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We generalize an integral Jensen–Mercer inequality to the class of n-convex functions using Fink’s identity and Green’s functions. We study the monotonicity of some linear functionals constructed from the obtained inequalities using the definition of n-convex functions at a point.
for a convex function , real numbers and positive real numbers , where , is one of the most important inequalities in many areas of mathematics and other areas of science. Many other inequalities can be derived from it and there are numerous of its variants, generalizations and refinements (see, for example [1,2]). One of these variants is the so-called Jensen–Mercer inequality,
for a convex function , real numbers and positive real numbers , which was introduced in  by A. McD. Mercer. It has been a main topic of our research (see, e.g., [4,5,6,7]) in different settings and for different classes of functions.
In the paper , we proved the following integral version.
Let be a continuous and monotonic function and be an interval such that . Let the function be either continuous or of bounded variation, and satisfying
Then, for every continuous convex function the inequality
holds. If φ is concave, then the reversed inequality (4) holds.
Theorem 1 is also valid when the condition (3) is replaced with the more strict condition that λ is a nondecreasing function such that .
The main goal of this paper is to present generalizations of inequality (4) to the class of n-convex functions. We achieve this by means of Fink’s identity and Green’s functions, which we introduce below.
A real-valued function defined on an interval is called n-convex if its nth order divided differences are nonnegative for all choices of distinct points in . Thus, 0-convex functions are nonnegative functions, 1-convex functions are increasing functions and 2-convex functions are convex functions. An n-convex function need not be n-times differentiable, however, if exists then is n-convex if and only if . For more information about n-convex functions, see  and also .
holds for every function such that is absolutely continuous on for some . For , we take the sum in (5) to be zero.
We consider Green’s functions defined on by
All five Green’s functions are continuous, symmetric and convex with respect to both the variables s and t.
It can be easily shown by integrating by parts that every function , can be represented in the following five forms. If , then
By easy calculation (see [11,12]), using representations (12)–(16), we can obtain the identity
for all . In the rest of the paper, for the sake of simplicity, let us denote
Since all five Green’s functions are continuous and convex, by Theorem 1, for all .
2. Main Results
We start with two identities which are very useful in obtaining generalizations of inequality (4) to the class of n-convex functions.
Let the functions g and λ be as in Theorem 1, and let the function K be defined by (6). Then, for every function such that is absolutely continuous on for some , the identity
Using (5) in the left hand of equality (19), we obtain
By regrouping and calculating all the first members in the above sums, we obtain (19). □
Let the functions g and λ be as in Theorem 1, and let the functions , and K be defined by (7)–(11) and (6), respectively. Then, for every function such that is absolutely continuous on for some , the identity
Using (21) in (17) and applying Fubini’s theorem, we easily obtain (20). □
The following generalizations of inequality (4) for n-convex functions hold.
Let the functions g and λ be as in Theorem 1, and let the function K be defined by (6). Let be a n-convex function such that is absolutely continuous on for some .
If n is even then (23) holds. Moreover, if for , for and for then the right-hand side of (23) is nonnegative and (4) holds.
Since is absolutely continuous on and is n-convex, exist almost everywhere on and . This fact and assumption (22) show that the third member on the right-hand side of (19) is nonnegative, so inequality (23) immediately follows from identity (19).
For a fixed consider the function defined by
It is continuous for and, since
it is convex on for even n. Therefore, by Theorem 1,
i.e., inequality (22) holds and, by , inequality (23) holds.
Further, the function is convex on for any k, and the function is convex (concave) on for even (odd) k. Hence, by Theorem 1,
for all ,
for and the reversed inequality (25) holds for . Hence, the second claim in immediately follows. □
Under the assumptions of Theorem 2, the right-hand side of (23) can be written in the form
where the function is defined by
Therefore, if n is even and the function is convex on then, by Theorem 1, the right-hand side of (23) is nonnegative and (4) holds.
Let the functions g and λ be as in Theorem 1, and let the functions , and K be defined by (7)–(11) and (6), respectively. Let be an n-convex function such that is absolutely continuous on for some . Then, for all :
If n is even, then (27) holds. Moreover, if for , for and for then the right-hand side of (27) is nonnegative and (4) holds.
Similarly as in the proof of Theorem 2, we can assume that is n-times differentiable and . Hence, assumption (26) gives that the second member on the right-hand side of (20) is nonnegative, so inequality (27) immediately follows from identity (20).
Since , for (26) holds when n is even, and the reversed (26) holds when n is odd, while for (26) always holds. Therefore, immediately follows from . □
Under the assumptions of Theorem 3, the right-hand side of (27) can be written in the form
where the functions , are defined by
Therefore, if n is even and the functions , are convex on then, by Theorem 1, the right-hand side of (27) is nonnegative for all and (4) holds.
3. Related Results for -Convex Functions at a Point
The class of n-convex functions at a point was introduced by Pečarić et al. in  as follows.
Let I be an interval in , (interior of I), and . A function is said to be n-convex at point d if there exists a constant such that the function is -concave on and -convex on . It is shown that a function is n-convex on an interval if and only if it is n-convex at every point of its interior.
Additionally, the authors of  investigated the conditions which some linear functionals and have to fulfill so that the inequality holds for any function that is n-convex at point . In this section, we give inequalities of this type for the linear functionals obtained as the differences of the left- and right-hand sides of inequalities from the previous section.
In the rest of this section, we take function to be as in Theorem 1, functions and K defined by (7)–(11) and (6), respectively, and function such that is absolutely continuous on . Let be such that , and let be continuous and monotonic functions such that , .
With and we denote the difference of the left- and right-hand sides of inequality (23) for the function on the intervals and , respectively, i.e.,
For even from Theorem 2 it follows that when is n-convex, and when is n-concave.
Similarly, for with and we denote the difference of the left- and right-hand sides of inequality (27) for the function on the intervals and , respectively, i.e.,
where and ,
where and .
For even and from Theorem 3 it follows that when is n-convex, and when is n-concave.
Under the above assumptions, let be even and the function ϕ be -convex at point d.
If , and
If , and
Let . Since is -convex at point d, there exists such that the function is n-concave on and n-convex on . Therefore
From identity (19) for the function on the intervals and we have
Hence, assumption (28) is equivalent to and from (30) and (31) it follows
Notice that from the proof of Theorem 4 it follows that it is also valid when the assumptions (28) and (29) are replaced with the weaker assumptions that and for , respectively.
In the papers [11,12], we generalized the Jensen–Mercer inequality to the class of n-convex functions by using the Taylor polynomial and Green’s functions. Here, we do so using Fink’s identity and use these generalizations to obtain some related results for n-convex functions at a point. In a similar way as in , we can obtain Grüss and Ostrowski type inequalities in terms of Čebyšev functionals. Using the main ideas from , we can also generate new families of n-exponentially convex and exponentially convex functions by considering the linear functionals defined above.
This research was supported by the University of Split, FESB.
We would like to thank the reviewers for their effort to read the paper thoroughly and give us very useful suggestions on how to improve it.
Conflicts of Interest
The author declares no conflict of interest.
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