The Monotonicity Results for the Ratio of Certain Mixed Means and Their Applications

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Volume 2012, Article ID 540710,13pages doi:10.1155/2012/540710

Research Article

The Monotonicity Results for the Ratio of

Certain Mixed Means and Their Applications

Zhen-Hang Yang

Power Supply Service Center, Zhejiang Electric Power Company, Electric Power Research Institute, Hangzhou 310014, China

Correspondence should be addressed to Zhen-Hang Yang,[email protected]

Received 22 July 2012; Accepted 26 September 2012

Academic Editor: Edward Neuman

Copyrightq2012 Zhen-Hang Yang. 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.

We continue to adopt notations and methods used in the papers illustrated by Yang2009, 2010 to investigate the monotonicity properties of the ratio of mixed two-parameter homogeneous means. As consequences of our results, the monotonicity properties of four ratios of mixed Stolarsky means are presented, which generalize certain known results, and some known and new inequalities of ratios of means are established.

1. Introduction

Since the Ky Fan1inequality was presented, inequalities of ratio of means have attracted attentions of many scholars. Some known results can be found in2–14. Research for the properties of ratio of bivariate means was also a hotspot at one time.

In this paper, we continue to adopt notations and methods used in the paper13,14

to investigate the monotonicity properties of the functionsQifi1,2,3,4defined by

Q1f

p: g1f

p;a, b g1f

p;c, d,

Q2f

p: g2f

p;a, b g2f

p;c, d,

Q3f

p: g3f

p;a, b g3f

p;c, d,

Q4f

p: g4f

p;a, b g4f

p;c, d,

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where

g1f

pg1f

p;a, b:

Hf

p, qHf

2k−p, q, 1.2

g2f

pg2f

p;a, b:

Hf

p, pmHf

2k−p,2k−pm, 1.3

g3f

pg3f

p;a, b:

Hf

p,2m−pHf

2k−p,2m−2kp, 1.4

g4f

pg4f

p;a, b:

Hf

pr, psHf

2k−pr,2k−ps, 1.5

theq, r, s, k, m∈R,a, b, c, d∈Rwithb/a > d/c≥1,Hfp, qis the so-called two-parameter

homogeneous functions defined by15,16. For conveniences, we record it as follows.

Definition 1.1. Letf:R2

\ {x, x, x ∈ R} → R be a first-order homogeneous continuous

function which has first partial derivatives. Then,Hf :R2×R2 → Ris called a homogeneous

function generated byfwith parameterspandqifHf is defined by fora /b

Hf

p, q;a, b

_f_ap_{, b}p

faq_{, b}q

1p−q

, ifpqp−q/0,

Hf

p, p;a, bexp

ap_f

xap, bplnabpfyap, bplnb

fap_{, b}p

, if pq /0,

1.6

where fxx, yand fyx, ydenote first-order partial derivatives with respect to first and

second component offx, y, respectively.

If limy→xfx, yexits and is positive for allx∈R, then further define

Hf

p,0;a, b

_f_ap_{, b}p

f1,1 1/p

, ifp /0, q0,

Hf

0, q;a, b

faq_{, b}q

f1,1 1/q

, ifp0, q /0,

Hf0,0;a, b afx1,1/f1,1bfy1,1/f1,1, if pq0,

1.7

andHfp, q;a, a a.

Remark 1.2. Witkowski17proved that if the functionx, y → fx, yis a symmetric and

first-order homogeneous function, then for allp, qHfp, q;a, bis a mean of positive numbers

aandbif and only iffis increasing in both variables onR. In fact, it is easy to see that the condition “fx, yis symmetric” can be removed.

IfHfp, q;a, bis a mean of positive numbersaandb, then it is called two-parameter

homogeneous mean generated byf.

For simpleness,Hfp, q;a, bis also denoted byHfp, qorHfa, b.

The two-parameter homogeneous function Hfp, q;a, b generated by f is very

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LLx, y x−y/lnx−lnyifx, y >0 withx /yandLx, x xforfyields Stolarsky meansHLp, q;a, b Sp,qa, bdefined by

Sp,qa, b

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

q p

ap₋_bp

aq₋_bq

1/p−q

, if pqp−q/0,

L1/p_ap_{, b}p_, _if _{p /}₀_{, q}₀_,

L1/q_aq_{, b}q_, _if _{q /}₀_{, p}₀_,

I1/pap, bp, if pq /0,

√

ab, if pq0,

1.8

where Ix, y e−1_xx_/yy1/x−y _if _{x, y >} _{0, with} _{x /}_y_{, and} _I_{x, x} _x _{is the identric} exponentialmeansee18. SubstitutingAAx, y xy/2 forfyields Gini means

HAp, q;a, b Gp,qa, bdefined by

Gp,qa, b

⎧ ⎪ ⎨ ⎪ ⎩

ap_bp

aq_bq

1/p−q

, ifp /q,

Z1/pap, bp, ifpq,

1.9

whereZa, b aa/ab_bb/ab_see₁₉_.

As consequences of our results, the monotonicity properties of four ratios of mixed Stolarsky means are presented, which generalize certain known results, and some known and new inequalities of ratios of means are established.

2. Main Results and Proofs

In15,16,20, two decision functions play an important role, that are,

IIx, y ∂

2_ln_f_{x, y}

∂x∂y

lnfx, y_xy lnf_xy,

JJx, yx−y∂xI

∂x

x−yxI_x.

2.1

In14, it is important to another key decision function defined by

T3

x, y:−xyxI_xln3

x y

, whereIlnf_xy, xat, ybt. 2.2

Note that the functionTdefined by

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has well propertiessee15,16. And it has shown in14,3.4,16, Lemma 4the relation amongTt,Jx, yandT3x, y:

Tt t−3T3

x, y, wherexat, ybt, 2.4

Tt −Ct−3Jx, y, whereCxyx−y−1lnx−lny3>0. 2.5

Moreover, it has revealed in14,3.5that

T3

x, yT3

x y,1

T3

1,y

x

. 2.6

Now, we observe the monotonicities of ratio of certain mixed means defined by1.1.

Theorem 2.1. Suppose thatf:R×R → R is a symmetric, first-order homogenous, and

three-time diﬀerentiable function, andT31, ustrictly increase (decrease) withu >1 and decrease (increase)

with 0< u <1. Then, for anya, b, c, d >0 withb/a > d/c≥1 and fixedq≥0,k ≥0, butq, kare

not equal to zero at the same time,Q1f is strictly increasing (decreasing) inponk,∞and decreasing

(increasing) on−∞, k.

The monotonicity ofQ1f is converse ifq≤0,k≤0, butq, kare not equal to zero at the same

time.

Proof. Since fx, y > 0 forx, y ∈ R×R, soTt is continuous onp, qor q, p for

p, q∈R, then2.13in13holds. Thus we have

lng1f

p 1

2lnHf

p, q1

2lnHf

2k−p, q 1

2 1

0

Tt11dt1

2 1

0

Tt12dt, 2.7

where

t12tp 1−tq, t11t

2k−p 1−tq. 2.8

Partial derivative leads to

lng1f

p 1

2 1

0

tTt12dt− 1

2 1

0

tTt11dt

1

2 ₁

0

tT|t12|dt−

1 2

₁

0

tT|t11|dt

by13,2.7

1

2 ₁

0

t

_|_t₁₂_|

|t11|

Tvdv dt,

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and then

lnQ1f

plng1f

p;a, b−lng1f

p;c, d

1

2 1

0

t

_|t12|

|t11|

Tvdv dt−1

2 1

0

t

_|t12|

|t11|

Tv;c, ddv dt

1

0

t|t12| − |t11|

_|t12|

|t11|T

_v_;_{a, b}₋_T_v_;_{c, d}_dv

|t12| − |t11| dt

: ₁

0

t|t12| − |t11|h|t11|,|t12|dt,

2.10

where

hx, y: ⎧ ⎪ ⎨ ⎪ ⎩

_y

xTv;a, b−Tv;c, ddv

y−x , ifx /y, Tx;a, b−Tx;c, d, ifxy.

2.11

SinceT31, ustrictly increasedecreasewithu >1 and decreaseincreasewith 0< u <1,

2.4and2.6together withb/a > d/c≥1 yield

Tv;a, b−Tv;c, d v−3T3av, bv− T3cv, dv

v−3

T3

1,

b a

v

− T3

1,

d c

v

><0, forv >0,

2.12

and thereforehx, y> <0 forx, y >0. Thus, in order to prove desired result, it suﬃces to determine the sign of|t12| − |t11|. In fact, ifq≥0,k≥0, then fort∈0,1

|t12| − |t11|

t2 12−t211

|t12||t11|

4tq1−t kt t12t11

p−k

>0, ifp > k,

<0, ifp < k. 2.13

It follows that

lnQ1f

p

><0, ifp > k,

<>0, ifp < k. 2.14

Clearly, the monotonicity ofQ1f is converse ifq≤0,k≤0.

This completes the proof.

Theorem 2.2. The conditions are the same as those ofTheorem 2.1. Then, for anya, b, c, d >0 with

b/a > d/c≥1 and fixedm, kwithk≥0,km≥0, butm, kare not equal to zero at the same time,

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The monotonicity ofQ2f is converse ifk ≤0 andkm≤0, butm, kare not equal to zero at

the same time.

Proof. By2.13in13we have

lng2f

p 1

2lnHf

p, pm1

2lnHf

2k−p,2k−pm

1

2 ₁

0

Tt22dt

1 2

₁

0

Tt21dt,

2.15

where

t22 tp 1−t

pm, t21t

2k−p 1−t2k−pm. 2.16

Direct calculation leads to

lng2f

p 1

2 ₁

0

Tt22dt−

1 2

₁

0

Tt21dt

1 2

₁

0

_|_t₂₂_|

|t21|

Tvdv dt, 2.17

and then

lnQ2f

plng2f

p;a, b−lng2f

p;c, d

1

2 1

0

_|t22|

|t21|

Tv;a, bdv dt−1

2 1

0

_|t22|

|t21|

Tv;c, ddv dt

1

2 1

0

|t22| − |t21|h|t21|,|t22|dt,

2.18

wherehx, yis defined by2.11. As shown previously,hx, y><0 forx, y >0 ifT31, u

strictly increasedecreasewithu > 1 and decreaseincreasewith 0 < u < 1; it remains to determine the sign of|t22| − |t21|. It is easy to verify that ifk≥0 andkm≥0, then

|t22| − |t21|

t2 22−t221

|t22||t21|

4km1−t

|t22||t21|

p−k

>0, ifp > k,

<0, ifp < k. 2.19

Thus, we have

lnQ2f

p

><0, ifp > k,

<>0, ifp < k. 2.20

Clearly, the monotonicity ofQ2f is converse ifk≤0 andkm≤0.

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b/a > d/c≥1 and fixedm >0, 0≤k≤2m,Q3f is strictly increasing (decreasing) inponk,∞

and decreasing (increasing) on−∞, k.

The monotonicity ofQ2f is converse ifm <0, 2m≤k≤0.

Proof. From2.13in13, it is derived that

lng3f

p 1

2lnHf

p,2m−p1

2lnHf

2k−p,2m−2kp

1

2 1

0

Tt32dt1

2 1

0

Tt31dt,

2.21

where

t32

tp 1−t2m−p, t31

t2k−p 1−t2m−2kp. 2.22

Simple calculation yields

lng3f

p 1

2 1

0

2t−1Tt32−Tt31

dt 1

2 1

0

2t−1

_|t32|

|t31|

Tv;a, bdv dt.

2.23

Hence,

lnQ3f

plng3f

p;a, b−lng3f

p;c, d

1

2 ₁

0

2t−1

_|t32|

|t31|

Tv;a, b−Tv;c, ddv dt

1

2 ₁

0

2t−1|t32| − |t31|h|t31|,|t32|dt,

2.24

wherehx, yis defined by2.11. It has shown thathx, y > <0 forx, y > 0 ifT31, u

strictly increasedecreasewithu > 1 and decreaseincreasewith 0 < u <1, and we have also to check the sign of2t−1|t32| − |t31|. Easy calculation reveals that ifm >0, 0≤k≤2m,

then

2t−1|t32| − |t31| 2t−1

t2₃₂−t2₃₁

|t32||t31|

42t−12tk 1−t2m−k

|t32||t31|

p−k

>0, ifp > k, <0, ifp < k,

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which yields

lnQ3f

p

><0, ifp > k,

<>0, ifp < k. 2.26

It is evident that the monotonicity ofQ3f is converse ifm <0, 2m≤k≤0.

Thus the proof is complete.

b/a > d/c ≥ 1 and fixedk, r, s ∈Rwith rs /0,Q4f is strictly increasing (decreasing) inpon

k,∞and decreasing (increasing) on−∞, kifkrs>0.

The monotonicity ofQ4f is converse ifkrs<0.

Proof. By2.13in13, lnHfpr, pscan be expressed in integral form

lnHf

pr, ps

⎧ ⎨ ⎩

1

r−s

r sT

ptdt, ifr /s,

Tpr, ifr s.

2.27

The caser s /0 has no interest since it can come down to the case ofm0 inTheorem 2.2. Therefore, we may assume thatr /s. We have

lng4f

p ln

Hf

pr, psHf

2k−pr,2k−ps

1

2 1

r−s

r

s

Tptdt1

2 1

r−s

r

s

T2k−ptdt,

2.28

and then

lng4f

p 1

2 1

r−s

_r

s

tTptdt−1

2 1

r−s

_r

s

tT2k−ptdt

1

2 1

r−s

_r

s

tTpt−T2k−pt.

2.29

Note thatTtis evensee13,2.7and sotTpt−T2k−ptis odd, then make use of Lemma 3.3 in13,lng4fpcan be expressed as

lng4f

p 1

2

rs

|r| − |s|

_|_r_|

|s|t

Tpt−T2k−ptdt

1

2

rs

|r| − |s|

_|_r_|

|s|t

_|_t₄₂_|

|t41|

Tvdv dt,

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where

t42pt, t41

2k−pt. 2.31

Hence,

lnQ4f

plng4f

p;a, b−lng4f

p;c, d

1

2

rs

|r| − |s|

_|r|

|s|t

_|t42|

|t41|

Tv;a, b−Tv;c, ddv dt

1

2

rs

|r| − |s|

_|_r_|

|s|t|t42| − |t41|h|t41|,|t42|dt,

2.32

wherehx, yis defined by2.11. We have shown thathx, y><0 forx, y >0 ifT31, u

strictly increasedecreasewithu > 1 and decreaseincreasewith 0 < u <1, and we also have

sgn|t42| − |t41| sgn

t2₄₂−t2₄₁sgnksgnp−k. 2.33

It follows that

sgnQ₄_fp sgnrssgnksgnp−ksgnh|t41|,|t42|

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

><0, ifkrs>0, p > k, <>0, ifkrs>0, p < k, <>0, ifkrs<0, p > k, ><0, ifkrs<0, p < k.

2.34

This proof is accomplished.

3. Applications

As shown previously,Sp,qa, b HLp, q;a, b, whereL Lx, yis the logarithmic mean.

Also, it has been proven in14thatT₃1, u<0 ifu >1 andT₃1, u>0 if 0< u <1. From the applications of Theorems2.1–2.4, we have the following.

Corollary 3.1. Leta, b, c, d > 0 withb/a > d/c ≥ 1. Then, the following four functions are all

strictly decreasing (increasing) onk,∞and increasing (decreasing) on−∞, k:

iQ1Lis defined by

Q1L

p

Sp,qa, bS2k−p,qa, b

Sp,qc, dS2k−p,qc, d

, 3.1

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iiQ2Lis defined by

Q2L

p

Sp,pma, bS2k−p,2k−pma, b

Sp,pmc, dS2k−p,2k−pmc, d

, 3.2

for fixedm, kwithk≥≤0 andkm≥≤0, butm, kare not equal to zero at the same

time,

iiiQ3Lis defined by

Q3L

p

Sp,2m−pa, bS2k−p,2m−2kpa, b

Sp,2m−pc, dS2k−p,2m−2kpc, d

, 3.3

for fixedm ><0,k∈0,2 m 2m,0.

ivQ4Lis defined by

Q4L

p

Spr,psa, bS2k−pr,2k−psa, b

Spr,psc, dS2k−pr,2k−psc, d

, 3.4

for fixedk, r, s∈Rwithkrs><0.

Remark 3.2. Letting in the first result ofCorollary 3.1,qkyields Theorem 3.4 in13since

Sp,kS2k−p,kSp,2k−p. Lettingq1,k0 yields

Ga, b

Gc, d Q1L∞<

Sp,1a, bS−p,1a, b

Sp,1c, dS−p,1c, d

< Q1L0

La, b

Lc, d. 3.5

Inequalities3.5in the case ofdcwere proved by Alzer in21. By lettingq1,k 1/2 fromQ1L1/2> Q1L1> Q1L2, we have

Aa, b Ga, b Ac, d Gc, d >

La, bIa, b

Lc, dIc, d >

Aa, bGa, b

Ac, dGc, d. 3.6

Inequalities3.6in the case ofdcare due to Alzer22.

Remark 3.3. Letting in the second result ofCorollary 3.1,m1,k0 yields Cheung and Qi’s

resultsee23, Theorem 2. And we have

Ga, b

Gc, d Q2L∞<

Sp,p1a, bS−p,−p1a, b

Sp,p1c, dS−p,−p1c, d

< Q2L0

La, b

Lc, d. 3.7

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Remark 3.4. In the third result ofCorollary 3.1, lettingkmalso leads to Theorem 3.4 in13. Putm1/2, k1/4. Then fromQ3L1/4> Q3L1/2, we obtain a new inequality

He1/2a, b

He1/2c, d

>

La, bI1/2a, b

Lc, dI1/2c, d

. 3.8

Puttingm1/2, k1/3 leads to another new inequality

A1/3a, b

A1/3c, d

>

S1/6,5/6a, bI1/2a, b

S1/6,5/6c, dI1/2c, d

. 3.9

Remark 3.5. Letting in the third result ofCorollary 3.1,k1/2 andr, s 1,0,1,1,2,1,

and we deduce that all the following three functions

p−→

Lpa, bL1−pa, b

Lpc, dL1−pc, d

, p−→

Ipa, bI1−pa, b

Ipc, dI1−pc, d

, p−→

Apa, bA1−pa, b

Apc, dA1−pc, d

,

3.10

are strictly decreasing on1/2,∞and increasing on−∞,1/2, whereLpL1/pap, bp,Ip

I1/p_ap_{, b}p_{, and} _A

p A1/pap, bp are thep-order logarithmic, identricexponential, and

power mean, respectively, particularly, so are the functionsLpL1−p,

IpI1−p,

ApA1−p.

4. Other Results

Let d c in Theorems2.1–2.4. Then,Hfp, q;c, d c and Tt;c, c 0. From the their

proofs, it is seen that the condition “T31, ustrictly increasesdecreases withu > 1 and

decreasesincreaseswith 0 < u < 1” can be reduce to “Tv > <0 forv > 0”, which is equivalent withJ x−yxI_x<>0, whereI lnf_xy, by2.4. Thus, we obtain critical theorems for the monotonicities ofgif,i1−4, defined as1.2–1.5.

Theorem 4.1. Suppose thatf:R×R → Ris a symmetric, first-order homogenous, and three-time

diﬀerentiable function andJ x−yxI_x < >0, whereI lnf_xy. Then, fora, b > 0 with

a /b, the following four functions are strictly increasing (decreasing) inponk,∞and decreasing

(increasing) on−∞, k:

ig1f is defined by1.2, for fixedq, k≥0, butq, kare not equal to zero at the same time;

iig2f is defined by1.3, for fixed m, kwithk≥0 andkm≥0, butm, kare not equal to

zero at the same time;

iiig3f is defined by1.4, for fixedm >0 and 0≤k≤2m;

ivg4f is defined by1.5, for fixedk, r, s∈Rwithkrs>0.

Iffis defined onR2

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Theorem 4.2. Suppose thatf:R2

\ {x, x, x∈R} → Ris a symmetric, first-order homogenous

and three-time diﬀerentiable function andJ x−yxI_x < >0, whereI lnf_xy. Then for

a, b >0 witha /bthe following four functions are strictly increasing (decreasing) inponk,2kand

decreasing (increasing) on0, k:

ig1f is defined by1.2, for fixedq, k >0;

iig2f is defined by1.3, for fixedm, kwithk >0 andkm >0;

iiig3f is defined by1.4, for fixedm >0 and 0≤k≤2m;

ivg4f is defined by1.5, for fixedk, r, s >0.

If we substituteL,A, andIforf, whereL,A, andIdenote the logarithmic, arithmetic, and identricexponentialmean, respectively, then fromTheorem 4.1, we will deduce some known and new inequalities for means. Similarly, letting inTheorem 4.2fx, y Dx, y

|x−y|,Kx, y xy|lnx/y|, wherex, y >0 withx /y, we will obtain certain companion ones of those known and new ones. Here no longer list them.

Disclosure

This paper is in final form and no version of it will be submitted for publication elsewhere.

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