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On some inequalities for functions with nondecreasing increments of higher order
Journal of Inequalities and Applications volume 2013, Article number: 8 (2013)
Abstract
We investigate a class of functions with nondecreasing increments of higher order. A generalization of Brunk’s theorem is proved for that class of functions. Also, we consider functions with nondecreasing increments of order three, we obtain the Levinson-type inequality, a generalization of Burkill-Mirsky-Pečarić’s results, and a result for the integral mean of a function with nondecreasing increments of higher order.
1 Introduction
Let denote the k-dimensional vector lattice of points , be real for , with the partial ordering if and only if for . We denote
where , and k-tuple is denoted by 0.
For , , a set is called an interval . The following definition of a function with nondecreasing increments is given in [1].
Definition 1.1 A real-valued function f on an interval is said to have nondecreasing increments if
whenever , , , .
In the same paper [1], Brunk gave some properties of that family of functions. The most remarkable result for functions with nondecreasing increments is the following Brunk theorem (see also [[2], p.266]).
Theorem 1.2 Let I be an interval in ; be a vector of functions where ’s (), are nondecreasing and continuous from the right on . Let H be continuous from the left and of bounded variation on with . Then
holds for every continuous function with nondecreasing increments if and only if
and
where .
More results about functions with nondecreasing increments can be found in papers [3] and [4]. The following theorem is the Jensen-Steffensen type inequality for a function with nondecreasing increments and it is proved in [4].
Theorem 1.3 Let be a function of bounded variation such that
and let be a continuous nondecreasing map from the real interval to the interval . If is a continuous function with nondecreasing increments, then
where is the vector .
The following theorem gives us a Jensen-type inequality for a function with nondecreasing increments when the finite sequence of k-tuples is monotone in means [3]. It is a Pečarić’s generalization of Burkill-Mirsky’s result. Firstly, let us describe a monotonicity in means. Let , , be positive numbers, be an interval in . A finite sequence is said to be nondecreasing in means with respect to weights if
where
If inequalities are reversed in (4), then is nonincreasing in means.
Theorem 1.4 Let I be an interval in , be a continuous function with nondecreasing increments and let be positive numbers. If
is nondecreasing or nonincreasing in means with respect to weights , then the Jensen-type inequality
holds.
In this paper, we extend the idea of functions with nondecreasing increments. Namely, we define a new class of functions with nondecreasing increments of higher order and prove a result similar to the above-mentioned Brunk theorem. In the third section, we consider functions with nondecreasing increments of order three. Finally, in the last section, a result for an arithmetic integral mean of a function with nondecreasing increments of higher order is given.
2 Functions with nondecreasing increments of order n
Let I be an interval from . Let us write
and inductively,
where , (). Using this notation with , , , a condition (1) from the definition of a function with nondecreasing increments becomes
Let us extend that definition to the following.
Definition 2.1 A real-valued function f on an interval is a function with nondecreasing increments of order n if
whenever , ().
Brunk observed that even if and , this does not imply continuity (see [1]). Indeed, every solution of Cauchy’s equation is a function with nondecreasing increments of order n with null increments, i.e., . If the n th partial derivatives exist, they are nonnegative. If f is a continuous function with nondecreasing increments of order n, it may be approximated uniformly on I by polynomials having nonnegative n th partial derivatives. To see this, let us set, for convenience, , . It is known that the Bernstein polynomials
converge uniformly to f on I as , if f is continuous. Furthermore, if f is a function with nondecreasing increments of order n, these polynomials have nonnegative n th partial derivatives, as may be shown by repeated application of the formula (see [1])
The aim of the rest of this section is to prove a result similar to Theorem 1.2. Let us introduce some further notations.
Let be positive integers and let . Let be a set of all permutations with repetitions whose elements are from the multiset
There are elements in the class .
For , , let be a set whose elements are described in the following way. We say that permutation belongs to the set iff there exist , and permutation σ of the multiset such that . Family of all classes is denoted with .
For illustration, we describe the above notation on one example. Let and . Classes are the following: , , , and . Let us describe the elements of the set . There are three different permutations of the multiset . These are
So, are , , , where and . If, for example, , then it contains all permutations with repetitions of elements , i.e., and it has elements.
In the following text, H is a function of bounded variation on with and . Let be a function such that
and
Further we write
where S is a multiset with elements from .
It is obvious that
and
Now, the following result holds.
Lemma 2.2 Let w be a fixed positive integer. Then
holds for every .
Proof We prove it using induction by m. For , using integration by parts, we have
Let us suppose that the statement holds for and let us apply integration by parts on the right-hand side of the formula.
□
Especially for , we have
where , ; , .
Example 2.3 If , , , then
Furthermore, if we suppose
then
Theorem 2.4 Let be a continuous function. Let H be a function of bounded variation on with and let f have continuous th partial derivatives, . Then the following statement holds: if
then
Proof For , we have
If we have for , and if we suppose that (7) holds for , then
by (5) and (6). □
Theorem 2.5 Let X be a nondecreasing continuous map from the real interval into an interval , and let H be a function of bounded variation on with . Then
for every continuous function f with nondecreasing increments of order n on I if and only if
for , and
for all , .
Proof Necessity: The validity of (8) for constant functions and implies (9). From (8) for and (), we have (10).
Inequality (11) is obtained from (8) on setting, for fixed and fixed ,
Sufficiency: Since f may be approximated uniformly on I by functions with continuous nonnegative n th partial derivatives, we may assume that the n th partials exist and are continuous and nonnegative. By Theorem 2.4 and (10), we have
By (11), each term in the sum is nonnegative so that (8) is verified. □
3 Functions with nondecreasing increments of order three
3.1 On inequalities of Levinson type
Levinson [5] proved that if a real-valued function f defined on has a nonnegative third derivative, then
for , , (), .
If , and , then Levinson’s inequality (12) becomes the famous Ky-Fan inequality
where , , and .
In [6] Pečarić showed that instead of variables the sum of which is equal to 2a, we can use variables the difference of which is constant, and that result becomes a source of some further generalizations [[2], pp.74, 75]. In fact, he proved that if f is a real-valued 3-convex function on and , (), 2n points on such that
and (), then (12) is valid.
The following theorem is a generalization of the Levinson inequality.
Theorem 3.1 Let be a function of bounded variation such that (2) holds, and let be a continuous and nondecreasing map from to an interval , . If f is a continuous function with nondecreasing increments of order three on , then
Proof If f is a function with nondecreasing increments of order three on J, then
i.e.,
If and , we have
i.e., the function is a function with nondecreasing increments of order two, i.e., it is a function with nondecreasing increments. Now, using Theorem 1.3, we obtain Theorem 3.1. □
Theorem 3.2 Let be a function of bounded variation such that (2) holds, and let f be a continuous function with nondecreasing increments of order three on . Let . If is a continuous and nondecreasing map, then
Proof Using (13) for , we have that the function is a function with nondecreasing increments, so from Theorem 1.3, we obtain Theorem 3.2. For , we have a result from [6]. □
Corollary 3.3 (i) Let X satisfy the assumptions of Theorem 3.1. Then
(ii) If X satisfies the assumptions of Theorem 3.2, then
where all components of X are nonnegative.
Proof The function is a function with nondecreasing increments of orders two and three for (). So, using Theorems 1.3, 3.1, and 3.2, we obtain Corollary 3.3. □
3.2 Generalization of Burkill-Mirsky-Pečarić result
In this subsection, we consider a sequence of k-tuples which is monotone in means.
Theorem 3.4 Let f be a continuous function with nondecreasing increments of order three on , , and let be positive numbers. If
is nondecreasing or nonincreasing in means with respect to positive weights (), then
holds.
Proof Since f is a function with nondecreasing increments of order three on J, so a function is a function with nondecreasing increments. Then by Theorem 1.4, we obtain the required result. □
Theorem 3.5 Let f be a continuous function with nondecreasing increments of order three on and let be positive numbers. Let . If
is nondecreasing or nonincreasing in means with respect to positive weights (), then
holds.
Proof By following the proof of Theorem 3.2, we obtain Theorem 3.5 by simply replacing ‘Theorem 1.3’ by ‘Theorem 1.4’ in the proof of Theorem 3.2. □
Corollary 3.6 (i) Let X satisfy the assumptions of Theorem 3.4. Then
(ii) If X satisfies the assumptions of Theorem 3.5. Then
where all components of X are nonnegative.
Proof We consider again the function which is a function with nondecreasing increments of orders two and three for (). So, using Theorems 1.4, 3.4, and 3.5, we obtain Corollary 3.6. □
4 Arithmetic integral mean
It is known that if , , is nonnegative and nondecreasing, then the function F,
is also a nondecreasing function on . Let us observe that F is an arithmetic integral mean of a function f on an interval . This result was generalized in [7] considering a real-valued function f for which holds for any . is defined as follows: , .
Here, we extend the above-mentioned result to functions with nondecreasing increments of higher order.
Theorem 4.1 Let the function be continuous and with nondecreasing increments of order n. Then the function
where and , is a function with nondecreasing increments of order n on .
Proof Let . Then
where we used the substitutions (, ), and where , . Now, we have
because if is a function with nondecreasing increments of order n, then the function is also a function with nondecreasing increments of order n. □
References
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Acknowledgements
This research work is funded by the Higher Education Commission, Pakistan. The research of the second and third authors was supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888 and 058-1170889-1050.
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JP made the main contribution in conceiving the presented research. JP and SV worked jointly on each section while ARK worked on first and third section and drafted the manuscript. All authors read and approved the final manuscript.
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Khan, A.R., Pečarić, J. & Varošanec, S. On some inequalities for functions with nondecreasing increments of higher order. J Inequal Appl 2013, 8 (2013). https://doi.org/10.1186/1029-242X-2013-8
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DOI: https://doi.org/10.1186/1029-242X-2013-8