2.1 A new inequality for generalized weighted mean
By taking the
\(\{ \alpha_{k} \}\) to be any weights obeying equation (
6) we generalize Conjecture
1 to the following theorem.
Without loss of generality we can assume the sequence
\(\{ b_{k} \}\) is given in increasing order. In that case condition (
7) is equivalent to
$$ \forall(i,j),\quad 1 \le i,j \le M, i \le j \quad\Rightarrow\quad \alpha'_{i} / \alpha '_{j} \ge \alpha_{i}/ \alpha_{j}. $$
(10)
Let us prove first the following lemma.
We prove now Theorem
1 under the
\(\{b_{k}\}\)-increasing condition:
Observe that Conjecture
1 is a particular case of equation (
21) when
\(k_{1}=k_{2}=1, C_{2}=C_{1}\).
Other interesting cases are when
\(k_{1} = 1, k_{2} = 0\):
$$ \mathcal{H}\bigl( \bigl\{ b_{k} ( b_{k} + C) \bigr\} \bigr) \le\mathcal{H}\bigl( \{ b_{k} \}\bigr) \mathcal{H}\bigl( \bigl\{ (b_{k} + C) \bigr\} \bigr), $$
(22)
and when
\(k_{1} = 1, k_{2} = 0, C_{1} = 0\):
$$ \mathcal{H}\bigl( \bigl\{ b_{k}^{2} \bigr\} \bigr) \le \bigl( \mathcal{H}\bigl( \{ b_{k} \}\bigr) \bigr)^{2}, $$
(23)
a result that can be immediately derived from the inequality between power means with orders −1 and −2 [
6]. Finally, taking
\(k_{1} = 1, k_{2} = 1, C_{1} = 0\):
$$ \frac{\mathcal{H}( \{ b_{k}^{2} \})}{\mathcal{H}( \{ b_{k} \})} \le\frac{\mathcal{H}( \{ b_{k} (b_{k} + C)\})}{\mathcal{H}( \{ (b_{k} + C) \})} . $$
(24)
Observe that for
\(\gamma= 1\) we reproduce again equation (
23). Observe also that taking
\(\gamma= 0\) we get the well-known inequality
$$\begin{aligned} \mathcal{H}\bigl( \{ b_{k} \}\bigr) \le \mathcal{A} \bigl( \{ b_{k} \}\bigr), \end{aligned}$$
(28)
where
\(\mathcal{A}( \{ b_{k} \}) = ( \mathcal{H}( \{ b_{k}^{-1} \}) )^{-1}\) is the arithmetic mean of
\(\{ b_{k} \}\).
Let us see now that Theorem
1, for the weighted harmonic mean, also holds for the weighted arithmetic mean.
Corollary
6 can easily be extended to the following one.
\(\mathcal{A}( \{ b_{k}^{\gamma} \}) = 1/\mathcal{H}( \{ b_{k}^{-\gamma} \})\) and reciprocally, Corollary
6, applied to sequences
\(\{ b_{k}^{\gamma_{1}} \}, \{ b_{k}^{\gamma_{2}} \}, \gamma_{1}, \gamma_{1} \ge0\), together with Corollary
4, allow us to establish the following corollary.
Observe that we can only guarantee that equation (
51) hold when both
\(\gamma_{1}, \gamma_{2}\) are of the same sign. For instance, taking
\(\gamma_{1}= \gamma= -\gamma_{2}\) we can easily check that equation (
51) would read
\(\mathcal{H}( \{ b_{k}^{\gamma} \}) \ge \mathcal{A}( \{ b_{k}^{\gamma} \})\), which is false in general (rather what is true is the inverse inequality).
Consider now \(H(\{p_{k}\})= -\sum_{k} p_{k}\log p_{k} \), the Shannon entropy of \(\{p_{k}\}\).
In information theory [
8], the value
\(-\log p_{i}\) is considered as the information of result
i, thus Corollary
10 says that the expected value of information is less than or equal to its average value.
The Tsallis entropy with entropic index
q of probability distribution
\(\{p_{k}\}\) [
9,
10] is defined as
$$ S_{q} \bigl(\{p_{k}\}\bigr) = \frac{1}{q-1} \Biggl( 1- \sum_{k=1}^{M} p_{k}^{q} \Biggr). $$
(56)
Rényi entropy of order
\(\beta\ge0\) [
11] is defined as
$$ H_{\beta} \bigl(\{p_{k}\}\bigr) = \frac{1}{1- \beta} \log \Biggl( \sum_{k=1}^{M} p_{k}^{q} \Biggr). $$
(62)
Let us see now that Theorem
1 extends to a weighted geometric mean.
Let us see now that Theorem
1 also extends to weighted generalized (or power) mean:
Theorem
5 below extends Theorem
1 to the quasi-arithmetic or Kolmogorov generalized weighted mean.
The mean
\(\operatorname{logsumexp} (\{ b_{k} \})\) is defined as
$$ \operatorname{logsumexp} \bigl(\{ b_{k} \}\bigr) = \log \biggl( \sum_{k} {e}^{b_{k} } \biggr). $$
(77)
We can state then the following corollary to Theorem
5.
Observe now that the condition
\(\alpha'_{i} / \alpha'_{j} \ge\alpha_{i}/ \alpha_{j}\) appearing Theorems
1-
5 is equivalent to the one of decreasing quotients:
$$ \alpha'_{1} / \alpha_{1} \ge \alpha'_{2}/ \alpha_{2} \ge\cdots\ge \alpha'_{M}/ \alpha_{M}. $$
(84)
Thus we can immediately extend the previous Theorems
1-
5 to the following one.
Finally, we consider the following theorem.
Observe that Corollaries
3-
15 can be considered as applications of Theorem
7.
2.2 Relationship to majorization
Consider the sequences
\(\{x_{k}\}\),
\(\{y_{k} \}\), and renumber the indices so that
\(\{x_{1} \ge x_{2} \ge\cdots\ge x_{M} \ge0 \}, \{y_{1} \ge y_{2} \ge \cdots\ge y_{M} \ge0 \}\). The sequence
\(\{ x_{k} \}\) is said to major sequence
\(\{ y_{k} \}\) [
5,
12], and we write
\(\{ x_{k} \} \succ\{ y_{k} \}\), when the following inequalities hold:
$$\begin{aligned} &x_{1} \ge y_{1}, \\ &x_{1} + x_{2} \ge y_{1} + y_{2}, \\ &\cdots \\ &x_{1} + x_{2} + \cdots+ x_{M-1} \ge y_{1} + y_{2} +\cdots+ y_{M-1}, \\ &x_{1} + x_{2} + \cdots+ x_{M-1}+ y_{M} = y_{1} + y_{2}+ \cdots+ y_{M-1} + y_{M}. \end{aligned}$$
(88)
In general, the
\(\alpha'_{k}, \alpha_{k} \) sequences fulfilling condition in equation (
7) do not major each other, and we can find examples of sequences that major each other but do not fulfill equation (
7), consider for instance the sequences
\(\{ 3,2,1,1\}, \{2,2,2,1\}\), they do not fulfill equation (
7) but
\(\{3,2,1,1\} \succ\{2,2,2,1 \}\). We will find now when both conditions, majorization and equation (
7), coincide. Let us prove first the following lemma.
We can then state Theorem
8.
Observe that the weights in Corollary
6 are such that
\(\{ \alpha_{k} \} \succ\{ \alpha'_{k}\}\), and in this way Corollary
6 can be proved by direct application of Lemma
1 in [
12].
A similar theorem can be proved for decreasing weights.
We can also obtain similar results for convex functions than in Theorem
10 for Theorems
11,
12, and
13.
2.3 Not a necessary condition
We can see with counterexamples that the sufficient condition equation (
7) appearing on all Theorems
1-
5 is not a necessary condition for the inequality of the means for any strictly positive sequence
\(\{b_{k}\}\) for
\(M\ge3\) (although it is easy to prove it is a necessary condition for
\(M=2\)).
Using the (unnormalized) weights
\(\{\alpha_{k}\}= \{ 1,2,\ldots,1,2\}\),
\(\{\alpha'_{k}\}= \{2,1,\ldots,2,1\}\) for
\(M \ge4\) even and
\(\{\alpha_{k}\}= \{1,2,\ldots,1,2,2\}\),
\(\{\alpha'_{k}\}= \{2,1,\ldots,2,1,2\}\) for
\(M\ge3\) odd, we can see that equation (
32) does not hold but on the other side equation (
33) holds for any strictly positive increasing sequence
\(\{b_{k}\}\). For instance, for
M even,
$$ 2 b_{1} + b_{2} + \cdots +2 b_{M-1} + b_{M} \le b_{1} + 2 b_{2} + \cdots + b_{M-1} + 2 b_{M} , $$
(104)
because as the
\(\{b_{k}\}\) are in increasing order then
\(2 b_{1} + b_{2} \le b_{1} + 2 b_{2}, \ldots,2 b_{M-1} + b_{M} \le b_{M-1} + 2 b_{M}\).
We leave it to the reader to check with the other means considered in this paper.