An Extension of Stolarsky Means to the Multivariable Case

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Volume 2009, Article ID 432857,14pages doi:10.1155/2009/432857

Research Article

An Extension of Stolarsky Means to the Multivariable Case

Slavko Simic

Mathematical Institute SANU, Kneza Mihaila 36, 11000 Belgrade, Serbia

Correspondence should be addressed to Slavko Simic,[email protected] Received 10 July 2009; Accepted 23 September 2009

Recommended by Feng Qi

We give an extension of well-known Stolarsky means to the multivariable case in a simple and applicable way. Some basic inequalities concerning this matter are also established with applications in Analysis and Probability Theory.

Copyrightq2009 Slavko Simic. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

There is a huge amount of papers investigating properties of the so-called Stolar- sky or extended two-parametric mean value, defined for positive values of x, y, as

E_r,s x, y

: r

x^s−y^s s

x^r−y^r

_1/s−r

,

rsr−s x−y

/0.

1.1

Emeans can be continuously extended on the domain

r, s;x, y

| r, s∈R;x, y∈R

1.2

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by the following:

Er,s

x, y

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ r

x^s−y^s s

x^r −y^r

_1/s−r

rsr−s/0;

exp

−1

s x^slogx−y^slogy x^s−y^s

, rs /0;

x^s−y^s s

logx−logy _1/s

, s /0, r0;

√xy, rs0;

x, yx >0,

1.3

and in this form are introduced by Keneth Stolarsky in1.

Most of the classical two-variable means are special cases of the classE. For example, E1,2 xy/2 is the arithmetic mean,E0,0 √

xy is the geometric mean, E0,1 x− y/logx−logyis the logarithmic mean,E_1,1 x^x/y^y^1/x−y/eis the identric mean, and so forth. More generally, therth power meanx^ry^r/2^1/ris equal toE_r,2r.

Recently, several papers are produced trying to define an extension of the classEto n, n >2 variables. Unfortunately, this is done in a highly artificial modecf.2–4, without a practical background. Here is an illustration of this point; recently Merikowski 4has proposed the following generalization of the Stolarsky meanE_r,sto several variables:

E_r,sX:

LX^s LX^r

_1/s−r

, r /s, 1.4

whereX x1, . . . , x_nis ann-tuple of positive numbers and

LX^s: n−1!

En−1

n i1

x^su_i ⁱdu₁· · ·du_n−1. 1.5

The symbolE_n−1stands for the Euclidean simplex which is defined by

E_n−1:{u1, . . . , u_n−1:u_i ≥0, 1≤i≤n−1; u₁· · ·u_n−1≤1}. 1.6

In this paper, we give another attempt to generalize Stolarsky means to the multivariable case in a simple and applicable way. The proposed task can be accomplished by founding a “weighted” variant of the classE, wherefrom the mentioned generalization follows naturally.

In the sequel, we will need notions of the weighted geometric meanG Gp, q;x, y and weightedrth power meanS_r S_rp, q;x, y, defined by

G:x^py^q; Sr :

px^rqy^r_1/r

, 1.7

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where

p, q, x, y∈R; pq1; r ∈R/{0}. 1.8

Note thatSr^r >G^r forx /y, r /0,and lim_r_→₀S_r G.

1.1. Weighted Stolarsky Means

We introduce here a classWof weighted two-parameters means which includes the Stolarsky classEas a particular case. Namely, forp, q, x, y∈R, pq1, rsr−sx−y/0, we define

WWr,s

p, q;x, y :

r² s²

Ss^s−G^s Sr^r−G^r

_1/s−r

r²

s²

px^sqy^s−x^psy^qs px^rqy^r−x^pry^qr

_1/s−r

. 1.9

Various properties concerning the means W can be established; some of them are the following:

W_r,s

p, q;x, y

W_s,r

p, q;x, y

; Wr,s

p, q;x, y Wr,s

q, p;y, x

; Wr,s

p, q;y, x

xyWr,s

p, q;x⁻¹, y⁻¹

; W_ar,as

p, q;x, y

W_r,s

p, q;x^a, y^a1/a

, a /0.

1.10

Note that

W2r,2s

1 2,1

2;x, y

r²

s²

x^2sy^2s−2√xy2s

x^2ry^2r−2√xy2r

1/2s−r

r²

s²

x^s−y^s₂ x^r−y^r₂

1/2s−r

E

r, s;x, y .

1.11

In the same manner, we get

Wr,s

2 3,1

3;x³, y³

2x^sy^s 2x^ry^r

_1/s−r

E

r, s;x, y₂

;

W_r,s 3

4,1 4;x⁴, y⁴

3x^2s− xy_s

y^2s 3x^2r−

xy_r y^2r

_1/s−r

E

r, s;x, y2

.

1.12

The weighted means from the classWcan be extended continuously to the domain

D

r, s;x, y

| r, s∈R;x, y∈R

. 1.13

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This extension is given by W_r,s

p, q;x, y

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ r²

s²

px^sqy^s−x^psy^qs px^rqy^r−x^pry^qr

_1/s−r

, rsr−s

x−y /0;

2px^sqy^s−x^psy^qs pqs²log²

x/y _1/s

, s

x−y

/0, r0;

exp −2

s px^slogxqy^slogy−

plogxqlogy x^psy^qs px^sqy^s−x^psy^qs

, s

x−y

/0, rs;

x^p1/3y^q1/3, x /y, rs0;

x, xy.

1.14 Note that those means are homogeneous of order 1, that is, W_r,sp, q;tx, ty tWr,sp, q;x, y, t >0, symmetric inr, s,Wr,sp, q;x, y Ws,rp, q;x, ybut are not symmetric inx, yunlesspq1/2.

1.2. Multivariable Case

A natural generalization of weighted Stolarsky means to the multivariable case gives

W_r,sp; x

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

⎛

⎜⎝r²

pix^s_i − x^p_iⁱs s²

p_ix^r_i − x_i^pⁱ_r

⎞

⎟⎠

1/s−r

, rss−r/0;

⎛

⎜⎝2 s²

pix^s_i − x^p_iⁱ_s p_ilog²x_i−

p_i logx_i2

⎞

⎟⎠

1/s

, r0, s /0;

exp

⎛

⎜⎝−2 s

pix^s_i logxi−

pi logxi x_i^pⁱ_s p_ix^s_i −

x_i^pⁱ_s

⎞

⎟⎠, rs /0;

exp

⎛

⎜⎝

pilog³xi−

pi logxi

₃

3

p_ilog²x_i−

p_i logx_i₂

⎞

⎟⎠, rs0,

1.15

wherex x1, x₂, . . . , x_n∈Rⁿ, n ≥2,p is an arbitrary positive weight sequence associated withx andWr,sp; x0 aforx0 a, a, . . . , a.

We also write·,·instead of_n

1·,_n

1·.

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The above formulae are obtained by an appropriate limit process, implying continuity.

For example, applying

t^s1slogts²

2log²ts³

6log³to s³

s−→0, 1.16

we get

W_0,0p; x lim

s→0W_s,0p; x lim

s→0

⎛

⎜⎝2 s²

p_ix^s_i − x_i^pⁱs

pilog²xi−

pi logxi

₂

⎞

⎟⎠

1/s

lim

s→0

⎛

⎜⎝ 2 s²

p_ilog²x_i−

p_i logx_i2

×

p_is

p_ilogx_i s²

2

p_ilog²x_i s³

6

p_ilog³x_i

−

p_islog x^p_iⁱ

s²

2

log² x^p_iⁱ

s³

6

log³ x^p_iⁱ

o

s³⎞

⎟⎠

1/s

lim

s→0

⎛

⎜⎝1

p_ilog³x_i−

p_i logx_i3

3

p_ilog²x_i−

p_i logx_i₂s1o1

⎞

⎟⎠

1/s

exp

⎛

⎜⎝

pilog³xi−

pi logxi

₃

3

pilog²xi−

pi logxi

₂

⎞

⎟⎠.

1.17

Remark 1.1. Analogously to the former considerations, one can define a class of Stolarsky means innvariablesE_r,sx;nas

E_r,sx;n:W_nr,nsp0,x, 1.18

wherep0{1/n}ⁿ₁.

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Therefore,

Er,sx;n r²

s² _n

1x^ns_i −n_n

1x^s_i _n

1x^nr_i −n_n

1x^r_i

_1/ns−r

, rsr−s/0. 1.19

Details are left to the readers.

2. Results

The following basic assertion is of importance.

Proposition 2.1. The expressionsWr,sp; xare actual means, that is, for arbitrary weight sequence p one has

min{x1, x2. . . , xn} ≤Wr,sp; x≤max{x1, x2, . . . , xn}. 2.1

Our main result is contained in the following.

Proposition 2.2. The meansW_r,sp,xare monotone increasing in both variablesrands.

Passing to the continuous variable case, we get the following definition of the class W_r,sp, x.

Assuming that all integrals exist,

W_r,sp,x

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ r²

ptx^stdt−exp s

ptlogxtdt s²

ptx^rtdt−exp r

ptlogxtdt

_1/s−r

, rss−r/0;

2 s²

ptx^stdt−exp s

ptlogxtdt ptlog²xtdt−

ptlogxtdt₂ 1/s

, r0, s /0;

exp −2

s

ptx^stlogxtdt−

ptlogxtdt exp

s

ptlogxtdt ptx^stdt−exp

s

ptlogxtdt

,

rs /0;

exp

⎛

⎜⎝

ptlog³xtdt−

ptlogxtdt3

3

ptlog²xtdt−

ptlogxtdt₂

⎞

⎟⎠, rs0,

2.2 where xt is a positive integrable function and pt is a nonnegative function with ptdt1.

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From our former considerations, a very applicable assertion follows.

Proposition 2.3. W_r,sp,xis monotone increasing in eitherrors.

3. Applications

3.1. Applications in Analysis

As an illustration of the above, we give the following proposition.

Proposition 3.1. The functionws,defined by

ws:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 12

πs²Γ1s−e^−γs _1/s

, s /0;

exp

−γ−4ξ3 π²

, s0,

3.1

is monotone increasing fors∈−1,∞.

In particular, fors∈−1,1,one has

Γ1−se^−γs Γ1se^γs− πs

sinπs ≤1−πs⁴

144 , 3.2

whereΓ·, ξ·, γstands for the Gamma function, Zeta function, and Euler’s constant, respectively.

3.2. Applications in Probability Theory

For a random variableXand an arbitrary probability distribution with support on−∞,∞, it is well known that

Ee^X≥e^EX. 3.3

Denoting the central moment of orderkbyμk μkX : EX−EX^k, we improve this inequality to the following propsositions.

Proposition 3.2. For an arbitrary probability law with support onR, one has

Ee^X ≥

1μ₂ 2

exp

μ₃ 3μ₂

e^EX. 3.4

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Proposition 3.3. One also has that

Ee^sX−e^sEX s²σ_X²/2

_1/s

3.5

is monotone increasing ins.

3.3. Shifted Stolarsky Means

Especially interesting is studying the shifted Stolarsky meansE^∗, defined by E^∗_r,s

x, y : lim

p→0Wr,s

p, q;x, y

. 3.6

Their analytic continuation to the wholer, splane is given by

E_r,s^∗ x, y

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ r²

x^s−y^s

1slog

x/y

s²

x^r−y^r

1r log

x/y

_1/s−r

, rsr−s

x−y /0;

2 s²

x^s−y^s

1slog x/y log²

x/y

_1/s

, s

x−y

/0, r0;

exp −2

s

x^s−y^s

logx−sy^s logy log x/y x^s−y^s

1slog x/y

, s

x−y

/0, rs;

x^1/3y^2/3, r s0;

x, xy.

3.7

Main results concerning the meansE^∗are contained in the following propositions.

Proposition 3.4. MeansE^∗_r,sx, yare monotone increasing in eitherrorsfor each fixedx, y∈R. Proposition 3.5. MeansE^∗_r,sx, yare monotone increasing in eitherxor y for eachr, s∈R.

A well known result of Qi 5 states that the meansEr,sx, y are logarithmically concave for each fixedx, y > 0 andr, s ∈ 0,∞; also, they are logarithmically convex for r, s∈−∞,0.

According to this, we propose the following proposition.

Open Question

Is there any compact intervalI, I⊂Rsuch that the meansE^∗_r,sx, yare logarithmically convex concaveforr, s∈Iand eachx, y∈R?

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A partial answer to this problem is given in what follows.

Proposition 3.6. On any interval I which includes zero andr, s∈I, (i)E^∗_r,sx, yare not logarithmically convex (concave);

(ii)Wr,sp, q;x, yare logarithmically convex (concave) if and only ifpq1/2.

4. Proofs

For the proof of Proposition 2.1, we apply the following assertion on Jensen functionals Jfp,xfrom6.

Theorem 4.1. Letf, g : I → Rbe twice continuously diﬀerentiable functions. Assume that g is strictly convex andφis a continuous and strictly monotonic function onI. Then the expression

φ⁻¹ Jn

p,x;f Jn

p,x;g

n≥2 4.1

represents a mean value of the numbersx1, . . . , xn, that is,

min{x1, . . . , x_n} ≤φ⁻¹ J_n

p,x;f Jn

p,x;g

≤max{x1, . . . , x_n} 4.2

if and only if the relation

ft φtgt 4.3

holds for eacht∈I.

Recall that the Jensen functionalJnp,x;fis defined on an intervalI, I⊆Rby

Jn

p,x;f :ⁿ

1

pifxi−f _n

1

pixi

, 4.4

wheref:I → R,x x1, x₂, . . . , x_n∈Iⁿ,andp{pi}ⁿ₁ is a positive weight sequence.

The famous Jensen’s inequality asserts that

Jn

p,x;f

≥0, 4.5

wheneverfis astrictlyconvex function onI, with the equality case if and only ifx1 x2

· · ·x_n.

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Proof ofProposition 2.1. Define the auxiliary functionhsxby

h_sx:

⎧⎪

⎪⎨

⎪⎪

⎩

e^sx−sx−1

s² , s /0;

x²

2 , s0.

4.6

Since

h_sx

⎧⎪

⎨

⎪⎩ e^sx−1

s , s /0;

x, s0,

h_sx e^sx, s∈R,

4.7

we conclude thath_sxis a continuously twice diﬀerentiable convex function onR.

Denotingft:hst, gt:hrt, we realize that the condition4.3ofTheorem 4.1 is fulfilled withφt e^s−rt. Hence, applying Theorem 4.1, we obtain that logWr,sp, e^x represents a mean value, which is equivalent to the assertion ofProposition 2.1.

Proof ofProposition 2.2. We prove first a global theorem concerning log-convexity of the Jensen’s functional with a parameter, which can be very usablecf.7.

Theorem 4.2. Letf_sxbe a twice continuously diﬀerentiable function inxwith a parameters. If f_sxis log-convex insfors∈I : a, b; x∈K: c, d, then the Jensen functional

Jfw, x;s Js:

wifsxi−fs

wixi

, 4.8

is log-convex insfors∈I, x_i∈K, i1,2, . . ., wherew{wi}is any positive weight sequence.

At the beginning, we need some preliminary lemmas.

Lemma 4.3. A positive functionfis log-convex onIif and only if the relation

fsu²2f st

2

uwftw²≥0 4.9

holds for each realu, wands, t∈I.

This assertion is nothing more than the discriminant test for the nonnegativity of second-order polynomials. Other well known assertions are the followingcf8, pages 74, 97-98lemmas.

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Lemma 4.4Jensen’s inequality. Ifgxis twice continuously diﬀerentiable andgx≥0 onK, thengxis convex onKand the inequality

w_igxi−g w_ix_i

≥0 4.10

holds for eachxi∈K, i1,2, . . . ,and any positive weight sequence{wi}, wi 1.

Lemma 4.5. For a convexf, the expression

fs−fr

s−r 4.11

is increasing in both variables.

Proof ofTheorem 4.2. Consider the functionFxdefined as

Fx Fu, v, s, t;x:u²f_sx 2uvf_st/2x v²f_tx, 4.12

whereu, v∈R; s, t∈Iare real parameters independent of the variablex∈K.

Since

Fx u²f_sx 2uvf_st/2 x v²f_tx, 4.13

and by assumingf_sxis log-convex ins, it follows fromLemma 4.3thatFx≥0, x∈K.

Therefore, byLemma 4.4, we get w_iFxi−F

w_ix_i

≥0, x_i∈K, 4.14

which is equivalent to

u²Js 2uvJ st

2

v²Jt≥0. 4.15

According toLemma 4.3again, this is possible only ifJsis log-convex and the proof is done.

Now, the proof ofProposition 2.2easily follows.

From the above, we see thathsxis twice continuously diﬀerentiable and thath_sx is a log-convex function for each reals, x.

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ApplyingTheorem 4.2, we conclude that the form

Φhw, x;s Φs:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

wie^sxⁱ−e^s^wⁱ^xⁱ

s² , s /0, w_ix²_i −

w_ix_i²

2 , s0,

4.16

is log-convex ins.

ByLemma 4.5, withfs logΦs, we find out that

logΦs−logΦr

s−r log

Φs Φr

_1/s−r

4.17

is monotone increasing either insorr. Therefore, by changing variablex_i → logx_i, we finally obtain the proof ofProposition 2.2.

Proof ofProposition 2.3. The assertion of Proposition 2.3 follows fromProposition 2.2 by the standard argumentcf.8, pages 131–134. Details are left to the reader.

Proof ofProposition 3.1. The proof follows putting xt t, pt e^−t, t ∈ 0,∞ and applyingProposition 2.2. withr0. Corresponding integrals are

_∞

0

e^−tlogt−γ;

_∞

0

e^−tlog²tγ²π² 6 ;

_∞

0

e^−tlog³t−γ³−γπ²

2 −2ξ3, 4.18

with

Γ1−sΓ1s πs

sinπs. 4.19

Proof ofProposition 3.2. ByProposition 2.3, we get

W0,1p, e^x≥W0,0p, e^x, 4.20

that is,

Ee^X−e^EX μ₂/2 ≥exp

EX³−EX³ 3μ₂

. 4.21

Using the identityEX³−EX³μ33μ2EX, we obtain the proof ofProposition 3.2.

Proof ofProposition 3.3. This assertion is straightforward consequence of the fact that W_0,sp, e^xis monotone increasing ins.

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Proof ofProposition 3.4. Direct consequence ofProposition 2.2.

Proof ofProposition 3.5. This is left as an easy exercise to the readers.

Proof ofProposition 3.6. We prove only partii. The proof ofigoes along the same lines.

Suppose that 0∈a, b:Iand thatE_r,sp, q;x, yare log-convexconcaveforr, s∈I and any fixedx, y∈R. Then there should be ans, s >0 such that

Fs

p, q;x, y :W0,s

p, q;x, y W_0,−s

p, q;x, y

− W0,0

p, q;x, y₂

4.22

is of constant sign for eachx, y >0.

Substitutingx/y^s : e^w, w ∈ R, after some calculations, we get that the above is equivalent to the assertion thatFp, q;wis of constant sign, where

F p, q;w

:pe^wq−e^pw−e^2/3^1p^w

pe^−wq−e^−pw

. 4.23

Developing in power series inw, we get

F p, q;w

1 1620pq

1p 2−p

1−2p

w⁵O w⁶

. 4.24

Therefore,Fp, q;wcan be of constant sign for eachw∈Ronly ifp1/2q.

Suppose now thatIis of the formI: 0, aorI : −a,0, a >0. Then there should be ans, s /0, s∈Isuch that

W_0,0

p, q;x, y W_0,2s

p, q;x, y

− W_0,s

p, q;x, y₂

4.25

is of constant sign for eachx, y∈R.

Proceeding as before, this is equivalent to the assertion thatGp, q;wis of constant sign with

G p, q;w

:p³q³w⁶e^2/3p1w

pe^2wq−e^2pw

−

pe^wq−e^pw₄

. 4.26

However,

G p, q;w

2 405p⁴q⁴

1p 1q

q−p

w¹¹O w¹²

. 4.27

Hence, we conclude that Gp, q;w can be of constant sign for suﬃciently small w, w ∈ Ronly if p q 1/2. Combining this with Feng Qi theorem, the assertion from Proposition 3.6follows.

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87–92, 1975.

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