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On the refinements of the JensenSteffensen inequality
Journal of Inequalities and Applications volume 2011, Article number: 12 (2011)
Abstract
In this paper, we extend some old and give some new refinements of the JensenSteffensen inequality. Further, we investigate the logconvexity 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.
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 ntuple, that is, such that p_{ i } > 0 for i = 1, ..., n. Then
Where
If f is strictly convex, then inequality (1) is strict unless x_{1} = ⋯ = x_{ n } .
The condition "p is a positive ntuple" can be replaced by "p is a nonnegative ntuple 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 nonnegative ntuple" 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 ntuple and p = (p_{1}, ..., p_{ n } ) a real ntuple 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 JensenSteffensen inequality. A refinement of the JensenSteffensen inequality was given in [3] (see also [[1], p.89]).
Theorem 1.3. Let x and p be two real ntuples such that a ≤ x_{1} ≤ ⋯ ≤ x_{ n } ≤ b and (3) hold. Then for every convex function f : [a, b] → ℝ
holds, where
P_{ k } are as in (2) and
Note that the function G_{ n } defined in (6) is in fact the difference of the righthand and the lefthand side of the Jensen inequality (1).
In this paper, we present a new refinement of the JensenSteffensen inequality, related to Theorem 1.3. Further, we investigate the logconvexity and the exponential convexity of functionals defined as differences of the lefthand and the righthand 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 JensenSteffensen inequality
The aim of this section is to give a new refinement of the JensenSteffensen 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 ntuple and p = (p_{1}, ..., p_{ n } ) a real ntuple such that (3) holds. Then for a convex function f : I → ℝ we have
where
For a concave function f, the inequality signs in (9) reverse.
Proof. The claim is that for a convex function f,
holds for every k = 2, ..., n. This inequality is equivalent to
where
If 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
while if x is strictly decreasing we get 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 ntuple x is assumed to be monotonic instead of increasing. The proof is in fact analogous to the proof of Theorem 2.2.
Let us observe inequalities (4) and (9). Motivated by them, we define two functionals
where functions F_{ k } and are as in (5) and (10), respectively, x = (x_{1}, ..., x_{ n } ) ∈ I^{n} is a monotonic ntuple and p = (p_{1}, ..., p_{ n } ) is a real ntuple 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 ntuple and p = (p_{1}, ..., p_{ n } ) a real ntuple 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 ntuple and p = (p_{1}, ..., p_{ n } ) a real ntuple 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]. □
Remark 2.6. If the inverse of the function f"/g" exists, then (16) gives
3. Logconvexity and exponential convexity of the JensenSteffensen 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.
Definition 1. A function h : I → ℝ is exponentially convex on I if it is continuous and
holds for every n ∈ ℕ, α_{ i } ∈ ℝ and x_{ i } such that x_{ i } + x_{ j } ∈ I, i, j = 1, ..., n.
Proposition 3.1. Function h : I → ℝ is exponentially convex if and only if h is continuous and
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 semidefinite matrix. Particularly,
Corollary 3.3. If h : I → (0, ∞) is an exponentially convex function, then h is a logconvex function, that is, for every x, y ∈ I and every λ ∈ [0, 1] we have
Lemma 3.4. A function h : I → (0, ∞) is logconvex in the Jsense on I, that is, for every x, y ∈ I,
holds if and only if the relation
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 logconvexity 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 logconvex in Jsense 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 logconvex in Jsense on I.

(ii)
If the function s ↦ Φ_{ i }(x, p, f_{ s } ) is continuous on I, then it is logconvex on I.

(iii)
If the function s ↦ Φ_{ i }(x, p, f_{ s } ) is differentiable on I, then for every s, q, u, v ∈ I such that s ≤ u and q ≤ v, we have
(19)
where
and Ξ is the family functions f_{ s } belong to.
Proof. (i) For α, β ∈ ℝ and s, q ∈ I, we define a function
Applying Lemma 3.4 for the function s ↦ [y_{0}, y_{1}, y_{2}; f_{ s } ] which is logconvex in Jsense on I by assumption, yields that
which in turn implies that g is a convex function on I and therefore we have Φ_{ i }(x, p, g) ≥ 0 (i = 1, 2). Hence,
Now using Lemma 3.4 again, we conclude that the function s ↦ Φ_{ i }(x, p, f_{ s } ) is logconvex in Jsense 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 logconvex on I.

(iii)
Since by (ii) the function s ↦ Φ_{ i }(x, p, f_{ s } ) is logconvex on I, that is, the function s ↦ log Φ_{ i }(x, p, f_{ s } ) is convex on I, applying (8) we get
(21)
for s ≤ u, q ≤ v, s ≠ q, u ≠ v, and therefore conclude that
If s = q, we consider the limit when q → s in (21) and conclude that
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)
If n ∈ ℕ and s _{1}, ..., s_{ n } ∈ I are arbitrary, then the matrix
is a positive semidefinite 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)
If the function s ↦ Φ_{ i }(x, p, f_{ s } ) is positive and differentiable on I, then for every s, q, u, v ∈ I such that s ≤ u and q ≤ v, we have
(23)
where μ_{ s, q } (x, Φ_{ i }, Ω) is defined in (20).
Proof. (i) Let α_{ j } ∈ ℝ (j = 1, ..., n) and consider the function
for n ∈ ℕ, where , s_{ j } ∈ I, 1 ≤ j, k ≤ n and . Then
and since is exponentially convex by assumption it follows that
and so we conclude that g is a convex function. Now we have
which is equivalent to
which in turn shows that the matrix is positive semidefinite, so (22) is immediate.

(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 logconvex, 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 logconvex in Jsense 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 JensenSteffensen 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.
Example 4.1. Consider a family of functions
defined by
We havewhich shows that g_{ s } is convex on ℝ for every s ∈ ℝ andis 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.
For this family of functions, μ_{ s, q } (x, Φ_{ i }, Ξ) (i = 1, 2) from (20) become
Example 4.2. Consider a family of functions
defined by
Here, which shows that f_{ s } is convex for x > 0 andis 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
Since the function ξ ↦ ξ^{sq} is invertible for s ≠ q, we then have
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.
Now, by substitutions, ,from (24) we get
where.
We define a new mean (for i = 1, 2) as follows:
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
We know that
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.
Example 4.3. Consider a family of functions
defined by
Exponential convexity ofon (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
Example 4.4. Consider a family of functions
defined by
Exponential convexity ofon (0, ∞) is given by Example 3 in Jakšetić and Pečarić (submitted).
In this case, μ_{ s, q } (x, Φ_{ i }, Ξ) (i = 1, 2) defined in (20) for x_{ j } > 0 (j = 1, ..., n) are
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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 05811708891050 (Iva Franjić) and 11711708890888 (Josip Pečarić).
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JP made the main contribution in conceiving the presented research. IF and JP worked on the results from Section 2, while SK and JP worked jointly on the results of Sections 3 and 4. IF and SK drafted the manuscript. All authors read and approved the final manuscript.
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Franjić, I., Khalid, S. & Pečarić, J. On the refinements of the JensenSteffensen inequality. J Inequal Appl 2011, 12 (2011). https://doi.org/10.1186/1029242X201112
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DOI: https://doi.org/10.1186/1029242X201112
Keywords
 JensenSteffensen inequality
 refinements
 exponential and logarithmic convexity
 mean value theorems