PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 86(100) (2009), 35–39 DOI: 10.2298/PIM0900035S

A NOTE ON DIFFERENCES OF POWER MEANS

Slavko Simić

Communicated by Stevan Pilipović

Abstract. We give some new inequalities concerning the differences of power means.

1. Introduction

Let x̃_{n} = {x_{i}}_{1}^{n}, p̃_{n} = {p_{i}}_{1}^{n} denote two sequences of positive real numbers with ∑ _{1}^{n} p_{i} = 1. From the Theory of Convex Means (cf. [1], [2], [3]), it is well known that for t > 1,

(1)  ∑_{1}^{n} p_{i}x_{i}^{t} ⩾ ( ∑_{1}^{n} p_{i}x_{i}) ^{t},

and vice versa for 0 < t < 1, The equality sign in (1) occurs if and only if all members of x̃_{n} are equal (cf. [1]).

In this article we shall consider the difference

d_{t} = d_{t}^{(n)} = d_{t}^{(n)} (x̃_{n}, p̃_{n}) := ∑_{1}^{n} p_{i}x_{i}^{t} − ( ∑_{1}^{n} p_{i}x_{i}) ^{t}, t > 1,

and thus generated sequence d = {d_{m}}_{m⩾2} of non-negative real numbers.

By the above, if all members of the sequence x̃_{n} are equal, then all members of d are zero; hence this trivial case will be excluded in the sequel.

An interesting fact is that there exists an explicit constant c_{m}, independent of the sequences x̃_{n} and p̃_{n}, such that d_{m−1} d_{m+1} ⩾ c_{m}(d_{m})^{2}, m ⩾ 3.

On the contrary, we show that there is no constant C_{m}, depending only on m, such that d_{m−1} d_{m+1} ⩽ C_{m}(d_{m})^{2}.

Nontrivial lower bound for d_{m} and corresponding integral inequalities will also be given.

Finally we posed an open problem concerning the above matter.

2000 Mathematics Subject Classification: Primary 26D15.

Key words and phrases: power means; logarithmic convexity; best possible bounds.

36 SIMIĆ

2. Results Denote by S_{+} the space of all positive sequences. Our main result is Theorem 1. Let p̃_{n}, x̃_{n} ∈ S_{+} and d_{m} = d_{m}^{(n)} := ∑ _{1}^{n} p_{i}x_{i}^{m} − (∑ _{1}^{n} p_{i}x_{i})^{m};

m ∈ ℕ. Then

(2)  d_{m−1} d_{m+1} ⩾ c_{m}(d_{m})^{2}, m ⩾ 3,

with the best possible constant c_{m} = 1 − \frac{2}{m(m−1)} .

This inequality is very precise. For example

d_{2}^{(2)}d_{4}^{(2)} − \frac{2}{3}(d_{3}^{(2)})^{2} = \frac{1}{3}(p_{1}p_{2})^{2}(1 + p_{1}p_{2})(x_{1} − x_{2})^{6}.

Non-trivial lower bound for d_{m} follows.

Theorem 2. For d_{m} defined as above, we have

d_{m} ⩾ \binom{m}{2} \frac{(d_{3}/3)^{m−2}}{(d_{2})^{m−3}} , m ⩾ 2.

Applying the standard procedure (cf. [1, p. 131]), we pass from finite sums to definite integrals and obtain

Theorem 3. Let f(t), p(t) be non-negative, continuous and integrable functions for t ∈ [a, b], with ∫_{a}^{b}p(t) dt = 1. Denote

D_{m} = D_{m}(a, b; f, p) := ∫_{a}^{b}p(t) f^{m}(t) dt − ( ∫_{a}^{b}p(t) f(t) dt) ^{m}.

Then

(i) D_{m−1} D_{m+1} ⩾ (1 − \frac{2}{m(m−1)} )(D_{m})^{2}, m ⩾ 3;
(ii) If f(t) ≠ C, t ∈ [a, b], we have

D_{m} ⩾ \binom{m}{2} \frac{(D_{3}/3)^{m−2}}{(D_{2})^{m−3}} , m ⩾ 2.

3. Proofs We start with an interesting formula. For p̃_{n}, x̃_{n} ∈ S_{+} , making a shift x_{i} →

x_{i} + t, we obtain

d_{m}(t) := ∑_{1}^{n} p_{i}(x_{i} + t)^{m} − ( ∑_{1}^{n} p_{i}(x_{i} + t)) ^{m} = ∑_{1}^{n} p_{i}(t + x_{i})^{m} − ( t + ∑_{1}^{n} p_{i}x_{i}) ^{m}.

Developing, we get

(3)  d_{m}(t) = ∑_{2}^{n} d_{i} \binom{m}{i} t^{m−i} .

Therefore d_{m}(t) belongs to the class of Appell polynomials i.e., d′_{m}(t) = md_{m−1} (t) (cf [3], [4]).

If the properties of this class of polynomials lead to the proof of Theorem 1 is left to the readers to examine. For example, by (1), d_{4}(t) is non-negative for each t ∈ ℝ. Hence by (3),

d_{4}(t) = d_{4} + 4d_{3}t + 6d_{2}t^{2} ⩾ 0.

Putting t = −\frac{1}{3} \frac{d_{3}}{d_{2}} , we obtain (2) with m = 3.

In this article we turn the other way, noting that (2) can be rewritten in the form

\frac{d_{m−1}}{(m − 1)(m − 2)} \frac{d_{m+1}}{(m + 1)m} ⩾ ( \frac{d_{m}}{m(m − 1)} )^{2}, m ⩾ 3.

Hence, (2) is equivalent to the assertion that \frac{d_{m}}{m(m−1)} is log-convex for m ⩾ 3.

Definition. A sequence of positive numbers {c_{m}} is log-convex (c_{m} ∈ LC) if c_{m−1} c_{m+1} ⩾ (c_{m})^{2}.

We quote here some useful lemmas from log-convex theory (cf [3]).

Lemma 3.1. A positive sequence {c_{m}} is log-convex if and only if the inequality c_{m−1} u^{2} + 2c_{m}uv + c_{m+1} v^{2} ⩾ 0 holds for each real u, v.

Lemma 3.2. Let a_{m}, b_{m} ∈ LC and A, B, C be arbitrary positive constants. Then: (i) AC^{m+B} a_{m} ∈ LC; (ii) Aa_{m} + Bb_{m} ∈ LC.

Now we are able to produce a proof of Theorem 1 by induction on n.

Proof of Theorem 1. For n = 2 we have to prove that

(4)  \frac{p_{1}x_{1}^{m} + p_{2}x_{2}^{m} − (p_{1}x_{1} + p_{2}x_{2})^{m}}{m(m − 1)} ∈ LC,

holds for each positive x_{1}, x_{2}, p_{1}, p_{2} with p_{1} + p_{2} = 1. To this end, we need the following simple assertion

Lemma 3.3. If A ⩾ B > 0, then \frac{A^{m} −B^{m}}{m} ∈ LC, holds for m ⩾ 2.

Now, for fixed x_{1}, x_{2}, p_{1}, p_{2} and arbitrary ξ ⩾ 1 put A = ξ, B = p_{1}ξ + p_{2}; note that A ⩾ B since p_{1} + p_{2} = 1. By lemmas 1, 3 and 2(i), for arbitrary u, v ∈ ℝ, m ⩾ 3, we get

(5) p_{1}x_{2}^{m−1} ( \frac{ξ^{m−2} − (p_{1}ξ + p_{2})^{m−2}}{m − 2} ) u^{2} + 2p_{1}x_{2}^{m}( \frac{ξ^{m−1} − (p_{1}ξ + p_{2})^{m−1}}{m − 1} ) uv

+ p_{1}x_{2}^{m+1} ( \frac{ξ^{m} − (p_{1}ξ + p_{2})^{m}}{m} ) v^{2} ⩾ 0.

Integrating (5) with respect to ξ over ξ ∈ [1, x_{1}/x_{2}], we obtain

\frac{p_{1}x_{1}^{m−1} + p_{2}x_{2}^{m−1} − (p_{1}x_{1}+p_{2}x_{2})^{m−1}}{(m − 1)(m − 2)} u^{2} + 2\frac{p_{1}x_{1}^{m} + p_{2}x_{2}^{m} − (p_{1}x_{1}+p_{2}x_{2})^{m}}{m(m − 1)} uv
+\frac{p_{1}x_{1}^{m+1} + p_{2}x_{2}^{m+1} − (p_{1}x_{1} + p_{2}x_{2})^{m+1}}{(m + 1)m} v^{2} ⩾ 0.

Therefore by Lemma 1 we conclude that (4) is true.

38 SIMIĆ

Let T := \frac{1}{1−p_{n}} ∑ _{1}^{n−1} p_{i}x_{i}. Then

\frac{d_{m}^{(n)}}{m(m − 1)} = (1 − p_{n})\frac{d_{m}^{(n−1)}}{m(m − 1)} + \frac{(1 − p_{n})T^{m} + p_{n}x_{n}^{m} − ((1 − p_{n})T + p_{n}x_{n})^{m}}{m(m − 1)} .

Since \frac{d_{m}^{(n−1)}}{m(m−1)} ∈ LC by induction hypothesis, by (4) and Lemma 2(ii), it follows that \frac{d_{m}^{(n)}}{m(m−1)} ∈ LC, and the proof is done. □

To see that the constant c_{m} = 1 − \frac{2}{m(m−1)} is best possible, consider the representation (3). Since variable t is independent of the sequences p̃_{n}, x̃_{n}, we have d_{m}(t) ∼ d_{2} \binom{m}{2}t^{m−2} (t → ∞). Hence

\frac{d_{m−1} (t)d_{m+1} (t)}{(d_{m}(t))^{2}} ∼ \frac{\binom{m−1}{2}t^{m−3} \binom{m+1}{2}t^{m−1}}{(\binom{m}{2}t^{m−2} )^{2}} = c_{m} (t → ∞).

Proof of Theorem 2. From (2) we get d_{m+1} /d_{m} ⩾ c_{m}(d_{m}/d_{m−1} ), m ⩾ 3. Hence

∏_{3}^{m} ( \frac{d_{k+1}}{d_{k}} ) ⩾ ∏_{3}^{m} \frac{(k + 1)(k − 2)}{k(k − 1)} ∏_{3}^{m} ( \frac{d_{k}}{d_{k−1}} ) ,

i.e.,

\frac{d_{m+1}}{d_{m}} ⩾ ( \frac{m + 1}{3(m − 1)} ) ( \frac{d_{3}}{d_{2}}) , m ⩾ 2.

Therefore, the conclusion follows from

\frac{d_{m}}{d_{2}} = ∏_{2}^{m−1} ( \frac{d_{k+1}}{d_{k}} ) ⩾ ∏_{2}^{m−1} ( \frac{k + 1}{k − 1} ) ∏_{2}^{m−1} ( \frac{d_{3}}{3d_{2}}) = \binom{m}{2} ( \frac{d_{3}}{3d_{2}}) ^{m−2} .  □

Proof of Theorem 3. Write d_{m}^{(n)} in the form

d_{m}^{(n)} = \frac{∑ _{1}^{n} p_{ni} x_{ni}^{m}}{∑ _{1}^{n} p_{ni}} − ( \frac{∑ _{1}^{n} p_{ni} x_{ni}}{∑ _{1}^{n} p_{ni}} ) ^{m},

with p_{ni} := p(a + i\frac{b−a}{n} ), x_{ni} := f(a + i\frac{b−a}{n} ). Passing to the limit, we obtain lim_{n→∞} d_{m}^{(n)} = D_{m} and from Theorems 1, 2 the assertions of Theorem 3 follow. □

There remains a problem of inverse inequality for the sequence d.

Question 1. Is there a constant C_{m}, independent of p̃_{n}, x̃_{n} ∈ S_{+} , such that d_{m−1} d_{m+1} ⩽ C_{m}(d_{m})^{2}, m ⩾ 2.

The answer to this question is negative.

Proof. We apply a special choice of the sequences p̃_{n}, x̃_{n} ∈ S_{+} . Namely, for fixed n ⩾ 2 let p_{i} := \binom{n−1}{i−1}/2^{n−1} ; x_{i} := (1 − t)^{i−1} (1 + t)^{n−i} , −1 < t < 1. We obtain a sequence d^{∗} = {d_{m}^{∗}(t)} with

d_{m}^{∗}(t) = ( \frac{(1 − t)^{m} + (1 + t)^{m}}{2} )^{n−1} − 1.

For sufficiently large n, we have

d_{2}^{∗}(1/\sqrt{2}) ∼ (3/2)^{n−1} ; d_{4}^{∗}(1/\sqrt{2}) ∼ (17/4)^{n−1} ; d_{3}^{∗}(1/\sqrt{2}) ∼ (5/2)^{n−1} .

Hence C_{3} ⩾ (51/50)^{n−1} → ∞ (n → ∞). □

Therefore, we have to reformulate the problem.

Question 2. Is there a constant C_{m,n} such that d_{m−1}^{(n)} d_{m+1}^{(n)} ⩽ C_{m,n} (d_{m}^{(n)} )^{2}, for each m, n ⩾ 2, independently of sequences p̃_{n}, x̃_{n} ∈ S_{+} ?

The best possible constant (if exists) is given by

C_{m,n} = sup\{ \frac{d_{m−1}^{(n)} d_{m+1}^{(n)}}{(d_{m}^{(n)} )^{2}} | p̃_{n}, x̃_{n} ∈ S_{+} \}

Examining the sequence d^{∗}, we conclude that C_{m,n} ⩾ (1 + C/m^{2})^{n−1} , where C is an absolute constant.

References

[1] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1978.

[2] D. S. Mitrinović, Analytic Inequalities, Springer-Verlag, Berlin, 1970.

[3] P. A. MacMahon, Combinatory Analysis, Chelsea, New York, 1960.

[4] S. Simić, Turan’s inequality for Appell polynomials, J. Inequal. Appl. (2006), 1–7.

Mathematical Institute SANU (Received 19 11 2008) Kneza Mihaila 36 (Revised 16 01 2009) Belgrade Serbia

    ssimic@mi.sanu.ac.rs