# A parameter-dependent refinement of the discrete Jensen's inequality for convex and mid-convex functions

- László Horváth
^{1}Email author

**2011**:26

https://doi.org/10.1186/1029-242X-2011-26

© Horváth; licensee Springer. 2011

**Received: **8 March 2011

**Accepted: **25 July 2011

**Published: **25 July 2011

## Abstract

In this paper, a new parameter-dependent refinement of the discrete Jensen's inequality is given for convex and mid-convex functions. The convergence of the introduced sequences is also studied. One of the proofs requires an interesting convergence theorem with probability theoretical background. We apply the results to define some new quasi-arithmetic and mixed symmetric means and study their monotonicity and convergence.

## Keywords

## 1 Introduction and the main results

The considerations of this paper concern

(A_{1}) an arbitrarily given real vector space *X*, a convex subset *C* of *X*, and a finite subset {*x*_{1},..., *x*_{
n
} } of *C*, where *n* ≥ 1 is fixed;

(A_{2}) a convex function *f* : *C* → ℝ, and a discrete distribution *p*_{1},..., *p*_{
n
} , which means that *p*_{
j
} ≥ 0 with
;

(A_{3}) a mid-convex function *f* : *C* → ℝ, and a discrete distribution *p*_{1},..., *p*_{
n
} with rational *p*_{
j
} (1 ≤ *j* ≤ *n*).

For a variety of applications, the discrete Jensen's inequalities are important:

(b) *If (A* _{1} *) and (A* _{3} *) are satisfied, then (2) also holds*.

Let ℕ := {0, 1, 2,...} and let ℕ_{+} := {1, 2,...}.

Various attempts have been made to refine inequality (2) in the following ways: Assume either (A_{1}) and (A_{2}) or (A_{1}) and (A_{3}). Let *m* ≥ 2 be an integer, and let *I* denote either the set {1,..., *m*} or the set ℕ_{+}.

*B*

_{ k })

_{k∈I}such that

*B*

_{ k }=

*B*

_{ k }(

*f*,

*x*

_{ i },

*p*

_{ i }) (

*k*∈

*I*) is a sum whose index set is a subset of {1,...,

*n*}

^{ k }and

*C*

_{ k })

_{k∈I}such that

*C*

_{ k }=

*C*

_{ k }(

*f*,

*x*

_{ i },

*p*

_{ i }) (

*k*∈

*I*) is a sum whose index set is a subset of {1,...,

*k*}

^{ n }and

The next two typical results belong to the group of refinements of type (B).

is due to Pečarić and Svrtan [3]. In a recent work, [4] Horváth and Pečarić define a lot of new sequences, they generalize and give a uniform treatment a number of well-known results from this area, especially (5) and (6) are extended. Horváth develops a method in [5] to construct decreasing real sequences satisfying (3). His paper contains some improvements of the results in [4] and gives a new approach of the topic. The description of the sequences in [4, 5] requires some work, so we do not go into the details. The problem (B) has been considered for the classical Jensen's inequality by Horváth [6].

In this paper, we establish a new solution of the problem (C). The constructed sequence (*C*_{
k
} (*λ*))_{k≥0}depends on a parameter *λ* belonging to [1, ∞[, and we can use arbitrary discrete distribution *p*_{1},..., *p*_{
n
} , not just the appropriate discrete uniform distribution. We give the limit of the sequence under fixed parameter. We also study the convergence of the sequence when the parameter varies and *k* ∈ ℕ is fixed. Finally, some applications are given which concern the theme of means.

The next theorems are the main results of this paper. We need some further hypotheses:

(A_{4}) Let *λ* ≥ 1.

(A_{5}) Let *λ* ≥ 1 be rational.

First, we give a refinement of the discrete Jensen's inequality (2).

**Theorem 1**

*Suppose either (A*

_{1}

*), (A*

_{2}

*), and (A*

_{4}

*) or (A*

_{1}

*), (A*

_{3}

*), and (A*

_{5}

*). Introduce the sets*

**Remark 2** *(a) It follows from the definition of S*_{
k
} *that S*_{
k
} ⊂ {0,..., *k*} ^{
n
} (*k* ∈ ℕ).

Finally, we establish two convergence theorems.

**Theorem 3** *Suppose (A*_{1}*), (A*_{2}*), and (A*_{4}*). Suppose × is a normed space and f is continuous. Then*,

*(b) The function λ* → *C*_{
k
} (*λ*) (*λ* ≥ 1) *is continuous for every k* ∈ ℕ.

The proof of Theorem 3(a) requires a lemma (see Lemma 15), which is interesting in its own right. Probability theoretical technique will be used to handle this problem.

**Remark 4** *In the previous theorem, it suffices to consider the case when (A*_{1}*), (A*_{2}*), and (A*_{4}*) are satisfied. Really, if f is mid-convex and continuous, then convex*.

We come now to the second convergence theorem.

## 2 Discussion and applications

Suppose either (A_{1}), (A_{2}), and (A_{4}) or (A_{1}), (A_{3}), and (A_{5}). First, we give three special cases of (8).

*f*is strictly convex (strictly mid-convex) that is strict inequality holds in (1) whenever

*x*≠

*y*and 0

*< α <*1. In this case, equality is satisfied in (2) if and only if

*x*

_{1}= ··· =

*x*

_{ n }, and therefore, it comes from the third part of the proof of Theorem 1 that

if not all *x*_{
i
} are equal.

*f*is strictly convex (strictly mid-convex), then the analysis of the proof of Theorem 1 shows that

whenever not all *x*_{
i
} are equal.

*X*is a normed space and

*f*is continuous (see Remark 4), then Theorem 3(b) and Theorem 5 insure that the range of the function

*λ*→

*C*

_{ k }(

*λ*) (

*k*∈ ℕ

_{+}) is the interval

**Conjecture 6** *Suppose either (A*_{1}*), (A*_{2}*), and (A*_{4}*) or (A*_{1}*), (A*_{3}*), and (A*_{5}*)*.

*The function λ* → *C*_{
k
} (*λ*) (*λ* ≥ 1) *is increasing for every k* ∈ ℕ.

Next, we define some new quasi-arithmetic means and study their monotonicity and convergence. About means see [8].

**Definition 7**

*Let I*⊂ ℝ

*be an interval, let x*

_{ j }∈

*I*(1 ≤

*j*≤

*n*),

*let p*

_{1},...,

*p*

_{ n }

*be a discrete distribution, and let g, h*:

*I*→ ℝ

*be continuous and strictly monotone functions. Let λ*≥ 1.

*We define the quasi-arithmetic means with respect to (8) by*

Some other means are also needed.

**Definition 8**

*Let I*⊂ ℝ

*be an interval, let x*

_{ j }∈

*I*(1 ≤

*j*≤

*n*),

*and let p*

_{1},...,

*p*

_{ n }

*be a discrete distribution. For a continuous and strictly monotone function z*:

*I*→ ℝ,

*we introduce the following mean*

We now prove the monotonicity of the means (11) and give limit formulas.

**Proposition 9** *Let I* ⊂ ℝ *be an interval, let x*_{
j
} ∈ *I* (1 ≤ *j* ≤ *n*), *let p*_{1},..., *p*_{
n
} *be a discrete distribution, and let g, h* : *I* → ℝ *be continuous and strictly monotone functions. Let λ* ≥ 1. *Then*,

*if either h* ○ *g*^{-1}*is convex and h is increasing or h* ○ *g*^{-1}*is concave and h is decreasing*.

*if either h* ○ *g*^{-1}*is convex and h is decreasing or h* ○ *g*^{-1}*is concave and h is increasing*.

*for each fixed k* ∈ ℕ_{+}.

**Proof**. Theorem 1 can be applied to the function *h*○*g*^{-1}, if it is convex (-*h*○*g*^{1}, if it is concave) and the *n*-tuples (*g* (*x*_{1}),..., *g*(*x*_{
n
} )), then upon taking *h*^{-1}, we get (a) and (b). (c) comes from Theorems 3(a) and 5. ■

As a special case, we consider the following example.

**Example 10**

*If I*:=]0, ∞[,

*h*:= ln,

*and g*(

*x*):=

*x*(

*x*∈]0, ∞[),

*then by Proposition 9(b), we have the following inequality*.

*for every x*

_{ j }

*>*0 (1 ≤

*j*≤

*n*),

*λ*≥ 1 ,

*and k*∈ ℕ

_{+}

Finally, we investigate some mixed symmetric means.

The power means of order *r* ∈ ℝ are defined as follows:

**Definition 11**

*Let x*

_{ j }∈]0, ∞[ (1 ≤

*j*≤

*n*),

*and let p*

_{1},...,

*p*

_{ n }

*be a discrete distribution with p*

_{ j }

*>*0 (1 ≤

*j*≤

*n*).

If *r* ≠ 0, then the power means of order *r* belong to the means (12) (*z* : ]0, ∞[→ ℝ, *z*(*x*) := *x*^{
r
} ), while we get the power means of order 0 by taking limit. Supported by the power means, we can introduce mixed symmetric means corresponding to (8):

**Definition 12**

*Let x*

_{ j }∈]0, ∞[ (1 ≤

*j*≤

*n*),

*and let p*

_{1},...,

*p*

_{ n }

*be a discrete distribution with p*

_{ j }

*>*0 (1 ≤

*j*≤

*n*).

*Let λ*≥ 1,

*and k*∈ ℕ.

*We define the mixed symmetric means with respect to (8) by*

*where t* ∈ ℝ.

The monotonicity and the convergence of the previous means are studied in the next result.

**Proposition 13** *Let x*_{
j
} ∈]0, ∞[ (1 ≤ *j* ≤ *n*), *and let p*_{1},..., *p*_{
n
} *be a discrete distribution with p*_{
j
} *>* 0 (1 ≤ *j* ≤ *n*). *Let λ* ≥ 1, *and k* ∈ ℕ. *Suppose s, t* ∈ ℝ *such that s* ≤ *t. Then*,

*for each fixed k* ∈ ℕ_{+}.

**Proof**. Assume *s*, *t* ≠ 0. Then, Proposition 9 (b) can be applied with *g*, *h* :]0, ∞[→ ℝ, *g*(*x*) := *x*^{
t
} , and *h*(*x*) := *x*^{
s
} . If *s* = 0 or *t* = 0, the result follows by taking limit. ■

## 3 Some lemmas and the proofs of the main results

**Proof**. The lowest common denominator is *i*_{1}!... *i*_{
n
} !. Combined with
, the result follows. ■

The proof of Theorem 1.

**Proof**. (a) We separate the proof of this part of the theorem into three steps.

Let *λ* ≥ 1 be fixed.

II. Next, we prove that *C*_{
k
} (*λ*) ≤ *C* _{k+1}(*λ*) (*k* ∈ ℕ).

which is just *C*_{k+1}(*λ*).

hence (15) implies (14). ■

The proof of Theorem 3 (a) is based on the following interesting result. The *σ*-algebra of Borel subsets of ℝ ^{
n
} is denoted by
.

**Lemma 15**

*Let p*

_{1},...,

*p*

_{ n }

*be a discrete distribution with n*≥ 2,

*and let λ >*1.

*Let l*∈ {1,...,

*n*}

*be fixed. e*

_{ l }

*denotes the vector in*ℝ

^{ n }

*that has*0

*s in all coordinate positions except the lth, where it has a*1.

*Let q*

_{1},...,

*q*

_{ n }

*be also a discrete distribution such that q*

_{ j }

*>*0 (1 ≤

*j*≤

*n*)

*and*

**Proof**. To prove the result, we can obviously suppose that *l* = 1.

For the sake of clarity, we shall denote the element (*i*_{1},..., *i*_{
n
} ) of *S*_{
k
} by (*i*_{1k},..., *i*_{
nk
} ) (*k* ∈ ℕ_{+}).

_{ k }:= (ξ

_{1k},...,

*ξ*

_{ nk }) (

*k*∈ ℕ

_{+}) be a (ℝ

^{ n }, )-random variable on a probability space such that ξ

_{ k }has multinomial distribution of order

*k*and with parameters

*q*

_{1},...,

*q*

_{ n }. A fundamental theorem of the statistics (see [9], Theorem 5.4.13), which is based on the multidimensional central limit theorem and the Cochran-Fisher theorem, implies that

where *F*_{n-1}means the distribution function of the Chi-squared distribution (*χ*^{2}-distribution) with *n* - 1 degrees of freedom.

*< ε*< 1. Since

*F*

_{n-1}is continuous and strictly increasing on ]0, ∞[, there exists a unique

*t*

_{ ε }

*>*0 such that

*j*≤

*n*. If (23) is false, then (22) yields that we can find a positive number

*ρ*, a strictly increasing sequence (

*k*

_{ u })

_{u≥1}and points

contrary to (24).

where *k* ∈ ℕ_{+}. The sum of these sequences is just the studied sequence in (17).

*ε*

_{1}

*>*0, we can find an integer that for all

and this proves the convergence claim (17).

The proof is now complete. ■

The proof of Theorem 3.

The case *p*_{
l
} = 0 is trivial.

(b) Elementary considerations show this part of the theorem.

The proof is complete. ■

The proof of Theorem 5.

where *t*_{
j
} (1 ≤ *j* ≤ *n*) is also rational if *f* is mid-convex.

By the definition of the set *S*_{
k
} , (0,..., 0, *k*, 0,..., 0) (the vector has 0s in all coordinate positions except the *l* th) is the only element of *S*_{
k
} for which *i*_{
l
} = *k* (1 ≤ *l* ≤ *n*). By using the boundedness of *f* on *G*, the previous assumptions imply the result, bringing the proof to an end. ■

## Declarations

### Acknowledgements

This study was supported by the Hungarian National Foundations for Scientific Research Grant No. K73274.

## Authors’ Affiliations

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