Notes on the Schur-convexity of the Extended Mean Values

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Notes on the Schur-convexity of the Extended Mean Values

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Qi, Feng, Sándor, József, Dragomir, Sever S and Sofo, Anthony (2002) Notes on the Schur-convexity of the Extended Mean Values. RGMIA research report collection, 5 (1).

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NOTES ON THE SCHUR-CONVEXITY OF THE EXTENDED MEAN VALUES

FENG QI, J ´OZSEF S ´ANDOR, SEVER S. DRAGOMIR, AND ANTHONY SOFO

Abstract. In this article, the Schur-convexities of the weighted arithmetic mean of function and the extended mean values are proved. Moreover, some inequalities involving the arithmetic mean, the harmonic mean, the logarithmic mean, and comparison between the extended mean values and the generalized weighted mean with two parameters and constant weight are obtained.

1. Introduction

It is well-known that, in 1975, the extended mean valuesE(r, s;x, y) were defined in [17] by K. B. Stolarsky as follows:

E(r, s;x, y) = r

s·y^s−x^s y^r−x^r

^1/(s−r)

, rs(r−s)(x−y)6= 0; (1) E(r,0;x, y) =

1

r· y^r−x^r lny−lnx

^1/r

, r(x−y)6= 0; (2) E(r, r;x, y) = 1

e^1/r x^x^r

y^y^r

^1/(x^r−y^r)

, r(x−y)6= 0; (3) E(0,0;x, y) =√

xy, x6=y; (4)

E(r, s;x, x) =x, x=y;

wherex, y >0 andr, s∈R.

Date: November 28, 2001.

2000Mathematics Subject Classification. Primary 26B25; Secondary 26D07, 26D20.

Key words and phrases. Extended mean values, Schur-convexity, inequality, generalized weighted mean values, weighted arithmetic mean of function.

The first author was supported in part by NSF (#10001016) of China, SF for the Prominent Youth of Henan Province, SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800), SF for Pure Research of Natural Science of the Education Department of Henan Province (#1999110004), Doctor Fund of Jiaozuo Institute of Technology, China.

This paper was typeset usingAMS-L^ATEX.

1

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The monotonicity of extended mean values E(r, s;x, y) has been researched in much literature, please refer to [1,4,9, 13, 14,16]. It can be stated as follows.

Theorem A. The extended mean valuesE(r, s;x, y) is increasing in bothxandy and in bothr ands.

The comparison of the extended mean values was reseached in [4,7].

Theorem B. Let r, s, u, v be real numbers with r6=s,u6=v, then the inequality

E(r, s;a, b)≤E(u, v;a, b) (5)

is satisfied for alla, b >0 if and only if

r+s≤u+v and e(r, s)≤e(u, v), (6)

where

e(x, y) =









 x−y

ln^x_y forxy >0 andx6=y, 0 forxy= 0

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if either0≤min{r, s, u, v} or max{r, s, u, v} ≤0, or e(x, y) =|x| − |y|

x−y forx, y∈Randx6=y (8)

if min{r, s, u, v}<0<max{r, s, u, v}.

In [11], the first author verified the logarithmic convexity of the extended mean valuesE(r, s;x, y) with two parametersrand sas follows.

Theorem C. For all fixedx, y >0ands∈[0,+∞) (orr∈[0,+∞), respectively), the extended mean values E(r, s;x, y) are logarithmically concave in r (or in s, respectively) on [0,+∞); For all fixed x, y >0 ands ∈(−∞,0] (or r∈ (−∞,0], respectively), the extended mean valuesE(r, s;x, y) are logarithmically convex inr (or ins, respectively) on(−∞,0].

For completeness, we list the definition of Schur-convex of function as follows.

Definition 1 ([8, pp. 75–76]). A function f with n arguments defined on Iⁿ is Schur-convex on Iⁿ if f(x) ≤ f(y) for each two n-tuples x = (x1, . . . , xn) and y= (y1, . . . , yn) inIⁿ such thatx≺y holds, whereIis an interval with nonempty interior.

The relationship of majorizationx≺y means that

k

X

i=1

x_[i] ≤

k

X

i=1

y_[i],

n

X

i=1

x_[i] =

n

X

i=1

y_[i], (9)

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NOTES ON THE SCHUR-CONVEXITY OF THE EXTENDED MEAN VALUES 3

where 1≤k≤n−1,x[i] denotes theith largest component inx.

A functionf is Schur-concave if and only if−f is Schur-convex.

The generalized weighted mean of positive sequencea= (a₁,· · · , a_n) was defined in [10] as follows.

Definition 2. For a positive sequencea= (a1,· · ·, an) withai>0 and a positive weightw= (w1,· · ·, wn) withwi>0 for 1≤i≤n, the generalized weighted mean of positive sequenceawith two parametersrandsis defined as

Mn(w;a;r, s) =









 Pⁿ

i=1wia^r_i Pn

i=1w_ia^s_i

^1/(r−s)

, r−s6= 0;

exp Pn

i=1w_ia^r_ilna_i Pn

i=1wia^r_i

, r−s= 0.

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The monotonicity of the generalized weighted mean of positive sequence a = (a₁,· · ·, a_n) was proved in [10] and can be stated as follows.

Theorem D. The generalized weighted mean Mn(w;a;r, s) of positive sequence a= (a1,· · ·, an), with positive weightw= (w1,· · · , wn)and two parameters rand s, is increasing in bothr ands.

The first author proved in [12] the following Schur-convexity of the extended mean valuesE(r, s;x, y).

Theorem E. For fixed x, y >0 andx6=y, the extended mean values E(r, s;x, y) are Schur-concave on[0,+∞)×[0,+∞), the first quadrants, and Schur-convex on (−∞,0]×(−∞,0], the third quadrants, with(r, s), respectively.

The following necessary and sufficient condition was stated in [6, p. 57] and [8, p. 333] and was cited in [3].

Theorem F. A continuously differentiable functionf onI²(whereIbeing an open interval)is Schur-convex if and only if it is symmetric and satisfies that

∂f

∂y −∂f

∂x

(y−x)>0 for allx, y∈I,x6=y. (11)

In [3], the Schur-convexity of the arithmetic mean of function was obtained as follows.

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Theorem G. Let f be a continuous function on I. Then the arithmetic mean of function f (or the integral arithmetic mean),

φ(u, v) =





 1 v−u

Z v u

f(t) dt, u6=v,

f(r), u=v,

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is Schur-convex (Schur-concave)onI² if and only if f is convex (concave) onI.

Meanwhile, the Schur-convexity of the logarithmic mean values was verified in the paper [3].

In this article, as a subsequent paper of [12], our main purpose is to prove the Schur-convexities of the weighted arithmetic mean of function and the extended mean valuesE(r, s;x, y) with respect to (x, y) for fixed (r, s), and then we obtain the following

Theorem 1. Let f be a continuous function on I, let p be a positive continuous weight onI. Then the weighted arithmetic mean of functionf with weightpdefined by

F(x, y) =









 Ry

xp(t)f(t) dt Ry

xp(t) dt , x6=y,

f(x), x=y

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is Schur-convex (Schur-concave)onI² if and only if inequality Ry

x p(t)f(t) dt Ry

xp(t) dt ≤ p(x)f(x) +p(y)f(y)

p(x) +p(y) (14)

holds (reverses)for allx, y∈I.

Theorem 2. Let x >0 andy >0 be positive real numbers andr∈R. (1) If r≤0, then

L(x^r, y^r)≥[G(x, y)]^r≥A(x, y)H(x^r−1, y^r−1), (15) the equalities in (15)hold only ifx=y orr= 0.

(2) If r≥ ³₂, we have

L(x^r, y^r)≥A(x, y)H(x^r−1, y^r−1), (16) the equality in (16)holds only ifx=y.

(3) If r∈(0,1], inequality (16)reverses without equality unless x=y.

(4) Otherwise, the validity of inequality (16) may not be certain.

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Theorem 3. For fixed point(r, s)such thatr, s6∈(0,³₂) (orr, s∈(0,1], resp.), the extended mean values E(r, s;x, y) is Schur-concave (or Schur-convex, resp.) with (x, y)on the domain (0,∞)×(0,∞).

Corollary 1. Letx, y >0. Then (1) ifr, s∈(0,1], we have

E(r, s;x, y)≤M₂((1,1); (x, y);r−1, s−1), (17) where M₂((1,1); (x, y);r−1, s−1) denotes the generalized weighted mean of positive sequence(x, y)with two parametersr−1ands−1and constant weight(1,1) defined in Definition2;

(2) ifr, s6∈(0,³₂), inequality (17)reverses;

(3) otherwise, the validity of inequality (17)may not be certain.

2. Proofs of Theorems Proof of Theorem 1. The function F is obviously symmetric.

Straightforward computation gives us ∂F

∂y −∂F

∂x

(y−x) =

p(y)f(y) +p(x)f(x) p(x) +p(y) −

Ry

xp(t)f(t) dt Ry

xp(t) dt

p(x) +p(y) Ry

xp(t) dt . (18)

The proof follows from Theorem F.

Proof of Theorem 2. For r= 0, it is easy to see that equality in (16) holds for all x, y >0.

Case 1. Forr <0, sets=−r >0, then inequality (16) can be rewritten as L 1

x^s, 1 y^s

≥ x+y

2 H 1

x^s+1, 1 y^s+1

, (19)

which is equivalent to

y^s−x^s

s(lny−lnx)x^sy^s ≥ x+y

x^s+1+y^s+1. (20)

From the logarithmic mean inequalityL(a, b)≥√

abfora, b >0 (see [15]), we have y^s−x^s

s(lny−lnx) ≥√

x^sy^s. (21)

Since the function u(t) = t^s+1 is convex on (0,∞) for s >−1, from definition of convex function it follows that

x^s+1+y^s+1

2 ≥x+y

2 s+1

(22)

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fors >−1. Combining (22) with the arithmetic-geometric mean inequality yields that

x^s+1+y^s+1≥(x+y)x+y 2

s

≥(x+y)(√

xy)^s, (23)

then we have

√1

x^sy^s ≥ x+y

x^s+1+y^s+1. (24)

Therefore, from (20), (21) and (24), it follows that y^s−x^s

s(lny−lnx)x^sy^s ≥ 1

√x^sy^s ≥ x+y

x^s+1+y^s+1, (25)

which implies inequality (15) forr <0.

Case 2. Ifr >0, without loss of generality, assumey > x >0, then inequality (16) becomes

(y^r−x^r)(y^r−1+x^r−1)≤r(x+y)x^r−1y^r−1lny

x. (26)

Dividing on both sides of (26) byx^2r−1 produces y^r

x^r −1

y^r−1 x^r−1+ 1

≤r

1 + y x

y^r−1 x^r−1lny

x. (27)

Let ^y_x=t >1 and define a functionp(t) on (1,∞) such that

p(t) = (1−t^r)(1 +t^r−1) +r(1 +t)t^r−1lnt. (28) Direct and standard calculating leads to

p⁰(t) =t^r−2[(2r−1)(1−t^r) +r(r−1 +rt) lnt],t^r−2g(t), g⁰(t) =r(r−1) +r²t+r(1−2r)t^r+r²tlnt

t , h(t)

t , h⁰(t) =r²[2 + lnt+ (1−2r)t^r−1],

h⁰⁰(t) =r²[1 + (1−2r)(r−1)t^r−1]

t ,r²w(t)

t .

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Case 2.1. Forr∈[¹₂,1] the function w(t)>0 andh⁰⁰(t)>0, thenh⁰(t) increases.

Since h⁰(1) = r²(3−2r) > 0, we have h⁰(t)> 0, and then h(t) increases. Since h(1) = 0, thush(t)>0, andg⁰(t)>0, and then g(t) is increasing. Fromg(1) = 0 it follows that g(t)> 0, which means that p⁰(t) >0 and p(t) increases. Further, sincep(1) = 0, we obtainp(t)>0 forr∈[¹₂,1] andt∈(1,∞). This implies that inequality (16) is reversed forr∈[¹₂,1].

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Case 2.2. Forr≥ ³₂, the functionw(t) decreases andw(1) =r(3−2r)≤0, and then w(t)≤0, and h⁰⁰(t)≤0 and h⁰(t) decreases. Since h⁰(1) ≤0, we haveh⁰(t)≤0, andh(t) is decreasing. From h(1) = 0 it follows that h(t)≤0, and g⁰(t)≤0, and then g(t) is dereasing. The fact thatg(1) = 0 yieldsg(t)≤0, and p⁰(t)≤0, and thenp(t) is decreasing. The fact thatp(1) = 0 results inp(t)≤0. This means that inequality (16) holds forr≥ ³₂.

Case 2.3. For 0< r < ¹₂, it is easy to see that the functionw(t) is increasing. Since w(1) =r(3−2r)>0, we obtainw(t)>0, andh⁰⁰(t)>0, and thenh⁰(t) increases strictly. The fact thath⁰(1) =r²(3−2r)>0 leads toh⁰(t)>0, andh(t) increases.

Meanwhile,h(1) = 0 producesh(t)>0, andg⁰(t)>0, and theng(t) is increasing.

since g(1) = 0, thus g(t) > 0 and p⁰(t) > 0, and then p(t) is increasing. From p(1) = 0, it follows thatp(t)>0, that is, inequality (16) reverses forr∈(0,¹₂).

Case 2.4. For r ∈ (1,³₂), the function w(t) has a zero t0 = [(r−1)(2r−1)]¹ ^1/(r−1). Rearranging equality w(1) = (1−2r)(r−1) + 1 = r(3 −2r) > 0 yields that 0<(r−1)(2r−1) = 1−w(1)<1, hence we havet0>1.

In the case oft∈(1, t0), we havew(t)>0 andh⁰⁰(t)>0, sincew(t) is decreasing for all t > 1 and r ∈ (1,³₂). By the same arguments as in Case 2.1, we obtain that inequality (16) is reversed when ^y_x ∈ 1,1/[(r−1)(2r−1)]^1/(r−1)

, where r∈(1,³₂).

In the case of t ∈ (t0,∞), we have w(t) < 0 and h⁰⁰(t) < 0, and then h⁰(t) decreases. It is easy to see that limt→∞h⁰(t) = −∞. Therefore, there exists a pointt1such thatt1≥t0andh⁰(t)<0 fort∈(t1,∞). On the interval (t1,∞), the functionh(t) decreases and lim_t→∞h(t) =−∞. Similarly, there exists a numbert2

such thatt2≥t1andh(t)<0 andg⁰(t)<0 fort∈(t2,∞). On the interval (t2,∞), the function g(t) decreases and lim_t→∞g(t) = −∞. Then there exists another number t3 ≥ t2 such that g(t) <0 and p⁰(t)< 0, and thenp(t) is decreasing on the interval (t3,∞). Since lim_t→∞p(t) =−∞, then there exists a number t4 ≥t3

such thatp(t) is negative on the interval (t4,∞). This means that, for ^y_x ∈(t4,∞) andr∈(1,³₂), inequality (16) holds. Note that the numbersti, 0≤i≤4, are all dependent onrundoubtedly.

Thus, forr∈(1,³₂), the validity of inequality (16) depends on values of the ratio

y

x, that is, inequality (16) cannot hold for allx, y >0. The proof is complete.

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Proof of Theorem 3. Forx, y >0 andt∈R, let us define a functiong by

g(t),g(t;x, y),







(y^t−x^t)

t , t6= 0;

lny−lnx, t= 0.

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It is easy to see thatg can be expressed in integral form as g(t;x, y) =

Z y x

u^t−1du, (31)

and

g⁽ⁿ⁾(t) = Z y

x

(lnu)ⁿu^t−1du. (32)

Therefore, in [1,11,15,16], the extended mean valuesE(r, s;x, y) were represented in terms ofg by

E(r, s;x, y) =











g(s;x, y) g(r;x, y)

1/(s−r)

, (r−s)(x−y)6= 0;

exp

∂g(r;x, y)/∂r g(r;x, y)

, r=s, x−y6= 0.

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To prove the Schur-convexity of the extended mean values, from Theorem1, it suffices to prove the following inequality

g(r;x, y) g(s;x, y) =

Ry xt^r−1dt Ry

xt^s−1dt = s(y^r−x^r)

r(y^s−x^s) <x^r−1+y^r−1

x^s−1+y^s−1, (34) which is equivalent to the monotonicity withtof function _xt−1^g(t;x,y)+y^t−1, this is further reduced to the reversed inequality of (16), since

d dt

g(t;x, y) (x^t−1+y^t−1)

= [lny−lnx]

A(x, y)H(x^t−1, y^t−1)−L(x^t, y^t)

t(x^t−1+y^t−1) . (35)

Therefore, the proof of Theorem3follows.

Proof of Corollary1. This follows from standard argument by combining (34) and Theorem2 with Definition2 and definition of the extended mean values.

3. Open Problems At last, we propose the following open problem.

Open Problem. Under what conditions do the following inequalities

f

xp(x) +yp(y) p(x) +p(y)

≤ Ry

xp(t)f(t) dt Ry

xp(t) dt ≤ p(x)f(x) +p(y)f(y)

p(x) +p(y) (36)

hold for allx, y∈I? where I denotes an interval on Randp(x)is positive.

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Acknowledgements. This paper was finalized during the first author’s visit to the RGMIA between November 1, 2001 and January 31, 2002, as a Visiting Professor with grants from the Victoria University of Technology and Jiaozuo Institute of Technology.

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(Qi) Department of Mathematics, Jiaozuo Institute of Technology, Jiaozuo City, Henan 454000, China

E-mail address:[email protected] or [email protected] URL:http://rgmia.vu.edu.au/qi.html

(S´andor)Department of Mathematics, Babes¸-Bolyai University, Str. Kogalniceanu, 3400 Cluj-Napoca, Romania

E-mail address:[email protected]

(Dragomir)School of Communications and Informatics, Victoria University of Tech- nology, P. O. Box 14428, Melbourne City MC, Victoria 8001, Australia

E-mail address:[email protected] URL:http://rgmia.vu.edu.au/SSDragomirWeb.html

(Sofo)School of Communications and Informatics, Victoria University of Technol- ogy, P. O. Box 14428, Melbourne City MC, Victoria 8001, Australia

E-mail address:[email protected]

URL:http://cams.vu.edu.au/staff/anthonys.html