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volume 6, issue 3, article 75, 2005.

Received 24 November, 2004;

accepted 31 May, 2005.

Communicated by:F. Qi

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Journal of Inequalities in Pure and Applied Mathematics

INEQUALITIES FOR WEIGHTED POWER PSEUDO MEANS

VASILE MIHESAN

Department of Mathematics Technical University of Cluj-Napoca Str. C.Daicoviciu nr.15

400020 Cluj-Napoca Romania.

EMail:Vasile.Mihesan@math.utcluj.ro

c

2000Victoria University ISSN (electronic): 1443-5756 227-04

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Inequalities for Weighted Power Pseudo Means

Vasile Mihesan

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J. Ineq. Pure and Appl. Math. 6(3) Art. 75, 2005

http://jipam.vu.edu.au

Abstract

In this paper we denote bym^[r]n the following expression, which is closely connected to the weighted power means of orderr, M_n^[r].

Letn≥2be a fixed integer and

m^[r]_n(x;p) =





 Pn

p1x^r₁−_p¹

1

Pn i=2p_ix^r_i¹_r

, r6= 0 x^P₁ⁿ^/p¹

.Qn

i=2x^p_iⁱ^/p¹, r= 0

(x∈Rr),

whereP_n=Pn

i=1p_iandR_rdenotes the set of the vectorsx= (x₁, x₂, . . . , x_n) for whichxi > 0 (i = 1,2, . . . , n),p = (p1, p2, . . . , pn), p1 > 0, pi ≥ 0 (i = 1,2, . . . , n)andP_nx^r₁>Pn

i=2p_ix^r_i.

Three inequalities are presented form^[r]n. The first is a comparison theorem. The second and the third is Rado type inequalities. The proofs show that the above inequalities are consequences of some well-known inequalities for weighted power means.

2000 Mathematics Subject Classification:26D15, 26E60.

Key words: Weighted power pseudo means, Inequalities.

1. Introduction

Lety = (y1, y2, . . . , yn)andq = (q1, q2, . . . , qn)be positiven-tuples, then the arithmetic and geometric means ofywith weightsqare defined by

A_n(y;q) = 1 Q_n

n

X

i=1

q_iy_i and G_n(y;q) =

n

Y

i=1

y^q_iⁱ

!_Qn¹ ,

where Q_n =

n

X

i=1

q_i.

If r is a real number, then the r-th power means of y with weights q, Mn^[r](y;q)is defined by

(1.1) M_n^[r](y;q) =











1 Qn

n

P

i=1

q_iy_i^r ¹_r

, r 6= 0;

_n Q

i=1

y^q_iⁱ _Qn¹

, r = 0.

Ifr, s∈R, r≤sthen [11]

(1.2) M_n^[r](y;q)≤M_n^[s](y,q)

is valid for all positive real numbersy_i andq_i (i = 1,2, . . . , n). Forr = 0 and s = 1 we obtain the clasical inequality between the weighted arithmetic and geometric means

(1.3) G_n=G_n(y;q) =

n

Y

i=1

y^q_iⁱ^/Qⁿ ≤ 1 Q_n

n

X

i=1

q_iy_i =A_n(y,q) = A_n.

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In this paper we denote by m^[r]n (y;q) the following expression which is closely connected toMn^[r](y;q).

Letn ≥2be an integer (considered fixed throughout the paper) and define

(1.4) m^[r]_n(x;p) =











Pn

p1x^r₁− _p¹

1

n

P

i=2

pix^r_i ¹_r

, r6= 0 x^P₁ⁿ^/p¹

_n Q

i=2

x^p_iⁱ^/p¹, r= 0

(x∈R_r)

whereP_n=Pn

i=1p_iandR_rdenotes the set of the vectorsx= (x₁, x₂, . . . , x_n) for which x_i > 0 (i = 1,2, . . . , n), p = (p₁, p₂, . . . , p_n), p₁ > 0, p_i ≥ 0 (i= 2,3, . . . , n)andP_nx^r₁ >Pn

i=2p_ix^r_i.

Although there is no general agreement in literature about what constitutes a mean value most authors consider the intermediate property as the main fea- ture. Since m^[r]n (x;p)do not satisfy this condition, this means that the double inequalities

1≤i≤nminx_i ≤m^[r]_n (x;p)≤ max

1≤i≤nx_i

are not true for all positive x_i, we call m^[r]n the weighted power pseudo means of orderr.

Forr = 1we obtain by (1.4) the pseudo arithmetic meansa_n(x,p)forr= 0 the pseudo geometric means,g_n(x,p), see [2]. In 1990, H. Alzer [2] published the following companion of inequality (1.3):

(1.5) a_n(x;p)≤g_n(x;p).

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For the special casep₁ = p₂ = · · ·= p_nthe inequality (1.5) was proved by S. Iwamoto, R.J. Tomkins and C.L. Wang [6].

Rado and Popoviciu type inequalities for pseudo arithmetic and geometric means were given in [2], [9], [10].

We note that inequality (1.5) is an example of a so called reverse inequality.

One of the first reverse inequalities was published by J. Aczél [1] who proved the following intriguing variant of the Cauchy-Schwarz inequality:

If x_i and y_i (i = 1,2, . . . , n) are real numbers with x²₁ > Pn

i=2x²_i and y₁² >Pn

i=2y_i², then (1.6) x₁y₁−

n

X

i=2

x_iy_i

!2

≥ x²₁−

n

X

i=2

x²_i

! y²₁ −

n

X

i=2

y_i²

! .

Further interesting reverse inequalities were given in [3], [5], [6], [7], [8], [11], [12].

The aim of this paper is to prove a comparison theorem and Rado type inequalities for the weighted power pseudo means.

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2. Comparison Theorem

Our first result is a comparison theorem for the weighted power pseudo means.

Theorem 2.1. If0≤r≤s, x∈R_s then x∈R_rand (2.1) m^[s]_n(x,p)≤m^[r]_n (x,p).

Ifr≤s≤0, x∈R_r then x∈R_sand

(2.2) m^[s]_n (x;p)≤m^[r]_n (x;p).

If r < 0 < s then R_r ∩R_s = ∅, hence m^[r]n(x,p), m^[s]n(x,p) cannot both be defined, they are not comparable.

Proof. To prove (2.1) leta=m^[s]n (x,p)>0, then we obtain by (1.4) and (1.2)

x₁ =

p₁a^s+Pn i=2p_ix^s_i P_n

¹_s

≥

p₁a^r+Pn i=2p_ix^r_i P_n

¹_r , hence

P_n p₁x^r₁−

Pn i=2pix^r_i

p₁ ≥a^r >0

which shows thatx∈R_r. Taking therth root, we obtain (2.1).

To prove (2.2) letb=m^[r]n (x,p)>0, then we obtain by (1.4) and (1.2), x₁ =

p₁b^r+Pn i=2p_ix^r_i P_n

¹_r

≤

p₁b^s+Pn i=2p_ix^b_i P_n

¹_s .

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Hence

x^s₁ ≥ p₁b^s+Pn i=2p_ix^s_i P_n

and

Pn

p₁x^s₁− Pn

i=2pix^s_i

p₁ ≥b^s >0,

which shows thatx∈R_s. Taking the(−s)th root, we obtain (2.2).

Ifr <0< swe infer forn = 2, p₁ =p₂ thatx₁, x₂ >0, x^r₁ > x^r₂, x^s₁ > x^s₂ hencex₁ < x₂ andx₁ > x₂, which is impossible.

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3. Rado Type Inequalities for Weighted Power Pseudo Means

The well-known extension of the arithmetic mean-geometric mean inequality (1.3) is the following inequality of Rado [11]:

(3.1) Q_n(A_n(y;q)−G_n(y;q))≥Qn−1(An−1(y;q)−Gn−1(y;q)).

The next proposition provides an analog of the Rado inequality (3.1) for pseudo arithmetic and geometric means [2].

Proposition 3.1. For all positive real numbersx_i (i = 1,2, . . . , n; n ≥ 2)we have

(3.2) g_n(x,p)−a_n(x;p)≥gn−1(x;p)−an−1(x;p).

The most obvious extension is to allow the means in the Rado inequality to have different weights [4]

Q_nA_n(y;q)− q_n pn

P_nG_n(y;p)≥Q_n−1A_n−1(y;q)− q_n pn

P_n−1G_n−1(y;p).

Using this inequality we obtain the following generalization of the inequality (3.2) [10].

Proposition 3.2. For all positive real numbersx_i (i = 1,2, . . . , n; n ≥ 2)we have

(3.3) g_n(x;p)−a_n(x;q)≥gn−1(x;p)−an−1(x;q).

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An extension of the Rado inequality for weighted power means is the following inequality [4]: Ifr, s, t∈Rsuch thatr/t≤1ands/t ≥1then

(3.4) Pn

M_n^[s](y;p)^t

− M_n^[r](y;p)^t

≥Pn−1

M_n−1^[s] (y;p)t

−

M_n−1^[r] (y,p)t . Using inequality (3.4) we obtain generalizations of the inequality of Rado type (3.2) for the weighted power pseudo means.

Theorem 3.3. Ifr≤1,x∈R_randx^r₁ ≤x^r_nthen

(3.5) m^[r]_n (x,p)−a_n(x,p)≥m^[r]_n−1(x,p)−an−1(x,p).

Ifs≥1,x∈R_s andx₁ ≤x_nthen

(3.6) a_n(x,p)−m^[s]_n (x,p)≥an−1(x,p)−m^[s]_n−1(x,p).

Proof. To prove (3.5) we put in (3.4)s = t = 1 and we obtain for r ≤ 1the inequality:

(3.7) P_n A_n(y;p)−M_n^[r](y;p)

≥Pn−1

An−1(y;p)−M_n−1^[r] (y;p) .

If we set in (3.7)y₁ =m^[r]n(x,p), y_i =x_i (i= 2,3, . . . , n)then we have:

P_n A_n(y;p)−M_n^[r](y;p)

=p₁ m^[r]_n (x;p)−a_n(x;p) ,

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which leads to inequality (3.5). We observe that forr≤1,x∈R_randx^r₁ ≤x^r_n we have

0< P_nx^r₁ −

n

X

i=2

p_ix^r_i ≤Pn−1x^r₁−

n−1

X

i=2

p_ix^r_i andm^[r]_n−1(x,p)exist.

To prove (3.6) we set in (3.4) r = t = 1 and we obtain for s ≥ 1 the inequality

(3.8) P_n M_n^(s)(y;p)−A_n(y;p)

≥Pn−1

M_n−1^[s] (y;p)−An−1(y;p) .

If we put in (3.8)y₁ =m^[s]n(x,p), y_i =x_i (i= 2,3, . . . , n)then we have Pn M_n^[s](y;p)−An(y;p)

=p1 an(x;p)−m^[s]_n (x;p) ,

which leads to inequality (3.6). Fors ≥ 1, x ∈ R_s andx₁ ≤ x_n, m^[s]_n−1(x;p) exist.

Theorem 3.4. If0< r≤s,x∈R_sandx₁ ≤x_nthen (3.9) m^[r]_n (x;p)^s

− m^[s]_n (x;p)^s

≥

m^[r]_n−1(x;p)s

−

m^[s]_n−1(x;p)s

and

(3.10) m^[r]_n (x;p)^r

− m^[s]_n (x;p)^r

≥

m^[s]_n−1(x;p)r

−

m^[s]_n−1(x;p)r

.

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Proof. To prove (3.9) we put in (3.4)t = s and we obtain for 0 < r ≤ sthe inequality

(3.11) P_n

M_n^[s](y;p)s

− M_n^[r](y;p)s

≥Pn−1

M_n−1^[s] (y,p)s

−

M_n−1^[r] (y,p)s .

If we set in (3.11)y₁ =m^[r]n(x;p), y_i =x_i (i= 2,3, . . . , n)then we have P_n

M_n^[s](y;p)^s

− M_n^[r](y;p)^s

=p₁

m^[r]_n (x;p)^s

− m^[s]_n (x;p)^s , which leads to inequality (3.9) If0< r≤s,x∈R_sthenx∈R_rand ifx₁ ≤x_n thenm^[r]_n−1(x;p)exists.

To prove (3.10) we set in (3.4) t = r and we obtain for 0 < r ≤ s the inequality

(3.12) P_n

M_n^[s](y;p)^r

− M_n^[r](y;p)^r

≥Pn−1

M_n−1^[s] (y;p) r

−

M_n−1^[r] (y;p) r

.

If we put in (3.12)y₁ =m^[s]n(x,p)y_i =x_i(i= 2,3, . . . , n)then we have P_n

M_n^[s](y;p)^r

− M_n^[r](y;p)^r

=p₁

m^[r]_n (x;p)^r

− m^[s]_n (x;p)^r , which leads to inequality (3.10).

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References

[1] J. ACZÉL, Some general methods in the theory of functional equations in one variable.New applications of functional equations (Russian), Uspehi Mat. Nauk (N.S.), 11(3) (69) (1956), 3–58.

[2] H. ALZER, Inequalities for pseudo arithmetic and geometric means, In- ternational Series of Numerical Mathematics, Vol. 103, Birkhauser-Verlag Basel, 1992, 5–16.

[3] R. BELLMAN, On an inequality concerning an indefinite form, Amer.

Math. Monthly, 63 (1956), 108–109.

[4] P.S. BULLEN, D.S. MITRINOVI ´C AND P.M. VASI ´C, Means and Their Inequalities, Reidel Publ. Co., Dordrecht, 1988.

[5] Y.J. CHO, M. MATI ´CANDJ. PE ˇCARI ´C, Improvements of some inequal- ities of Aczél’s type, J. Math. Anal. Appl., 256 (2001), 226–240.

[6] S. IWAMOTO, R.J. TOMKINS ANDC.L. WANG, Some theorems on re- verse inequalities, J. Math. Anal. Appl., 119 (1986), 282–299.

[7] L. LOSONCZI, Inequalities for indefinite forms, J. Math. Anal. Appl., 285 (1997),148–156.

[8] V. MIHE ¸SAN, Applications of continuous dynamic programing to inverse inequalities, General Mathematics, 2(1994), 53–60.

[9] V. MIHE ¸SAN, Popoviciu type inequalities for pseudo arithmetic and geometric means, (in press)

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[10] V. MIHE ¸SAN, Rado and Popoviciu type inequalities for pseudo arithmetic and geometric means, (in press)

[11] D.S. MITRINOVI ´C, Analytic Inequalities, Springer Verlag, New York, 1970.

[12] X.H. SUN, Aczél-Chebyshev type inequality for positive linear functional, J. Math. Anal. Appl., 245 (2000), 393–403.