On the refinements of the Jensen-Steffensen inequality

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.

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 = (x₁, ..., x_n) ∈ Iⁿ and p = (p₁, ..., p_n) be a positive n-tuple, that is, such that p_i > 0 for i = 1, ..., n. Then

Where

If f is strictly convex, then inequality (1) is strict unless x₁ = ⋯ = 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 = (x₁, ..., x_n) ∈ Iⁿ is a monotonic n-tuple and p = (p₁, ..., 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 x₁ = ⋯ = 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 ≤ x₁ ≤ ⋯ ≤ 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 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.

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 x₁ ≤ y₁, x₂ ≤ y₂, x₁ ≠ x₂, y₁ ≠ y₂, then the following inequality is valid

If the function f is concave, the inequality reverses.

The main result states.

Theorem 2.2. Let x = (x₁, ..., x_n) ∈ Iⁿ be a monotonic n-tuple and p = (p₁, ..., p_n) a real n-tuple 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 n-tuple 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 = (x₁, ..., x_n) ∈ Iⁿ is a monotonic n-tuple and p = (p₁, ..., 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 = (x₁, ..., x_n) ∈ [a, b]ⁿ be a monotonic n-tuple and p = (p₁, ..., p_n) a real n-tuple such that (3) holds. Let f ∈ C²[a, b] and Φ₁ and Φ₂ be linear functionals defined as in (13) and (14). Then there exists ξ ∈ [a, b] such that

where f₀(x) = x².

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

Theorem 2.5. Let x = (x₁, ..., x_n) ∈ [a, b]ⁿ be a monotonic n-tuple and p = (p₁, ..., p_n) a real n-tuple such that (3) holds. Let f, g ∈ C²[a, b] be such that g"(x) ≠ 0 for every x ∈ [a, b] and let Φ₁ and Φ₂ be linear functionals defined as in (13) and (14). If Φ₁ and Φ₂ 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

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 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 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 y₀, y₁, y₂ ∈ [a, b] is defined recursively by

Remark 3.5. The value [y₀, y₁, y₂; f] is independent of the order of the points y₀, y₁ and y₂. 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 y₁ → y₀ in (18), we get

provided that f' exists, and furthermore, taking the limits y_i → y₀, 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 ↦ [y₀, y₁, y₂; f_s] is log-convex in J-sense on I for every three mutually different points y₀, y₁, y₂ ∈ [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) 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

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 ↦ [y₀, y₁, y₂; f_s] which is log-convex in J-sense 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 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) Since by (ii) the function s ↦ Φ_i(x, p, f_s) is log-convex on I, that is, the function s ↦ log Φ_i(x, p, f_s) is convex on I, applying (8) we get

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 ↦ [y₀, y₁, y₂; f_s] is exponentially convex on I for every three mutually different points y₀, y₁, y₂ ∈ [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 s₁, ..., s_n ∈ I are arbitrary, then the matrix

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) 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

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 semi-definite, 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 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 y₀, y₁, y₂ ∈ [a, b] coincide, say y₁ = y₀, for a family of differentiable functions f_s such that the function s ↦ [y₀, y₁, y₂; 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].

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 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 ↦ [y₀, y₁, y₂; 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 and is exponentially convex by Example 1 given in Jakšetić and Pečarić (submitted). From Jakšetić and Pečarić (submitted), we have that s ↦ [y₀, y₁, y₂; 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 ∈ Ω₂ and g = f_q ∈ Ω₂ yields that there exists such that

Since the function ξ ↦ ξ^s-q is invertible for s ≠ q, we then have

which together with the fact that μ_{s, q}(x, Φ_i, Ω₂) is continuous, symmetric and monotonous (by (23)), shows that μ_{s, q}(x, Φ_i, Ω₂) 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, Ω₂) 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 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

Example 4.4. Consider a family of functions

defined by

Exponential convexity of on (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

The authors declare that they have no competing interests.

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.

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).

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