Abstract
In the year 2003, McD Mercer established an interesting variation of Jensen’s inequality and later in 2009 Mercer’s result was generalized to higher dimensions by M. Niezgoda. Recently, Asif et al. has stated an integral version of Niezgoda’s result for convex functions. We further generalize Niezgoda’s integral result for functions with nondecreasing increments and give some refinements with applications. In the way, we generalize an important result, Jensen-Boas inequality, using functions with nondecreasing increments. These results would constitute a valuable addition to Jensen-type inequalities in the literature.
1. Introduction and Preliminaries
Let us start with Jensen’s inequality for convex functions, one of the most celebrated inequalities in mathematics and statistics (for detailed discussion and history, see [1, 2]). Throughout the paper, we assume that and are intervals in , and is an interval in and is a -dimensional rectangle for integer . Also for weights , , we would use .
Proposition 1. Let and let be nonnegative real numbers such that . If is a convex function, then the following inequality holds:
In [3], McD Mercer proved the following variant of Jensen’s inequality, which we will refer to as Mercer’s inequality.
Proposition 2. Under the assumptions of Proposition 1, the following inequality holds:where
There are many versions, variants, and generalizations of Propositions 1 and 2; see for example [4–7]. Here we state an integral version of Jensen’s inequality from [1, pages 58-59] which will be needed in our main result.
Proposition 3. Let be a continuous function. If the function is nondecreasing and bounded and , then for every continuous convex function the inequalityholds.
In our construction for next proposition, we recall the definitions of majorization.
For fixed , denote two real -tuples and letbe their ordered components.
Definition 4. For , when , is said to be majorized by or majorizes .
This notion and notation of majorization were introduced by Hardy et al. in [8].
The following extension of inequality (2) was given by Niezgoda in [6] which is referred to as Niezgoda’s inequality.
Proposition 5. Let be a continuous convex function. Suppose that with and is a real matrix such that for all and .
If majorizes each row of , that is, then we have the inequalitywhere with nonnegative weights .
The present paper is organized as follows: after some preliminaries, in Section 2, we recall definition of functions with nondecreasing increments and their properties and note that some inequalities from Section 1 which held true for convex functions also hold for functions with nondecreasing increments. In Section 3, we give an integral generalization of Niezgoda’s inequality. In the process, we will use an integral majorization result of Pečarić [9] and prove a result which gives the Jensen-Boas inequality on disjoint set of subintervals for functions with nondecreasing increments. In Section 4, we will discuss some refinements of the main results we proved in Section 3. The last part of this section is devoted to the applications of some related results.
2. Introduction to Functions with Nondecreasing Increments
In 1964, Brunk defined an interesting class of multivariate real valued functions [10] known as functions with nondecreasing increments.
Definition 6. A real valued function on a -dimensional rectangle , where is a fixed positive integer, is said to have nondecreasing increments if where partial order is defined by
In the same paper [10], Brunk gave some examples and properties of the functions which we discuss below.
2.1. Examples of Functions with Nondecreasing Increments
(i)The simplest example of a function with nondecreasing increments is a constant function.(ii)Lines of the form , where and whose direction cosines are nonnegative, also belong to the family of functions with nondecreasing increments.(iii)An important continuous function with nondecreasing increments is defined by : Another useful continuous function with nondecreasing increments is defined by .(iv)An interesting and widely used example of such functions is the Cauchy functional equation
2.2. Properties of Functions with Nondecreasing Increments
Functions with nondecreasing increments possess the following properties:(i)A function with nondecreasing increments is not necessarily continuous.(ii)If the first partial derivatives of a function exist for , then has nondecreasing increments if and only if each of these partial derivatives is nondecreasing in each argument.(iii)If the second partial derivatives of a function exist for , then has nondecreasing increments if and only if each of these partial derivatives is nonnegative.(iv)If a function with nondecreasing increments is continuous for , where , then the function defined by is convex.
We define here a special type of functions which belong to the class of functions with nondecreasing increments and which themselves contain the class of convex functions. These functions are called Wright convex functions [1, page 7].
Definition 7. We say is a Wright convex function if with and we have
Remark 8. It is easy to see that, in one-dimensional case, functions with nondecreasing increments are Wright convex functions. Also, continuous Wright convex functions are convex functions. Thus, the class of convex functions is a proper subclass of the Wright convex functions.
Now we state some results that will be needed to derive our main results. The following proposition gives Jensen’s inequality for functions with nondecreasing increments [11].
Proposition 9. Let be a continuous function with nondecreasing increments, let be a nonnegative -tuple such that , and let , where , be such that or . Then, it holds that
We now state Jensen-Steffensen’s inequality for functions with nondecreasing increments [12].
Proposition 10. is a nondecreasing continuous function and is of bounded variation satisfying If is a continuous function with nondecreasing increments, then the following inequality holds:where .
At this stage, we prove Jensen-Boas inequality for functions with nondecreasing increments as follows.
Theorem 11. Let be a continuous and monotonic (either nonincreasing or nondecreasing) map in each of the intervals . Let be continuous or of bounded variation satisfyingfor all , and . If is a continuous function having nondecreasing increments in each of the intervals , then we have the following inequality:
Proof. Using Jensen’s inequality (13) for nonnegative -tuple and Jensen-Steffensen’s inequality (15), if we have for .
If we consider and , then we can write using this fact, we have
The following proposition represents an integral majorization result which would be needed in our next main result [1, page 328].
Proposition 12. Let be two nonincreasing continuous functions and let be a function of bounded variation. If hold, then for every continuous function with nondecreasing increments the following inequality holds:
Remark 13. If are two nondecreasing continuous functions such that then again inequality (22) holds. In this paper, we will state our results for nonincreasing and satisfying the assumption of Proposition 12, but they are still valid for nondecreasing and satisfying the above condition (see, e.g., [13, page 584]).
3. Generalized Jensen-Mercer Inequality
Here we state a result needed in the main theorems of this section. The following lemma shows that the subintervals in the Jensen-Boas inequality (see Theorem 11) can be disjoint for the inequality of type (15) to hold.
Lemma 14. Let be continuous or a function of bounded variation and let be a partition of the interval ; and . If then, for every function which is continuous and monotonic (either nonincreasing or nondecreasing) in each of the intervals and every continuous function with nondecreasing increments , the following inequality holds:
Proof. Denote . Due to (16), if then is a null-measure on and , while otherwise . Denote and Notice that and, due to Proposition 10, Therefore, taking into account the discrete Jensen’s inequality (13),
The following theorem is our main result of this section and it gives a generalization of Proposition 5.
Theorem 15. Let , , , and be a function of bounded variation such that for all and and .
Furthermore, let be a measure space with positive finite measure , let be a nonincreasing continuous function, and let be a measurable function such that the mapping is nonincreasing and continuous for each :Then, for a continuous function with nondecreasing increments , the following inequality holds:
Proof. Using Fubini’s theorem, inequality (30), and Jensen’s integral inequality (4), we haveApplying Lemma 14 and Proposition 12, respectively, we have
The special case of Theorem 15 can be found in [14] which may be stated as follows.
Corollary 16. Let , , , and be a function of bounded variation such that for all and and .
Furthermore, let be a measure space with positive finite measure , let be a nonincreasing continuous function, and let be a measurable function such that the mapping is nonincreasing and continuous for each : Then, for a continuous convex function , the following inequality holds:
4. Refinements
Let be a measure space with positive finite measure . Throughout this section, we assume that with , and we use the following notations: The following refinement of (31) is valid.
Theorem 17. Under the assumptions of Theorem 15, the following refinement is valid for every continuous function with nondecreasing increments :where
Proof. Using discrete Jensen’s inequality (13) for functions with nondecreasing increments, we have for any , which proves the first inequality in (37).
By inequality (31), we also have for any , which proves the second inequality in (37).
Remark 18. Direct consequences of the previous theorem are the following two inequalities:
The special case of Theorem 17 can be found in [14] which may be stated as follows.
Corollary 19. Under the assumptions of Corollary 16, the following refinement is valid for every continuous convex function :where
4.1. Applications
Haluška and Hutník discussed a class of generalized weighted quasiarithmetic means in the integral form using the integral form of Jensen’s inequality [15]. In their work, they used the definition of quasiarithmetic nonsymmetrical weighted mean proposed by Feng [16] which we state below.
Let , where , be an interval. Let denote the vector space of all real Lebesgue measurable functions defined on the interval with the classical Lebesgue measure, and let denote the positive cone of , that is, the vector space of all real positive Lebesgue integrable functions on . Let denote the finite -norm of a function .
Definition 20. Let and be a real continuous and strictly monotone function. The generalized weighted quasiarithmetic mean of a function with respect to weight function is a number , wherewhere denotes the inverse of the function .
In what follows, is always a real continuous and strictly monotone function (in accordance with Definition 20). Means include many commonly used two-variable integral means as particular cases when taking the suitable functions , , and . For instance,(a)for (the identity function), we obtain the weighted arithmetic mean: (b)for , we have the weighted harmonic mean: (c)for , we get the weighted power mean of order : The case corresponds to the weighted geometric mean.
Under the assumptions of Corollary 19, we define the following notations where . Throughout this section, we also assume that and have the natural domain.
Arithmetic Mean. It is as follows:
Geometric Mean. It is as follows:
Harmonic Mean. It is as follows:
Power Mean. It is as follows: We now define a relationship between arithmetic and geometric means.
Theorem 21. Consider .
Proof. In (43), let to get In our notation, we have Further simplification gives us Using the property of gives us which can be written as Finally,
Here we obtain another relationship between geometric and arithmetic means.
Theorem 22. Consider .
Proof. Take ; if we replace with and with in (43), then we get Using the property of , we have In out notations, we have Finally,
The following theorem states a relationship between geometric and harmonic means.
Theorem 23. Consider .
Proof. Take ; replace with and with in (43) to get In our notations, we have which gives us Multiplying the last inequality above by , we get which can be written as Using the property of , we have Simplifying the above, we get Finally, we get
Now we define another relationship between geometric and harmonic means.
Theorem 24. Consider .
Proof. In (43), take and replace with and with to get Using the property of , we have Using the property of , we have In our notations, we have Finally,
Now we define a relationship between power mean and arithmetic mean.
Theorem 25. (i) For , we have(ii) For , the above inequalities are reversed.
Proof. (i) In (43), take and replace with and with to get In out notations, we have which can be written as Finally, (ii) If , then is concave, so the inequalities in (76) are reversed.
The following theorem gives another relationship between power and arithmetic means.
Theorem 26. Let , . (i)If , then(ii)For , the above inequalities are reversed.
Proof. (i) In (43), take and replace with and with to get In our notations, we have Further simplification gives usFinally, (ii) For , the function is concave, so the inequalities in (81) are reversed.
Competing Interests
The authors declare that they have no competing interests.