Properties of some functionals associated with h-concave and quasilinear functions with applications to inequalities
© Nikolova and Varošanec; licensee Springer. 2014
Received: 11 October 2013
Accepted: 7 January 2014
Published: 24 January 2014
We consider quasilinearity of the functional , where Φ is a monotone h-concave (h-convex) function, v and g are functionals with certain super(sub)additivity properties. Those general results are applied to some special functionals generated with several inequalities such as the Jensen, Jensen-Mercer, Beckenbach, Chebyshev and Milne inequalities.
1 Introduction and preliminaries
In [1–4] Dragomir researched functionals which arise from quasilinear functionals related to the classical inequalities. For example, he considered the functionals (in ), , (both in ), (in ), and finally, (in ), where v is additive, g is super(sub)additive, Φ is a concave (convex) function and p and q are real numbers with some properties. In each paper he applied the given results about a composite functional to some of the classical inequalities such as: the Jensen, Hölder or Minkowski inequalities.
In this paper we generalize his results. We investigate similar functionals related to an h-convex function Φ under assumptions, which are weaker than the assumptions in the above-mentioned papers. In this introductory section, we give definitions and basic properties of several classes of functions.
, imply .
for any .
If , then a functional f is simply called superadditive (subadditive).
for any and .
In particular, if , then we simply say that f is positive homogeneous on C of order k. If , we call it positive homogeneous.
It is easy to see that and K is multiplicative. Moreover, we have either or .
for all .
If inequality (1) is reversed, then h is said to be a submultiplicative function. If equality holds in (1), then h is said to be a multiplicative function.
superadditive and L-subadditive with for ,
subadditive and L-superadditive with for ,
L-subadditive with for .
A function is called starshaped if for any , .
In the sequel I and J are intervals in R, and functions h and f are non-negative functions defined on J and I, respectively.
Definition 2 
If the inequality is conversed, then f is called an h-concave function.
Some properties of the h-convex functions are given in papers  and . It is evident that this concept of h-convexity generalizes the concepts of classical non-negative convexity (for ); s-convexity in the second sense (for , ) [7, 8]; P-functions (for ) ; and Godunova-Levin functions (for ) .
Example 3 It is known (see ) that the function is s-convex in the second sense if
( and ) or ( and ).
The function is s-concave in the second sense if
(, ) or (, ).
Proposition 4 Let and be s-convex, , . Then , , and f is -superadditive.
If , we get that f is starshaped and hence superadditive. □
Here we extract some properties proved in , which we use in this paper. Firstly, we give a result which allows us to use weights α, β, the sum of which is not exactly 1 (as we have to use when we are working with convex functions) (see [, Theorem 12]).
- (a)Let f be h-convex on I and . If the function h is supermultiplicative, then the inequality(2)
- (b)Let f be h-concave on I. If the function h is submultiplicative, then the inequality(3)
holds for all and all with .
In fact, in the original paper  in part (b) we have assumption that but a careful introspect of the proof gives us that non-negativity of the function f is a sufficient assumption.
The next proposition gives us a tool for generating new h-convex functions.
Let f be convex on I and . If f is non-increasing and , then is s-convex.
Let f be concave on I. If f is non-decreasing and , then is s-concave.
If f is non-decreasing, , then is h-concave with .
which means that is an h-concave function.
The other cases can be proved similarly. □
Another way to generate new s-convex (s-concave) functions is using of the following statements: If f is non-negative convex, then , is s-convex. If f is non-negative concave, , then is s-concave.
- (iii)Consider the function
where . This function is non-decreasing, convex on , and starshaped on , . Here we will show that this function is s-concave on , .
since , . After considering all possibilities, we conclude that f is s-concave.
The paper is arranged as follows. In the following section we prove some results about the functional , where Φ is a monotone h-convex (h-concave) function. In the third section some applications of those results to the Jensen, Jensen-Mercer and Beckenbach functionals are given. Specific assumption in the third section is that the function v is additive. The fourth section is devoted to some new superadditive mappings such as the Chebyshev and Milne functionals. A characteristic assumption in that section is that v is superadditive.
2 Functionals associated with monotone h-concave and h-convex functions
The above-proved lemma is a generalization of a result from  in which f is superadditive and positive homogeneous of order s.
- (i)If h is submultiplicative, is an L-superadditive (L-subadditive) functional on C and is h-concave and non-decreasing (non-increasing), then the functional defined by
If h is supermultiplicative, g is L-subadditive, Φ is h-convex and non-decreasing with , then is -subadditive.
Hence is -superadditive. The proofs of the other cases follow in a similar manner. □
A superadditive and non-negative functional has the following boundedness property.
Corollary 10 Let h be a non-negative submultiplicative function which is -positive homogeneous. Let C be a convex cone in the linear space X, and let be L-superadditive and -positive homogeneous on C. Let and assume that there exist such that and . Let .
Proof Note that . We observe that if v and g are -positive homogeneous functionals, then is a K-positive homogeneous functional and, by Theorem 9, it follows that is a -superadditive functional on C. By applying Lemma 8 we get the result. □
Corollary 11 Let h be a non-negative submultiplicative function which is positive homogeneous of order . Let C be a convex cone in the linear space X and be L-superadditive and positive homogeneous of order on C. Let and assume that there exist such that and .
Proof Put in the previous corollary , , and . □
Remark 12 If , then the assumption about homogeneity of h can be omitted and the statement of Theorem 9 still holds, namely we get superadditivity (subadditivity) of .
If g is superadditive (subadditive), Φ is h-concave and non-decreasing (non-increasing), where h is submultiplicative, then the functional is superadditive.
If g is superadditive (subadditive), Φ is h-convex and non-increasing (non-decreasing), where h is supermultiplicative, then the functional is subadditive.
Comparing these statements with the results of Theorem 5 from paper , we see that if Φ is a non-negative function, then we have results for a wider class of functions Φ, i.e., for h-concave or h-convex functions.
The case , gives results for concave Φ, as it is in , but for v and g superadditive and -positive homogeneous. The case when v is only superadditive is important for applications - see the application to the Chebyshev and Milne functionals.
Moreover, Corollary 10 under assumptions that v is additive and , , becomes the same as Corollary 1(a) from .
More about Corollary 10: If , , and we use as an example , , is concave non-decreasing, then we get the result of Corollary 1 from . However, we can also use the functions from Example 7 to get new results.
3 Case 1: function v is additive
Applications to Jensen-type inequalities
Let f be a real mapping on a convex subset of a linear space. Let us fix and (), and let . is a convex cone.
As an application of the results from the previous section, we have the following theorem.
Proof Take and . The functionals v and g are positive homogeneous, v is additive and g is superadditive. Using Theorem 9 we get that the composite functional is superadditive on and k-positive homogeneous. Hence, we apply Lemma 8 and get the wanted inequalities. □
Remark 14 If , then we get results from . In the same paper Dragomir discussed applications involving the Hölder and Minkowski functionals, but we leave that direction of investigation at this moment. In the rest of this section, we want to point out applications to some other Jensen-type functionals.
On the Jensen-Steffensen conditions
The set is a cone. By the Jensen-Steffensen inequality [, p.57], the difference , where f is convex on I, is non-negative for each . Using a similar proof as for the Jensen functional on , we get that J is superadditive for a convex function f and applying Theorem 9 we obtain that the functional given by (4) is superadditive and the corresponding boundedness property holds.
The boundedness property of the Jensen functional under the Jensen-Steffensen conditions with an additional normalizing property was proved in  by using a different method.
Applications to the Jensen-Mercer functional
Mercer in paper  proved the Jensen-type inequality which includes boundary points of an interval. Namely, he stated the following result, which is nowadays called the Jensen-Mercer inequality.
If f is concave, then the reversed inequality holds.
It is easy to see that the functional JM is positive homogeneous, non-negative for a convex function f and non-positive for a concave function f. Superadditivity of the functional JM was considered in . In fact we have the following result.
Theorem 15 
If f is convex, then the Jensen-Mercer functional JM is superadditive on .
If f is concave, then JM is subadditive on .
Applying results of the second section, we have the following theorem.
is superadditive on .
Hence, by applying Theorem 9, we get the desired result. □
The proof follows from Corollary 10.
Applications to the Beckenbach functional
where and a, b, f satisfy assumptions of the Beckenbach inequality.
The above-mentioned theorem shows that .
Proposition 18 The functional is superadditive.
Proof Consider the functionals and . The functional v is additive and g is superadditive, and applying Theorem 9 we get that is superadditive. The boundedness property follows from Corollary 10. □
4 Case 2: function v is superadditive
In the previous section the function v was additive. Now, we will show some examples of applications with a superadditive function v.
Applications to the Chebyshev functional for sums
for any . If the above inequality is reversed, then n-tuples are called oppositely ordered.
The statement of the classical Chebyshev inequality is the following (see [, pp.197-204]).
holds. If and are oppositely ordered, then the reversed inequality holds.
In the following theorem, we consider a quasilinear property of the Chebyshev functional .
Theorem 20 If and are similarly ordered real n-tuples, , then the functional is superadditive in the variable . If and are oppositely ordered real n-tuples, then the functional is subadditive.
which means that is superadditive. If and are oppositely ordered, the proof is similar. □
Let us apply results from the second section to the functional .
If and are similarly ordered, then the functional is superadditive on .
If and are oppositely ordered, then the functional is superadditive on . If, additionally, the assumptions on h, , , M and m are satisfied as in case (i), then inequalities (5) hold with substitution .
Proof If and are similarly ordered, let us define v and g as and . These functionals are positive homogeneous of order 2 and superadditive. By Theorem 9 with we have that is superadditive, and by Corollary 10 for the functional we obtain inequality (5).
If and are oppositely ordered, then the functional is superadditive and non-negative, and we proceed as in the proof of case (i). □
If and are oppositely ordered, then the reversed inequalities in (6) and (7) hold.
Let us take , , . Since we can use the above monotonicity to obtain the following result.
If and are oppositely ordered, then the reversed inequalities in the above inequalities hold with substitution in the second result.
The Chebyshev functional for integrals
when f and g are similarly ordered. So we have superadditivity of this functional on the cone . If f and g are oppositely ordered, then the functional is superadditive. Here we will show only how Corollary 10 can be applied to this situation.
Proof Let the function v be defined by . It is superadditive and positive homogeneous of order . The function g will be the Chebyshev functional . It is also positive homogeneous of order , superadditive and non-negative. By Corollary 10 for the functional with , , we obtain the wanted inequality. □
If f and g are oppositely ordered, then the reversed inequalities hold.
Applications to the Milne functional
where are positive real numbers. Of course, it can be improved to non-negative weights.
The weighted Milne inequality means that . Also, it is easy to see that , i.e., is positive homogeneous of order 2.
Theorem 26 The functional is superadditive on .
which means that is superadditive and the proof is complete. □
is superadditive and it has boundedness property which follows from Corollary 10. As in Remark 25, we have the following chain of inequalities.
The research of the first author was partially supported by the Sofia University SRF under contract No. 184/2013. The research of the second author was supported by the Ministry of Science, Education and Sports of the Republic of Croatia under grants 058-1170889-1050.
- Dragomir SS: Inequalities for superadditive functionals with applications. Bull. Aust. Math. Soc. 2008, 77: 401-411.MATHMathSciNetGoogle Scholar
- Dragomir SS: Quasilinearity of some composite functionals with applications. Bull. Aust. Math. Soc. 2011, 83: 108-121. 10.1017/S0004972710000444MATHMathSciNetView ArticleGoogle Scholar
- Dragomir SS: The quasilinearity of a family of functionals in linear spaces with applications in to inequalities. Riv. Mat. Univ. Parma 2013, 4: 135-149.MATHGoogle Scholar
- Dragomir SS: Quasilinearity of some functionals associated with monotonic convex functions. J. Inequal. Appl. 2012., 2012: Article ID 276Google Scholar
- Varošanec S: On h -convexity. J. Math. Anal. Appl. 2007, 326: 303-311. 10.1016/j.jmaa.2006.02.086MATHMathSciNetView ArticleGoogle Scholar
- Bombardelli M, Varošanec S: Properties of h -convex functions related to the Hermite-Hadamard-Fejer inequalities. Comput. Math. Appl. 2009, 58: 1869-1877. 10.1016/j.camwa.2009.07.073MATHMathSciNetView ArticleGoogle Scholar
- Breckner WW: Stetigkeitsaussagen fur eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Raumen. Publ. Inst. Math. 1978, 23: 13-20.MathSciNetGoogle Scholar
- Hudzik H, Maligranda L: Some remarks on s -convex functions. Aequ. Math. 1994, 48: 100-111. 10.1007/BF01837981MATHMathSciNetView ArticleGoogle Scholar
- Pearce CEM, Rubinov AM: P -Functions, quasi-convex functions and Hadamard-type inequalities. J. Math. Anal. Appl. 1999, 240: 92-104. 10.1006/jmaa.1999.6593MATHMathSciNetView ArticleGoogle Scholar
- Godunova EK, Levin VI: Neravenstva dlja funkcii širokoga klassa, soderžaščego vypuklye, monotonnye i nekotorye drugie vidy funkcii. In Vyčislitel. Mat. i. Mt. Fiz. Mežvuzov. Sb. Nauč. Trudov.. MGPI, Moscow; 1985:138-142. (Russian)Google Scholar
- Barić J, Matić M, Pečarić J: On the bounds for the normalized Jensen functional and Jensen-Steffensen inequality. Math. Inequal. Appl. 2009, 12: 413-432.MathSciNetGoogle Scholar
- Dragomir SS: Bounds for the normalized Jensen’s functional. Bull. Aust. Math. Soc. 2006, 74: 471-478. 10.1017/S000497270004051XMATHView ArticleGoogle Scholar
- Dragomir SS, Pečarić J, Persson LE: Properties of some functionals related to Jensen’s inequality. Acta Math. Hung. 1996, 70: 129-143. 10.1007/BF00113918MATHView ArticleGoogle Scholar
- Pečarić JE, Proschan F, Tong YL: Convex Functions, Partial Orderings, and Statistical Applications. Academic Press, Boston; 1992.MATHGoogle Scholar
- Mercer AMcD: A variant of Jensen’s inequality. J. Inequal. Pure Appl. Math. 2003.,4(4): Article ID 73Google Scholar
- Krnić M, Lovričević N, Pečarić J: On some properties of Jensen-Mercer’s functional. J. Math. Inequal. 2012, 6: 125-139.MATHMathSciNetView ArticleGoogle Scholar
- Beckenbach EF: Superadditivity inequalities. Pac. J. Math. 1964, 14: 421-438. 10.2140/pjm.1964.14.421MATHMathSciNetView ArticleGoogle Scholar
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1952.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.