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# On the refinements of the Jensen-Steffensen inequality

- Iva Franjić
^{1}, - Sadia Khalid
^{2}Email author and - Josip Pečarić
^{2, 3}

**2011**:12

https://doi.org/10.1186/1029-242X-2011-12

© Franjićć et al; licensee Springer. 2011

**Received:**16 March 2011**Accepted:**21 June 2011**Published:**21 June 2011

## Abstract

In this paper, we extend some old and give some new refinements of the Jensen-Steffensen inequality. Further, we investigate the log-convexity and the exponential convexity of functionals defined via these inequalities and prove monotonicity property of the generalized Cauchy means obtained via these functionals. Finally, we give several examples of the families of functions for which the results can be applied.

2010 *Mathematics Subject Classification*. 26D15.

## Keywords

- Jensen-Steffensen inequality
- refinements
- exponential and logarithmic convexity
- mean value theorems

## 1. Introduction

One of the most important inequalities in mathematics and statistics is the Jensen inequality (see [[1], p.43]).

**Theorem 1.1**.

*Let I be an interval in*ℝ

*and f*:

*I*→ ℝ

*be a convex function. Let n*≥ 2,

**x**= (

*x*

_{1}, ...,

*x*

_{ n }) ∈

*I*

^{ n }

*and*

**p**= (

*p*

_{1}, ...,

*p*

_{ n })

*be a positive n-tuple, that is, such that p*

_{ i }

*>*0

*for i*= 1, ...,

*n. Then*

*If f is strictly convex, then inequality* (1) *is strict unless x*_{1} = ⋯ = *x*_{
n
} .

The condition "**p** is a positive *n*-tuple" can be replaced by "**p** is a non-negative *n*-tuple and *P*_{
n
} *>* 0". Note that the Jensen inequality (1) can be used as an alternative definition of convexity.

It is reasonable to ask whether the condition "**p** is a non-negative *n*-tuple" can be relaxed at the expense of restricting **x** more severely. An answer to this question was given by Steffensen [2] (see also [[1], p.57]).

**Theorem 1.2**.

*Let I be an interval in*ℝ

*and f*:

*I*→ ℝ

*be a convex function. If*

**x**= (

*x*

_{1}, ...,

*x*

_{ n }) ∈

*I*

^{ n }

*is a monotonic n-tuple and*

**p**= (

*p*

_{1}, ...,

*p*

_{ n })

*a real n-tuple such that*

*is satisfied, where P*_{
k
} *are as in* (2), *then* (1) *holds. If f is strictly convex, then inequality* (1) *is strict unless x*_{1} = ⋯ = *x*_{
n
} .

Inequality (1) under conditions from Theorem 1.2 is called the Jensen-Steffensen inequality. A refinement of the Jensen-Steffensen inequality was given in [3] (see also [[1], p.89]).

**Theorem 1.3**.

*Let*

**x**

*and*

**p**

*be two real n-tuples such that a*≤

*x*

_{1}≤ ⋯ ≤

*x*

_{ n }≤

*b and*(3)

*hold. Then for every convex function f*: [

*a*,

*b*] → ℝ

Note that the function *G*_{
n
} defined in (6) is in fact the difference of the right-hand and the left-hand side of the Jensen inequality (1).

In this paper, we present a new refinement of the Jensen-Steffensen inequality, related to Theorem 1.3. Further, we investigate the log-convexity and the exponential convexity of functionals defined as differences of the left-hand and the right-hand sides of these inequalities. We also prove monotonicity property of the generalized Cauchy means obtained via these functionals. Finally, we give several examples of the families of functions for which the obtained results can be applied.

In what follows, *I* is an interval in ℝ, *P*_{
k
} are as in (2) and
are as in (7). Note that if (3) is valid, since
, it follows that
satisfy (3) as well.

## 2. New refinement of the Jensen-Steffensen inequality

The aim of this section is to give a new refinement of the Jensen-Steffensen inequality. In the proof of this refinement, the following result is needed (see [[1], p.2]).

**Proposition 2.1**.

*If f is a convex function on an interval I and if x*

_{1}≤

*y*

_{1},

*x*

_{2}≤

*y*

_{2},

*x*

_{1}≠

*x*

_{2},

*y*

_{1}≠

*y*

_{2},

*then the following inequality is valid*

*If the function f is concave, the inequality reverses*.

The main result states.

**Theorem 2.2**.

*Let*

**x**= (

*x*

_{1}, ...,

*x*

_{ n }) ∈

*I*

^{ n }

*be a monotonic n-tuple and*

**p**= (

*p*

_{1}, ...,

*p*

_{ n })

*a real n-tuple such that*(3)

*holds. Then for a convex function f*:

*I*→ ℝ

*we have*

*For a concave function f, the inequality signs in* (9) *reverse*.

**x**is increasing then , while if

**x**is decreasing then for every

*k*. Furthermore, without loss of generality, we can assume that

**x**is strictly monotonic and that 0

*< P*

_{ k }

*< P*

_{ n }for

*k*= 1, ...,

*n -*1. Now, applying (8) for a convex function

*f*when

**x**is strictly increasing yields inequality

both of which are equivalent to (12). If *f* is concave, the inequalities reverse. Thus, the proof is complete. □

**Remark 2.3**. *A slight extension of the proof of Theorem 1.3 in*[3]*shows that Theorem 1.3 remains valid if the n-tuple* **x** *is assumed to be monotonic instead of increasing. The proof is in fact analogous to the proof of Theorem 2.2*.

where functions *F*_{
k
} and
are as in (5) and (10), respectively, **x** = (*x*_{1}, ..., *x*_{
n
} ) ∈ *I*^{
n
} is a monotonic *n*-tuple and **p** = (*p*_{1}, ..., *p*_{
n
} ) is a real *n*-tuple such that (3) holds. If function *f* is convex on *I*, then Theorems 1.3 and 2.2, joint with Remark 2.3, imply that Φ_{
i
}(**x**, **p**, *f*) ≥ 0, *i* = 1, 2.

Now, we give mean value theorems for the functionals Φ_{
i
}, *i* = 1, 2.

**Theorem 2.4**.

*Let*

**x**= (

*x*

_{1}, ...,

*x*

_{ n }) ∈ [

*a*,

*b*]

^{ n }

*be a monotonic n-tuple and*

**p**= (

*p*

_{1}, ...,

*p*

_{ n })

*a real n-tuple such that*(3)

*holds. Let f*∈

*C*

^{2}[

*a*,

*b*]

*and*Φ

_{1}

*and*Φ

_{2}

*be linear functionals defined as in*(13)

*and*(14).

*Then there exists ξ*∈ [

*a*,

*b*]

*such that*

*where f*_{0}(*x*) = *x*^{2}.

*Proof*. Analogous to the proof of Theorem 2.3 in [4]. □

**Theorem 2.5**.

*Let*

**x**= (

*x*

_{1}, ...,

*x*

_{ n }) ∈ [

*a*,

*b*]

^{ n }

*be a monotonic n-tuple and*

**p**= (

*p*

_{1}, ...,

*p*

_{ n })

*a real n-tuple such that*(3)

*holds. Let f*,

*g*∈

*C*

^{2}[

*a*,

*b*]

*be such that g*"(

*x*) ≠ 0

*for every x*∈ [

*a*,

*b*]

*and let*Φ

_{1}

*and*Φ

_{2}

*be linear functionals defined as in*(13)

*and*(14).

*If*Φ

_{1}

*and*Φ

_{2}

*are positive, then there exists ξ*∈ [

*a*,

*b*]

*such that*

*Proof*. Analogous to the proof of Theorem 2.4 in [4]. □

## 3. Log-convexity and exponential convexity of the Jensen-Steffensen differences

We begin this section by recollecting definitions of properties which are going to be explored here and also some useful characterizations of these properties (see [[5], p.373]). Again, *I* is an open interval in ℝ.

*holds for every n* ∈ ℕ, *α*_{
i
} ∈ ℝ *and x*_{
i
} *such that x*_{
i
} + *x*_{
j
} ∈ *I, i*, *j* = 1, ..., *n*.

*holds for every n* ∈ ℕ, *α*_{
i
} ∈ ℝ *and x*_{
i
} ∈ *I, i* = 1, ..., *n*.

**Corollary 3.2**.

*If h is exponentially convex, then the matrix*

*is a positive semi-definite matrix. Particularly*,

**Corollary 3.3**.

*If h*:

*I*→ (0, ∞)

*is an exponentially convex function, then h is a log-convex function, that is, for every x*,

*y*∈

*I and every λ*∈ [0, 1]

*we have*

**Lemma 3.4**.

*A function h*:

*I*→ (0, ∞)

*is*log-

*convex in the J-sense on I, that is, for every x*,

*y*∈

*I*,

*holds for every α*, *β* ∈ ℝ *and x*, *y* ∈ *I*.

**Definition 2**.

*The second order divided difference of a function f*: [

*a*,

*b*] → ℝ

*at mutually different points y*

_{0},

*y*

_{1},

*y*

_{2}∈ [

*a*,

*b*]

*is defined recursively by*

**Remark 3.5**.

*The value*[

*y*

_{0},

*y*

_{1},

*y*

_{2};

*f*]

*is independent of the order of the points y*

_{0},

*y*

_{1}

*and y*

_{2}.

*This definition may be extended to include the case in which some or all the points coincide*(

*see*[[1], p.16]).

*Namely, taking the limit y*

_{1}→

*y*

_{0}

*in*(18),

*we get*

*provided that f' exists, and furthermore, taking the limits y*

_{ i }→

*y*

_{0},

*i*= 1, 2,

*in*(18),

*we get*

*provided that f*″ *exists*.

Next, we study the log-convexity and the exponential convexity of functionals Φ _{
i
} (*i* = 1, 2) defined in (13) and (14).

**Theorem 3.6**.

*Let*ϒ = {

*f*

_{ s }:

*s*∈

*I*}

*be a family of functions defined on*[

*a*,

*b*]

*such that the function s*↦ [

*y*

_{0},

*y*

_{1},

*y*

_{2};

*f*

_{ s }]

*is log-convex in J-sense on I for every three mutually different points y*

_{0},

*y*

_{1},

*y*

_{2}∈ [

*a*,

*b*].

*Let*Φ

_{ i }(

*i*= 1, 2)

*be linear functionals defined as in*(13)

*and*(14).

*Further, assume*Φ

_{ i }(

**x**,

**p**,

*f*

_{ s }) > 0 (

*i*= 1, 2)

*for f*

_{ s }∈ ϒ.

*Then the following statements hold*.

- (i)
*The function s*↦ Φ_{ i }(**x**,**p**,*f*_{ s })*is log-convex in J-sense on I*. - (ii)
*If the function s*↦ Φ_{ i }(**x**,**p**,*f*_{ s })*is continuous on I, then it is log-convex on I*. - (iii)

*and* Ξ *is the family functions f*_{
s
} *belong to*.

*s*↦ [

*y*

_{0},

*y*

_{1},

*y*

_{2};

*f*

_{ s }] which is log-convex in J-sense on

*I*by assumption, yields that

*g*is a convex function on

*I*and therefore we have Φ

_{ i }(

**x**,

**p**,

*g*) ≥ 0 (

*i*= 1, 2). Hence,

*s*↦ Φ

_{ i }(

**x**,

**p**,

*f*

_{ s }) is log-convex in J-sense on

*I*.

- (ii)
If the function

*s*↦ Φ_{ i }(**x**,**p**,*f*_{ s }) is in addition continuous, from (i) it follows that it is then log-convex on*I*. - (iii)

The case *u* = *v* can be treated similarly. □

**Theorem 3.7**.

*Let*Ω = {

*f*

_{ s }:

*s*∈

*I*}

*be a family of functions defined on*[

*a*,

*b*]

*such that the function s*↦ [

*y*

_{0},

*y*

_{1},

*y*

_{2};

*f*

_{ s }]

*is exponentially convex on I for every three mutually different points y*

_{0},

*y*

_{1},

*y*

_{2}∈ [

*a*,

*b*].

*Let*Φ

_{ i }(

*i*= 1, 2)

*be linear functionals defined as in*(13)

*and*(14).

*Then the following statements hold*.

- (i)

*is a positive semi-definite matrix for i*= 1, 2.

*Particularly*,

- (ii)
*If the function s*↦ Φ_{ i }(**x**,**p**,*f*_{ s })*is continuous on I, then it is also exponentially convex function on I*. - (iii)

*where μ*_{
s, q
} (**x**, Φ_{
i
}, Ω) *is defined in* (20).

- (ii)
If the function

*s*↦ Φ_{ i }(**x**,**p**,*f*_{ s }) is continuous on*I*, then from (i) and Proposition 3.1 it follows that it is exponentially convex on*I*. - (iii)
If the function

*s*↦ Φ_{ i }(**x**,**p**,*f*_{ s }) is differentiable on*I*, then from (ii) it follows that it is exponentially convex on*I*. If this function is in addition positive, then Corollary 3.3 implies that it is log-convex, so the statement follows from Theorem 3.6 (iii). □

**Remark 3.8**. *Note that the results from Theorem 3.6 still hold when two of the points y*_{0}, *y*_{1}, *y*_{2} ∈ [*a*, *b*] *coincide, say y*_{1} = *y*_{0}, *for a family of differentiable functions f*_{
s
} *such that the function s* ↦ [*y*_{0}, *y*_{1}, *y*_{2}; *f*_{
s
} ] *is log-convex in J-sense on I, and furthermore, they still hold when all three points coincide for a family of twice differentiable functions with the same property. The proofs are obtained by recalling Remark 3.5 and taking the appropriate limits. The same is valid for the results from Theorem 3.7*.

**Remark 3.9**. *Related results for the original Jensen-Steffensen inequality regarding exponential convexity, which are a special case of Theorem 3.7, were given in*[6].

## 4. Examples

In this section, we present several families of functions which fulfil the conditions of Theorem 3.7 (and Remark 3.8) and so the results of this theorem can be applied for them.

*We have*
*which shows that g*_{
s
} *is convex on* ℝ *for every s ∈* ℝ *and*
*is exponentially convex by Example 1 given in Jakšetić and Pečarić (submitted). From Jakšetić and Pečarić (submitted), we then also have that s* ↦ [*y*_{0}, *y*_{1}, *y*_{2}; *g*_{
s
} ] *is exponentially convex*.

*Here*,
*which shows that f*_{
s
} *is convex for x > 0 and*
*is exponentially convex by Example 1 given in Jakšetić and Pečarić (submitted). From Jakšetić and Pečarić (submitted), we have that s* ↦ [*y*_{0}, *y*_{1}, *y*_{2}; *f*_{
s
} ] *is exponentially convex*.

*In this case, μ*

_{ s, q }(

**x**, Φ

_{ i }, Ξ) (

*i*= 1, 2)

*defined in*(20)

*for x*

_{ j }

*> 0*(

*j =*1,

*..., n*)

*are*

*If*Φ

_{ i }

*is positive, then Theorem 2.5 applied for f*=

*f*

_{ s }∈ Ω

_{2}

*and g*=

*f*

_{ q }∈ Ω

_{2}

*yields that there exists*

*such that*

*which together with the fact that μ*_{
s, q
} (x, Φ_{
i
}, Ω_{2}) *is continuous, symmetric and monotonous* (*by* (23)), *shows that μ*_{
s, q
} (x, Φ_{
i
}, Ω_{2}) *is a mean*.

*where*
.

*These new means are also monotonous. More precisely, for s, q, u, v*∈ ℝ

*such that s*≤

*u, q*≤

*v, s*≠

*u, q*≠

*v, we have*

*for s, q, u, v* ∈ *I such that s/t* ≤ *u/t, q/t* ≤ *v/t and t* ≠ 0, *since μ*_{
s, q
} (x, Φ_{
i
}, Ω_{2}) *are monotonous in both parameters, so the claim follows. For t* = 0, *we obtain the required result by taking the limit t* → 0.

*Exponential convexity of*
*on* (0,∞) *is given by Example 2 in Jakšetić and Pečarić (submitted)*.

*μ*

_{ s, q }(x, Φ

_{ i }, Ξ) (

*i*= 1, 2)

*defined in*(20)

*in this case for x*

_{ j }

*>*0 (

*j*= 1, ...,

*n*)

*are*

*Exponential convexity of*
*on* (0, ∞) *is given by Example 3 in Jakšetić and Pečarić (submitted)*.

## Declarations

### Acknowledgements

This research work was partially funded by Higher Education Commission, Pakistan. The research of the first and the third author was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grants 058-1170889-1050 (Iva Franjić) and 117-1170889-0888 (Josip Pečarić).

## Authors’ Affiliations

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