequality, which is famously known as the Jensen–Mercer inequality (see [
8]): if
is a convex function on
, then
is fulfilled for
,
with
. In case of
, inequality (
2) reads as
for
. Extensions of this result can be found in e.g., [
9,
10,
11].
The well-known refinement of Jensen’s inequality, the Hermite–Hadamard inequality
for convex functions, was proved by Hermite in 1883 and independently by Hadamard in 1893; see, e.g., [
12]. This inequality has been generalized by many researchers, taking into account various aspects such as general convexity and fractional operators. For Hermite–Hadamard–Mercer type results, see [
13,
14,
15,
16,
17,
18].
In general, the concept of convex and general convex functions plays a major role in the theory of integral inequalities. So far, many general convex classes have been described in the literature. A summary of many of these classes was given in [
19].
Definition 1. Let , and . If inequalityis fulfilled and , where , then function Φ is called -convex on I. In [
20,
21], the following definitions were presented.
Definition 2. Let and . If inequalityis fulfilled and , where , , then function Φ is called -convex modified of the first type on I and this set of functions will be denoted as Definition 3. Let and . If inequalityis fulfilled and , where , , then function Φ is called -convex modified of the second type on I and this set of functions will be denoted as Throughout the paper, for -convex modified functions of the first or, of the second type, we assume that and .
The following results are extended versions of Jensen–Mercer inequality (
3).
Theorem 1. Let be an integrable and -convex function. Then, the following Mercer’s type inequality holds:for , and , such that . Proof. Putting
and
, we have
. Now, using the
-convexity of
, we have
By adding the corresponding sides of the inequalities, we obtain
From the above, the desired inequality (
8) is easily obtained. □
Corollary 1. Let be an integrable -convex function. Then, from (8), we havefor , and . Remark 1. For , Corollary 1 leads to a correct version of Lemma 3.1 of [11]. Theorem 2. Let be an integrable and . Then, the following Mercer’s-type inequality holds:for , and such that . Proof. The proof is analogous to that of Theorem 1. Taking
,
and combining inequalities
results in inequality (
10). □
Corollary 2. Let be an integrable and . Then, from (10), we havefor , and Theorem 3. Let be an integrable and . Then, the following Mercer’s type inequality holds:for , and , such that . Proof. The proof is analogous to that of Theorem 1. Taking
,
and combining inequalities
yields inequality (
11). □
Corollary 3. Let be an integrable and . Then, from Theorem 3, we havefor , and Remark 2. For and , we have , moreover, Theorems 2 and 3 (or, Corollaries 2 and 3) become the Jensen–Mercer inequality for convex functions (3). Remark 3. Other variants of the Jensen–Mercer inequality (2), for different notions of convexity, can be found in [16,22,23,24,25]. In the remainder of this paper, we aim to give generalizations of Hermite–Hadamard inequality (
4) via non-conformable fractional integrals defined by Nápoles et al. in [
26].
Definition 4. Let and . For each function , we definefor every Definition 5. Let and . For each function , that is the linear spacelet us define the fractional integralsfor every . Here, for we have . Definition 6. More details on the fractional integral and the corresponding fractional derivative can be read in [26]. Fractional differential and integral computations have been widely used in many fields of applied sciences. The interested reader can read about the role of fractional calculus in the study of biological models and chemical processes in [
27,
28,
29].
2. Inequalities for Convex Functions
In this section, we obtain analogues of Hermite–Hadamard inequality (
4) for non-conformable fractional operators (
13) using Jensen–Mercer inequalities.
Remark 4. If in (2), we take and , then we have Theorem 4. Let . If and Φ is convex on , thenwhere and . Proof. If in (
14), we choose
and
, and multiply by
, then we can write the inequality
Now, by integrating the resulting inequality with respect to
on
and changing the variable, we obtain
After dividing both sides of the last inequality by
, we get the left inequality in (
15).
For the proof of the second inequality of (
15), keeping in mind that
is convex, one can write
By multiplying both sides of last inequality by
and by integrating with respect to
t on
and changing the variables, we obtain
By multiplying the last inequality by
and adding
to both sides, we get the right-hand side of (
15):
Thus, inequality (
15) is proved. □
Corollary 4. For under the assumptions of Theorem 4, we getfor all . This inequality was obtained by Kian and Moslehian in ([30], Theorem 2.1), and by Ögülmüs and Sarikaya in ([17], Remark 2.2). Theorem 5. Let . If and is convex on , then we havewhere and Proof. To prove inequality (
16), we use the left-hand side of (
14) and choose
,
to obtain the auxiliary inequality
More precisely, we use the equivalent inequality
Multiplying both sides of (
17) by
, integrating with respect to
t on
and changing the variables yields
It is easy to see that left-hand side of (
16) is proved. To prove the remaining part of (
16), we need the following inequalities:
and
By summing the above inequalities, we have
By multiplying both sides (
17) by
, integrating with respect to
t on
and changing the variables, we obtain
This inequality implies the remaining part of (
16) by keeping (
3) in mind. The proof is complete. □
Corollary 5. For , under the assumptions of Theorem 5, we havefor all . This inequality was obtained by Kian and Moslehian in ([30], Theorem 2.1), and by Ögülmüs and Sarikaya in ([17], Remark 2.2). Remark 5. If in (18), we choose and then we get the Hermite–Hadamard inequality (4). 3. Inequalities for General Convex Functions
By considering
-convexity modified in the first and the second sense, we give analogues of Hermite–Hadamard inequality (
4) for fractional operators (
13) using Jensen–Mercer inequalities proven for these classes. Before that, we recall the following identity obtained by Nápoles et al. in [
26] (see Lemma 1).
Lemma 1. Let be a differentiable function. If , then we havewhere and If in Lemma 1, we substitute in place of and in place of , we get the next equation.
Corollary 6. Under the assumptions of Lemma 1, we havewhere , and Theorem 6. Let be a differentiable function. If and , then the following inequality holds for all , :where is from Corollary 2. Proof. From Corollary 6 and modulus properties, we can write
Using
-convexity of the first sense of function
and Corollary 2, for integral
, we get
One can write for the second integral
similarly
By multiplying the last inequality by
and taking into account (
21), we obtain (
20). □
Corollary 7. If in Theorem 6, we choose and , then we have If, in addition, , then Theorem 7. Let be a differentiable function. If and , then the following inequality holds for all , :where is from Corollary 3. Proof. The proof is analogous to that of Theorem 7, but with the use of Corollary 3 instead of Corollary 2. □
Corollary 8. If in Theorem 7, we choose , and , then we have Theorem 8. Let be a differentiable function. If and , then for all , , with , the following inequality holds:where Proof. From Lemma 6 and modulus properties, we can write (
21). Using the well-known Hölder integral inequality and Corollary 2, since
, we get
Since
we can write similarly for the second integral
By adding inequalities (
25) and (
26), we get
Multiplying both sides of the last inequality by the expression
and keeping (
21) in mind yields (
24). The proof is complete. □
Corollary 9. If in Theorem 8, we choose , and , then we have Theorem 9. Let be a differentiable function. If and , then for all , , with , the following inequality holds:where Proof. The proof is analogous to that of Theorem 8, but with the use of Corollary 3 instead of Corollary 2. □
Corollary 10. If in Theorem 9, we choose , and , then we have Theorem 10. Let be a differentiable function. If and , then for all , , , we havewhere Proof. We first write (
21). Then, using the well-known power–mean integral inequality and Corollary 2, since
, for the integral
, we obtain
One can write for the second integral similarly
By adding inequalities (
28) and (
29), we obtain
Multiplying both sides of the last inequality by the expression
and keeping (
21) in mind, we get (
27). The proof is complete. □
Corollary 11. If in Theorem 10, we choose , and , then we have If, in addition, we suppose , then we get (22). Theorem 11. Let be a differentiable function. If and , then for all , , , we havewhere Proof. The proof is analogous to that of Theorem 10, but with the use of Corollary 3 instead of Corollary 2. □
Corollary 12. If in Theorem 11, we choose , and , then we have If, in addition, we suppose , then we get (23). 4. Applications
Throughout the paper, we examined the fractional integral sums
for
.
We demonstrate the scope and strength of our results through three examples, two related to trigonometric functions and one to arithmetic means.
First, consider a convex function. Let
,
, which is convex on
, and fix
. Then, according to Theorem 4, we have the inequality
for all
.
Second, we consider a non-convex function that has a convex derivative in absolute value. Let
,
, which has a convex derivative
on
, and fix
. Keeping Remark 2 in mind, applying Corollary 7 or Corollary 8 (with
x in place of
and
y in place of
) yields
for all
.
Finally, consider the convex function
,
with
, and fix
. Then, according to Theorem 4, we have
for
, from which we obtain an inequality of arithmetic means:
where
denotes the arithmetic mean
.
5. Conclusions
In the present work, we obtained interesting results pertaining to the Jensen–Mercer-type Hermite–Hadamard inequalities via non-conformable integrals, using the classical convex, -convex, and -convex modified functions. Thus, we presented various relevant fractional inequalities related to convex functions and differentiable functions of general convex derivative in absolute value.
As applications, we gave examples of functions for which our main inequalities can be applied, and we presented the resulting inequalities.
Our results are expected to provide motivation to generate further research on inequalities that includes other notions of convexity, such as new variants of the Hermite–Hadamard–Mercer inequalities obtained in this work. For example, instead of working with the operators of [
26], one can consider the following more general fractional integral:
Definition 7 ([
31])
. Let , such that Generalized fractional Riemann–Liouville integral of order and , , is given as follows:with , and . Obviously . By considering the kernel
, we have
and we get the
–Riemann–Liouville fractional integral in Definition 2.1 of [
32]. Furthermore, by setting
, we obtain the Katugampola fractional integral (see [
33]).