The article provides the generalization of Jensen’s inequality for convex functions on the line segments. The main and preliminary inequalities are expressed in discrete form using affine combinations that can be reduced to convex combinations. The resulting quasi-arithmetic means are used to extend the two well-known inequalities.
MSC:26A51, 26D15.
Hinweise
Competing interests
The author declares that they have no competing interests.
1 Introduction
First of all, we recapitulate the basic notations of affinity and convexity relating to the binomial combinations. Let be a real vector space.
A set is affine if it contains all binomial affine combinations in , that is, the combinations of points and coefficients of the sum . The affine hull of a set as the smallest affine set that contains is denoted . A function is affine if the equality
(1.1)
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holds for all binomial affine combinations in .
A set is convex if it contains all binomial convex combinations in , that is, the combinations of points and non-negative coefficients of the sum . The convex hull of a set as the smallest convex set that contains is denoted . A function is convex if the inequality
(1.2)
holds for all binomial convex combinations in .
Using mathematical induction, it can be proved that everything said above is true for all n-membered affine or convex combinations. In this way, Jensen (see [1]) has extended the inequality in equation (1.2).
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We will consider affine and convex combinations of the real numbers, the case where . Take real numbers a and b such that . Every number can be uniquely presented as the binomial affine combination
(1.3)
which is convex if, and only if, the number x belongs to the interval . Given the function , let be the function of the chord line passing through the points and of the graph of f. Applying the affinity of to the combination in equation (1.3), we get
(1.4)
If the function f is convex, then we have the inequality
(1.5)
and the reverse inequality
(1.6)
A wide area of convex analysis including convex functions and their inequalities is covered in [2]. The practical applications of convex analysis are presented in [3].
A brief scientific-historic background on Jensen’s inequality follows at the end of this introduction. Because of its attractiveness, Jensen’s and related inequalities were studied during the whole last century. So, there are Steffensen’s, Brunk’s, and Olkin’s inequality. In this century the research goes on, and we got Jensen-Mercer’s and Mercer-Steffensen’s inequality. For information as regards these inequalities, one may refer to papers [4‐7] and [8].
2 Generalization and reversal of Jensen-Mercer’s inequality
In order to generalize Jensen’s inequality, we firstly consider Jensen-Mercer’s inequality. A generalization of this inequality is given in Corollary 2.2, and the reverse inequality is specified in Corollary 2.4. The presentation of the section is based on the affine combinations in equation (2.1), and the simple inequality in equation (2.2).
Let be coefficients such that . Let be points. We consider the affine combination . Including the affine combination assuming that , we get the binomial form
(2.1)
Lemma 2.1Letbe coefficients such that . Letbe points such that .
Then the affine combinationis in , and every convex functionsatisfies the inequality
(2.2)
Proof The condition involves . Then the binomial combination of the right-hand side in (2.1) is convex since its coefficients and also . So, the combination belongs to .
Since the inequality in equation (2.2) is trivially true for , we suppose that . Applying the inequality in equation (1.5), and the affinity of , we get
(2.3)
(2.4)
(2.5)
because . □
The inequality in equation (2.2) can also be proved applying the convexity of the function f to the right-hand side of the representation in equation (2.1), wherein we use the inequality .
Corollary 2.2Letandbe coefficients such that . Letbe points such that all .
Then the affine combinationis in , and every convex functionsatisfies the inequality
(2.6)
As demonstrated above, to prove the inequality in equation (2.6) it is not necessary to use the Jensen inequality. Lemma 2.1 and Corollary 2.2 are also valid for because then the observed affine combinations with become convex, and the associated inequalities follow from Jensen’s inequality. The combinations including were observed in [[9], Corollary 11 and Theorem 12] additionally using a monotone function g.
If , then the inequality in equation (2.6) is reduced to Mercer’s variant of Jensen’s inequality obtained in [6]. Another generalization of Mercer’s result was achieved in [10] using the majorization assumptions. The importance and impact of the majorization theory to inequalities can be found in [11].
Lemma 2.3Letbe coefficients such that . Letbe points such that .
Then the affine combinationis not in , and every convex functionsatisfies the inequality
(2.7)
Proof The condition entails or , and the first coefficient of the binomial form of (2.1) satisfies if , or if . So, the combination does not belong to . Applying the inequality in (1.6), we get the series of inequalities as in equations (2.3), (2.4), and (2.5), but with the reverse inequality signs. □
It is not necessary to require in Lemma 2.3, it follows from the conditions and .
Corollary 2.4Letandbe coefficients such that . Letbe points such that anyand the convex combination .
Then the affine combinationis not in , and every convex functionsatisfies the inequality
(2.8)
3 Main results
The aim of the paper is to generalize the inequality in equation (2.2). The first step is already done in Corollary 2.2 in the expression of the inequality in equation (2.6), presenting the point c as the convex combination. We are going to proceed in the same way with the points a and b. The main theorem follows.
Theorem 3.1Letbe coefficients with the sums , , andsatisfyingand . Letbe points such that allwhere
(3.1)
Then the affine combination
(3.2)
belongs to , and every convex functionsatisfies the inequality
(3.3)
Iffis concave, then the reverse inequality is valid in equation (3.3). Iffis affine, then the equality is valid in equation (3.3).
Proof Note that . Prove the theorem for the convex function f.
If , the combination in equation (3.2) takes the convex form belonging to , and the inequality in equation (3.3) is reduced to Jensen’s inequality.
If , then, including points a, b, and
(3.4)
in equation (3.2), we get the combination which belongs to by Lemma 2.1. The inequality in equation (3.3) is trivially true for . So, we assume that and use the function . Applying the affinity of to the convex combination in equation (3.4), and respecting the inequalities , we have
(3.5)
Using the inequality in equation (2.4), applying Jensen’s inequality to and , and finally using the inequality in equation (3.5) respecting the minus, we get
completing the proof. □
Corollary 3.2Let all the assumptions of Theorem 3.1 be fully satisfied. Let , , andbe convex combinations having the common center with the affine combination in equation (3.2), that is,
(3.6)
Then every convex functionsatisfies the double inequality
(3.7)
Iffis concave, then the reverse inequalities are valid in equation (3.7). Iffis affine, then the equalities are valid in equation (3.7).
Proof Relying on the function and the Jensen inequality, we derive the series of inequalities
containing the double inequality in equation (3.7). □
Corollary 3.3Let all the assumptions of Theorem 3.1 be fully satisfied. Letbe a convex combination with pointssuch that
(3.8)
Then every convex functionsatisfies the inequality
(3.9)
We finish the section with the basic integral variant of the main theorem. In the corollary that follows we use segments with different endpoints.
Corollary 3.4Letbe coefficients such that . Letbe segments such that
Then the integral sum
(3.10)
belongs to , and every convex functionsatisfies the inequality
(3.11)
Iffis concave, then the reverse inequality is valid in equation (3.11). Iffis affine, then the equality is valid in equation (3.11).
Proof The sum of (3.10) is equal to the affine combination
(3.12)
belonging to by Lemma 2.1. Given the positive integer n, we can construct the affine combination
(3.13)
which approaches the integral sum in equation (3.10) as n approaches infinity. Let us show how it looks in the example of the first member of the above combination.
We make the partition where all subsegments have the same length , and the adjacent subsegments have a common endpoint. If we take points , then we have
(3.14)
suggesting to take .
Applying Theorem 3.1 to the affine combination in equation (3.13) and the convex function f, we get
(3.15)
and letting n to infinity, we obtain the integral inequality in equation (3.11). □
4 Application to quasi-arithmetic means
Let be an interval. Applying the Jensen inequality to the means, we use strictly monotone continuous functions such that the function is convex. In this case, we say that ψ is convex with respect to φ, or ψ is φ-convex. The same notation is used for the concavity. If the function ψ is φ-convex and φ-concave, we can say that it is φ-affine.
Suppose that
(4.1)
is an affine combination as in Theorem 3.1, and is a strictly monotone continuous function where . The discrete φ-quasi-arithmetic mean of the combination can be defined as the point
(4.2)
belonging to , because the affine combination enclosed in parentheses is located in . The quasi-arithmetic means defined in equation (4.2) are invariant with respect to the affinity, that is, the equality
(4.3)
holds for all pairs of real numbers and v.
The order of the pair of quasi-arithmetic means and depends on convexity of the function and monotonicity of the function ψ, as follows.
Corollary 4.1Letbe an affine combination as in equation (4.1) satisfying all the assumptions of Theorem 3.1. Letbe strictly monotone continuous functions where .
Ifψis eitherφ-convex and increasing orφ-concave and decreasing, then we have the inequality
(4.4)
Ifψis eitherφ-convex and decreasing orφ-concave and increasing, then we have the reverse inequality in equation (4.4).
Ifψisφ-affine, then the equality is valid in equation (4.4).
Proof We prove the case that the function ψ is φ-convex and increasing. Put the set . Applying the inequality in equation (3.3) of Theorem 3.1 to the affine combination
(4.5)
which is in the set , and the convex function , we get
Assigning the increasing function to the above inequality, we attain
which finishes the proof. □
The inequality in equation (4.4) may further be applied to the power means. The monotonicity of these power means is also valid. We will only derive the formula of the harmonic-geometric-arithmetic mean inequality.
Corollary 4.2Ifis an affine combination as in equation (4.1) satisfying all the assumptions of Theorem 3.1 with the addition that all , then we have the harmonic-geometric-arithmetic mean inequality
(4.6)
Proof Let , , and . Applying the inequality in equation (4.4), first to the pair and , and then to the pair and , we get representing the inequality in equation (4.6). □
The general approach to the theory of means and their inequalities was presented in [12]. Different forms of quasi-arithmetic means were considered in [13].
5 Application to other inequalities
The essential inequalities in equations (2.2) and (2.7) can be used to extend the Bernoulli and the Young inequality. Let us start with a simplified case of the mean inequality in equation (4.6).
Corollary 5.1If numberssatisfy , then the double inequality
(5.1)
holds for all numberssuch that .
Ifsatisfy , then the reverse double inequality is valid in equation (5.1) for allsuch that .
Proof The inequality in equation (5.1) follows from the inequality in equation (4.6) with , , and , , . So, it is actually a consequence of the inequality in equation (2.2).
The reverse inequality of equation (5.1) can be obtained from the inequality in equation (2.7) using the functions , , and . □
The previous corollary with a suitable choice of the coefficients and the points will be applied in the following two examples.
Example 5.2 If numbers satisfy , then the double inequality
(5.2)
holds for all numbers such that .
If satisfy , then the reverse double inequality is valid in equation (5.2) for all such that .
Proof Corollary 5.1 should be applied to the coefficients , , and the points , , . □
The inequality
(5.3)
applying to and follows from the inequality in equation (5.2) putting first and taking , then putting and taking . In the same way, from the reverse inequality of equation (5.2) we can obtain the reverse inequality of equation (5.3) respecting and . The right-hand sides of these inequalities are the well-known Bernoulli’s inequalities.
Example 5.3 If numbers and satisfy , then the double inequality
(5.4)
holds for all numbers such that .
If and satisfy , then the reverse double inequality is valid in equation (5.4) for all such that .
Proof Corollary 5.1 should be applied to the coefficients , , and the points , , . □
which applies to numbers with the sum , as well as to numbers . The right-hand side of the inequality in equation (5.5) represents the discrete form of Young’s inequality.
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