Abstract
In this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We essentially generalize and extend these results by using permutations of finite sets and bijections of the set of positive numbers. We get refinements of the discrete Jensen’s inequality for infinite convex combinations in Banach spaces. Similar results are rare. Finally, some applications are given on different topics.
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1 Introduction
Different variants of Jensen’s inequality and other inequalities have their origin in the notion of convexity. A real function f defined on a convex subset C of a real vector space is called convex if it satisfies
for all \(v_{1},v_{2}\in C\) and all \(\alpha \in \left[ 0,1\right] \).
The set of positive integers will be denoted by \(\mathbb {N}_{+}\).
The following versions of Jensen’s inequality are well known.
Theorem 1.1
(discrete Jensen’s inequalities, see [11] and [13]) (a) Let C be a convex subset of a real vector space V, and let \(f:C\rightarrow \mathbb {R}\) be a convex function. If \(p_{1},\ldots ,p_{n}\) are nonnegative numbers with \(\sum \nolimits _{i=1}^{n}p_{i}=1\), and \(v_{1},\ldots ,v_{n}\in C\), then
(b) Let C be a closed convex subset of a real Banach space V, and let \(f:C\rightarrow \mathbb {R}\) be a convex function. If \(p_{1},p_{2},\ldots \) are nonnegative numbers with \(\sum \nolimits _{i=1}^{\infty }p_{i}=1\), and \(v_{1},v_{2},\ldots \in C\) such that the series \(\sum \nolimits _{i=1}^{\infty }p_{i}v_{i} \) and \(\sum \nolimits _{i=1}^{\infty }p_{i}f\left( v_{i}\right) \) are absolutely convergent, then
To give refinements of the discrete Jensen’s inequality (1.1) is an extensively investigated theme with numerous methods and results (see e.g. the book [8] and references therein), and applications (see e.g. [5] and [6]). Then again, to the best of my knowledge, there are no refinements of the discrete Jensen’s inequality (1.2) in such generality. There are some refinements of (1.2) when C is an interval of \(\mathbb {R}\): either one estimates formulas in (1.2) in a suitable way (see [12]) or one can obtain results from refinements of integral Jensen’s inequality (see [7]).
The following refinement of (1.1) can be found in [9] (see also [1]).
Theorem 1.2
Let \(2\le k\le n\) be integers, and let \(p_{1},\ldots ,p_{n}\) and \(\lambda _{1},\ldots ,\lambda _{k}\) be positive numbers with \(\sum \nolimits _{i=1}^{n}p_{i}=1\) and \(\sum \nolimits _{i=1}^{k}\lambda _{i}=1\). If C is a convex subset of a real vector space V, \(f:C\rightarrow \mathbb {R}\) is a convex function, and \(v_{1},\ldots ,v_{n}\in C\), then
where \(i+j\) means \(i+j-n\) in case of \(i+j>n\).
It is easy to think that the previous result cannot be generalized for infinite sums, but we can observe that the middle term \(C_{dis}\) in (1.3) can be rewritten in the following form
where \(\pi _{j}\) \(\left( j=1,\ldots ,k\right) \) is the \(\left( j-1\right) \)-cyclic permutation of the set \(\left\{ 1,\ldots ,n\right\} \) to the right (all elements are moved to the right \(j-1\) times with elements overflowing from the right being inserted to the left).
In this paper we show that formulas like (1.4) refine both (1.1) and (1.2) by using either permutations of the set \(\left\{ 1,\ldots ,n\right\} \) or bijections from \(\mathbb {N}_{+}\) onto itself. On the one hand, an essential generalization of Theorem 1.2 is given, on the other hand, refinements of (1.2) are developed without assuming that V is a special Banach space. Finally, we give some applications concerning information theory, the norm function, Hölder’s inequality and the inequality of arithmetic and geometric means.
2 Main results
The positive part \(f^{+}\) and the negative part \(f^{-}\) of a real valued function f are defined in the usual way.
Let the set I denote either \(\left\{ 1,\ldots ,n\right\} \) for some \(n\ge 1\) or \(\mathbb {N}_{+}\). We say that the numbers \(\left( p_{i}\right) _{i\in I}\) represent a (positive) discrete probability distribution if \(\left( p_{i}>0\right) \) \(p_{i}\ge 0\) \(\left( i\in I\right) \) and \(\sum \nolimits _{i\in I}p_{i}=1\). A permutation \(\pi \) of I refers to a bijection from I onto itself.
We need the following hypotheses which are partitioned into two classes:
- (\(\hbox {H}_{{1}}\)):
-
Let \(k,n\ge 2\) be integers, and let \(p_{1},\ldots ,p_{n}\) and \(\lambda _{1},\ldots ,\lambda _{k}\) represent positive probability distributions.
- (\(\hbox {H}_{{2}}\)):
-
For each \(j=1,\ldots ,k\) let \(\pi _{j}\) be a permutation of the set \(\left\{ 1,\ldots ,n\right\} \).
- (\(\hbox {H}_{{3}}\)):
-
Let C be a convex subset of a real vector space V, and \(f:C\rightarrow \mathbb {R}\) be a convex function.
- (\(\hbox {C}_{{1}}\)):
-
Let the set J denote either \(\left\{ 1,\ldots ,k\right\} \) for some \(k\ge 2\) or \(\mathbb {N}_{+}\). Let \(p_{1},p_{2},\ldots \) and \(\left( \lambda _{j}\right) _{j\in J}\) represent positive probability distributions.
- (\(\hbox {C}_{{2}}\)):
-
For each \(j\in J\) let \(\pi _{j}\) be a permutation of the set \(\mathbb {N}_{+}\).
- (\(\hbox {C}_{{3}}\)):
-
Let C be a closed convex subset of a real Banach space \(\left( V,\left\| \cdot \right\| \right) \), and \(f:C\rightarrow \mathbb {R}\) be a convex function.
Theorem 2.1
-
(a)
Assume (\(\hbox {H}_{{1}}\)), (\(\hbox {H}_{{2}}\)) and (\(\hbox {H}_{{3}}\)). If \(v_{1},\ldots ,v_{n}\in C\), then
$$\begin{aligned} f\left( \sum \limits _{i=1}^{n}p_{i}v_{i}\right)\le & {} C_{per}=C_{per}\left( f,\mathbf {v,p},{{\varvec{\lambda }},{\varvec{\pi }}}\right) \nonumber \\ := & {} \sum \limits _{i=1}^{n}\left( \sum \limits _{j=1}^{k}\lambda _{j}p_{\pi _{j}\left( i\right) }\right) f\left( \frac{\sum \nolimits _{j=1}^{k} \lambda _{j}p_{\pi _{j}\left( i\right) }v_{\pi _{j}\left( i\right) }}{\sum \nolimits _{j=1}^{k}\lambda _{j}p_{\pi _{j}\left( i\right) }}\right) \nonumber \\\le & {} \sum \limits _{i=1}^{n}p_{i}f\left( v_{i}\right) . \end{aligned}$$(2.1) -
(b)
Assume (\(\hbox {C}_{{1}}\)), (\(\hbox {C}_{{2}}\)) and (\(\hbox {C}_{{3}}\)). If \(v_{1},v_{2} ,\ldots \in C\) such that the series \(\sum \nolimits _{i=1}^{\infty } p_{i}v_{i}\) and \(\sum \nolimits _{i=1}^{\infty }p_{i}f\left( v_{i}\right) \) are absolutely convergent, then
$$\begin{aligned} f\left( \sum \limits _{i=1}^{\infty }p_{i}v_{i}\right)\le & {} C_{per} =C_{per}\left( f,\mathbf {v,p},{{\varvec{\lambda }},{\varvec{\pi }}}\right) \nonumber \\ := & {} \sum \limits _{i=1}^{\infty }\left( \sum \limits _{j\in J}\lambda _{j}p_{\pi _{j}\left( i\right) }\right) f\left( \frac{\sum \nolimits _{j\in J}\lambda _{j}p_{\pi _{j}\left( i\right) }v_{\pi _{j}\left( i\right) }}{\sum \nolimits _{j\in J}\lambda _{j}p_{\pi _{j}\left( i\right) }}\right) \nonumber \\\le & {} \sum \limits _{i=1}^{\infty }p_{i}f\left( v_{i}\right) . \end{aligned}$$(2.2)
Proof
(a) By using Theorem 1.1 (a) and the fact that \(\pi _{j}\) is a permutation of the set \(\left\{ 1,\ldots ,n\right\} \),
The left hand side inequality can be proved similarly. Since
the discrete Jensen’s inequality implies that
(b) The proof is divided into four parts.
I. We first prove that the series
and
are convergent and
and
For each \(j\in J\) the series
is a rearrangement of the absolutely convergent series \(\sum \nolimits _{i=1} ^{\infty }p_{i}v_{i}\), and hence it is also absolutely convergent and
(i) If \(J=\left\{ 1,\ldots ,k\right\} \), then it follows trivially from (2.7) that
(ii) Assume \(J=\mathbb {N}_{+}\). The property of absolute convergence of (2.6) implies that
Therefore, as it is well known, the order of summation can be interchanged in the double sum
and hence
The series in (2.4) can be handled in a similar way.
II. (i) If \(J=\left\{ 1,\ldots ,k\right\} \), then Theorem 1.1 (a) gives us that for every \(n\in \mathbb {N}_{+}\)
(ii) Assume \(J=\mathbb {N}_{+}\). For each \(i\in \mathbb {N}_{+}\) the series \(\sum \nolimits _{j=1}^{\infty }\lambda _{j}p_{\pi _{j}\left( i\right) }\) is obviously convergent and
and hence the series
is absolutely convergent. Further, we know from part I that the series
is also absolutely convergent.
Consequently, Theorem 1.1 (b) shows that
We have seen in part I that the series (2.4) is convergent and its sum is \(\sum \nolimits _{i=1}^{\infty }p_{i}f\left( v_{i}\right) \). Thus by (2.8) or by (2.9), the second inequality in (2.2) will be proved if we succeed in showing that the series
is convergent.
It is known that the positive part of f is also convex. The convergence of (2.4) implies the convergence of the series
It now follows that we can copy the proofs of (2.8) and (2.9) with \(f^{+}\) instead of f. Taking account of the nonnegativity of \(f^{+}\), we obtain
From this it follows that the series (2.10) will be convergent (absolutely) if and only if
III. In this step we show that the series (2.12) is convergent and the first inequality in (2.2) holds assuming f is bounded below, that is \(f\left( v\right) \ge c\) \(\left( v\in C\right) \) for some nonpositive number c.
Since
the series (2.12) is convergent.
According to (2.11) and (2.12) the series (2.10) is absolutely convergent.
Since the series (2.3) and (2.10) are absolutely convergent and (2.5) holds, we can apply Theorem 1.1 (b), and obtain the first inequality in (2.2).
IV. At this point we abandon the lower boundedness hypothesis on f.
Let the function \(f_{n}:C\rightarrow \mathbb {R}\) be defined by
Then \(f_{n}\) \(\left( n\in \mathbb {N}_{+}\right) \) is convex and bounded below, and \(f_{n}\ge f\) \(\left( n\in \mathbb {N}_{+}\right) \). From this and from the results of part III, we get that for each \(n\in \mathbb {N}_{+}\) the series
is absolutely convergent and
Since the sequence \(\left( f_{n}\right) _{n\in \mathbb {N}_{+}}\) is decreasing and \(\lim \nolimits _{n\rightarrow \infty }f_{n}=f\) pointwise, the previous two assertions imply that B. Levi’s theorem can be applied, and it gives that the series (2.10) is absolutely convergent and the first inequality in (2.2) holds.
The proof is complete. \(\square \)
Remark 2.2
It can be seen that Theorem 2.1 (a) contains Theorem 1.2 as a special case.
3 Applications
We begin with some inequalities corresponding to information theory.
The following notion was introduced by Csiszár in [2] and [3].
Definition 3.1
Let \(f:\left] 0,\infty \right[ \rightarrow \left] 0,\infty \right[ \) be a convex function, and let \(\mathbf {r}:=\left( r_{1},\ldots ,r_{n}\right) \in \left] 0,\infty \right[ ^{n}\) and \(\mathbf {q}:=\left( q_{1},\ldots ,q_{n}\right) \in \left] 0,\infty \right[ ^{n}\). The f-divergence functional is
Based on this concept, we have introduced a new functional in [10], and this functional can be further generalized:
Definition 3.2
Let C be a convex subset of a real vector space V, and \(f:C\rightarrow \mathbb {R}\) be a convex function. If \(\mathbf {w}:=\left( w_{1},\ldots ,w_{n}\right) \in V^{n}\) and \(\mathbf {q}:=\left( q_{1} ,\ldots ,q_{n}\right) \in \left] 0,\infty \right[ ^{n}\) such that
then define
Proposition 3.3
Let \(k,n\ge 2\) be integers, and let \(\lambda _{1},\ldots ,\lambda _{k}\) represent a positive probability distribution. Assume (\(\hbox {H}_{{2}}\)) and (\(\hbox {H}_{{3}}\)). If \(\mathbf {w}:=\left( w_{1},\ldots ,w_{n}\right) \in V^{n}\) and \(\mathbf {q}:=\left( q_{1},\ldots ,q_{n}\right) \in \left] 0,\infty \right[ ^{n}\) such that
then
Proof
By applying Theorem 2.1 (a) with
we have
The proof is complete. \(\square \)
Remark 3.4
(a) It was proved in [4] that if \(f:\left] 0,\infty \right[ \rightarrow \left] 0,\infty \right[ \) is a convex function, and \(\mathbf {r}:=\left( r_{1},\ldots ,r_{n}\right) \in \left] 0,\infty \right[ ^{n}\) and \(\mathbf {q}:=\left( q_{1},\ldots ,q_{n}\right) \in \left] 0,\infty \right[ ^{n}\), then
From Proposition 3.3 we can obtain the following refinement of this inequality:
(b) Let \(f:=-\log \), where the base of \(\log \) is greater than 1, \(\mathbf {r}:=\left( 1,\ldots ,1\right) \), and \(\mathbf {q}:=\left( q_{1},\ldots ,q_{n}\right) \) represent a positive probability distribution. Then (3.3) gives
which is a refinement of a remarkable inequality for the Shannon entropy.
Next we establish inequalities for the norm function.
Proposition 3.5
Assume (\(\hbox {C}_{{1}}\)) and (\(\hbox {C}_{{2}}\)), and assume \(\left( V,\left\| \cdot \right\| \right) \) is a Banach space. If \(v_{1},v_{2} ,\ldots \in V\) such that the series \(\sum \nolimits _{i=1}^{\infty } p_{i}\left\| v_{i}\right\| ^{\alpha }\) is absolutely convergent for some \(\alpha \ge 1\), then
Proof
Since the function \(f:V\rightarrow \mathbb {R}\) defined by \(f\left( v\right) =\left\| v\right\| ^{\alpha }\) is convex, and the series \(\sum \nolimits _{i=1}^{\infty }p_{i}v_{i}\) is also absolutely convergent, the result follows from Theorem 2.1 (b). \(\square \)
Now we get a refinement of the discrete Hölder’s inequality.
Proposition 3.6
Let the set J denote either \(\left\{ 1,\ldots ,k\right\} \) for some \(k\ge 2\) or \(\mathbb {N}_{+}\), and let \(\left( \lambda _{j}\right) _{j\in J}\) represent a positive probability distribution. Let \(\left( w_{i}\right) _{i=1}^{\infty }\) be a sequence of positive numbers, and let \(\left( x_{i}\right) _{i=1}^{\infty }\) and \(\left( y_{i}\right) _{i=1}^{\infty }\) be sequences of nonnegative numbers such that the series \(\sum \nolimits _{i=1} ^{\infty }w_{i}x_{i}^{p}\) and \(\sum \nolimits _{i=1}^{\infty }w_{i}y_{i}^{q}\) are convergent, where \(p>1\) and \(q>1\) are conjugate exponents that is \(\frac{1}{p}+\frac{1}{q}=1\). Then
Proof
For each \(s>1\) the power function
is strictly convex. Let \(v_{1},v_{2}\ldots \) be positive numbers such that the series \(\sum \nolimits _{i=1}^{\infty }p_{i}v_{i}^{s}\) is convergent. Theorem 2.1 (b) can be applied to the function \(f_{s}\) and to the positive numbers \(v_{1},v_{2}\ldots \), and it yields that
If \(\sum \nolimits _{i=1}^{\infty }w_{i}y_{i}^{q}=0\), then (3.4) is obvious.
Otherwise, from the inequality (3.6) with the choices
(\(-f_{1/p}\) is convex) we obtain
and this delivers the desired conclusion.
The proof is complete. \(\square \)
Finally, we apply our results to get a refinement of the inequality of generalized arithmetic and geometric means.
Proposition 3.7
Assume (\(\hbox {C}_{{1}}\)) and (\(\hbox {C}_{{2}}\)). If \(v_{1},v_{2},\ldots \) are positive numbers such that the series \(\sum \nolimits _{i=1}^{\infty } p_{i}v_{i}\) and \(\sum \nolimits _{i=1}^{\infty }p_{i}\ln \left( v_{i}\right) \) are absolutely convergent, then
Proof
By applying Theorem 2.1 (b) to the convex function \(-\ln \), we obtain
and this is equivalent to (3.7). \(\square \)
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Open access funding provided by University of Pannonia (PE). The research of the author has been supported by Hungarian National Foundations for Scientific Research Grant No. K120186.
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Horváth, L. New refinements of the discrete Jensen’s inequality generated by finite or infinite permutations. Aequat. Math. 94, 1109–1121 (2020). https://doi.org/10.1007/s00010-019-00696-z
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DOI: https://doi.org/10.1007/s00010-019-00696-z