On the refinements of the Jensen-Steffensen inequality
© Franjićć et al; licensee Springer. 2011
Received: 16 March 2011
Accepted: 21 June 2011
Published: 21 June 2011
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 [, p.43]).
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  (see also [, p.57]).
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 .
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.
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 [, p.2]).
If the function f is concave, the inequality reverses.
The main result states.
For a concave function f, the inequality signs in (9) reverse.
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 inshows 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 = (x1, ..., x n ) ∈ I n 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.
where f0(x) = x2.
Proof. Analogous to the proof of Theorem 2.3 in . □
Proof. Analogous to the proof of Theorem 2.4 in . □
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 [, 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.
holds for every α, β ∈ ℝ and x, y ∈ I.
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).
The function s ↦ Φ i (x, p, f s ) is log-convex in J-sense on I.
If the function s ↦ Φ i (x, p, f s ) is continuous on I, then it is log-convex on I.
and Ξ is the family functions f s belong to.
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.
The case u = v can be treated similarly. □
If the function s ↦ Φ i (x, p, f s ) is continuous on I, then it is also exponentially convex function on I.
where μ s, q (x, Φ i , Ω) is defined in (20).
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.
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.
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 ↦ [y0, y1, y2; 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 ↦ [y0, y1, y2; f s ] is exponentially convex.
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.
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.
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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