- Open Access
Some generalizations of Aczél, Bellman's inequalities and related power sums
© Zhou; licensee Springer. 2012
- Received: 9 January 2012
- Accepted: 8 June 2012
- Published: 8 June 2012
In this paper, we establish some functional generalizations and refinements of Aczél's inequality and of Bellman's inequality. We also establish several mean value theorems for the related power sums.
Mathematics Subject Classification (2010): 26D15; 26D20; 26D99.
- Aczél's inequality
- Bellman's inequality
- mean value theorem
- power sums
with equality if and only if the sequences a i and b i are proportional.
The Aczél inequality (1) plays an important role in the theory of functional equations in non-Euclidean geometry. During the past years, many authors have given considerable attention to this inequality, its generalizations and applications [2–11].
where n is a positive integer, and p, q, a i , b i (i = 1, 2, ..., n) are positive numbers such that p− 1+ q-1 = 1, and .
Here n is a positive integer, and p ≥ 1, a i , b i (i = 1, 2, ..., n) are positive numbers such that and .
In this paper, inspired by the functional generalizations of the Cauchy-Schwarz inequality  and of the Hölder inequality [15, 16], we will establish some functional generalizations of the Aczél inequality and of the Bellman inequality. Refinements of these inequalities will also be presented.
As we seen, the following theorem is very useful to give results related to power sums and Aczél's inequality.
The inequality is reversed if f(x)/x is decreasing on ℝ+. The inequalities are strict if f(x)/x is strictly increasing or decreasing on ℝ+.
In order to establish the functional generalization of Aczél's inequality, we need the following lemma.
where , .
Rearranging the terms of (7) immediately leads to the second inequality of (6). This completes the proof. □
Remark 2.1. From the proof we have that the second inequality of (6) still holds if for p j > 0, j = 1, 2, ..., m.
From Theorem 2.1, by taking f j (x) = x, we get
The first inequality of (8) gives a lower bound of Aczél's inequality. And the second is a generalized Aczél inequality obtained in .
The following theorem is the functional generalization of Bellman's inequality.
Proof. The proof of the first inequality of (9) is similar to the proof of Theorem 2.1 and we omit it. The second inequality is an identity when p = 1. Hence we only need to prove the second inequality of (9) for p > 1 below.
which yields immediately the desired inequality. This completes the proof. □
Taking f j (x) = x in Theorem 2.2, we obtain
and reusing the corresponding theorems, we present these refinements below without proofs.
leading to a refinement of (6).
Remark 2.2. The third and fourth inequality of (11) still holds if for p j > 0, j = 1, 2, ..., m.
Taking f j (x) = x in Theorem 2.3, we get
leading (12) to a refinement of (8).
Remark 2.3. The third and fourth inequality of (12) still holds for p j > 0 if , j = 1, 2, ..., m. The inequality (13) is also obtained in .
Taking f j (x) = x in Theorem 2.4, we have the following.
The third and fourth inequality (15) is also obtained in .
As we seen, Theorem 1.1 is very useful to give results related to Aczél's inequality. It is also useful to give results related to power sums [9, 17]. In this section, we first present a generalized version of Theorem 1.1, then use it to establish our first generalized mean value theorem related to power sums. We conclude this section with a mean value theorem which generalize the recent result obtained by Pečarić and Rehman .
Lemma 3.1. Let n be a positive integer, and x i (i = 1, 2, ..., n) be positive numbers such that . If f : ℝ+ → ℝ is a function such that f(x)/x p is increasing on ℝ+. Then the inequality (4) holds for p ≥ 1, f ≥ 0 on ℝ+ or p ≤ 1, f ≤ 0 on ℝ+. If f(x)/x p is decreasing on ℝ+, the inequality (4) is reversed for p ≥ 1, f ≤ 0 on ℝ+ or p ≤ 1, f ≥ 0 on ℝ+.
Remark 3.1. If p ≠ 1, f ≠ 0 or f(x)/x is strictly increasing or decreasing on ℝ+, then strictly inequalities hold.
Proof. This Lemma is an easy corollary of Theorem 1.1, so we omit the proof. □
which immediately leads to (16). This completes the proof. □
We present the Cauchy type mean value theorem of Theorem 3.1 below without proof for the proof is quite standard and coincides with the proof of Theorem 3.14 in .
Remark 3.2. If p = 2, the conditions , could be removed, then Theorem 3.1 and Theorem 3.2 reduce to Theorem 3.13 and Theorem 3.14 of , respectively.
We conclude this section with a generalization of the mean value theorem obtained in , which is a special case of the following theorem with k = 1. As given in , this theorem is also a generalization of Theorem 3.1 with p = 2.
combining (26) and (27) immediately leads to (23). This completes the proof. □
Similarly, we present the Cauchy type mean value theorem of Theorem 3.3 below without proof. This theorem reduce to the Cauchy type mean value theorem of  with k = 1.
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