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

Authors: Zhen-Hang Yang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We continue to adopt notations and methods used
in the papers illustrated by Yang (2009, 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.


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## Body

## 1. Introduction

Since the Ky Fan [1] inequality was presented, inequalities of ratio of means have attracted attentions of many scholars. Some known results can be found in [2–14]. 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 paper [13, 14] to investigate the monotonicity properties of the functions Qif(i=1,2,3,4) defined by
(1.1)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),
where
(1.2)g1f(p)=g1f(p;a,b):=ℋf(p,q)ℋf(2k-p,q),(1.3)g2f(p)=g2f(p;a,b):=ℋf(p,p+m)ℋf(2k-p,2k-p+m),(1.4)g3f(p)=g3f(p;a,b):=ℋf(p,2m-p)ℋf(2k-p,2m-2k+p),(1.5)g4f(p)=g4f(p;a,b):=ℋf(pr,ps)ℋf((2k-p)r,(2k-p)s),
the q,r,s,k,m∈ℝ, a,b,c,d∈ℝ+ with b/a>d/c≥1, ℋf(p,q) is the so-called two-parameter homogeneous functions defined by [15, 16]. For conveniences, we record it as follows.Definition 1.1.
Letf: ℝ+2∖{(x,x),x∈ℝ+}→ℝ+ be a first-order homogeneous continuous function which has first partial derivatives. Then, ℋf:ℝ2×ℝ+2→ℝ+ is called a homogeneous function generated by f with parameters p and q if ℋf is defined by for a≠b(1.6)ℋf(p,q;a,b)=(f(ap,bp)f(aq,bq))1(p-q),ifpq(p-q)≠0,ℋf(p,p;a,b)=exp(apfx(ap,bp)lna+bpfy(ap,bp)lnbf(ap,bp)),ifp=q≠0,
where fx(x,y)and fy(x,y)denote first-order partial derivatives with respect to first and second component of f(x,y),respectively.
Iflimy→xf(x,y) exits and is positive for all x∈ℝ+, then further define
(1.7)ℋf(p,0;a,b)=(f(ap,bp)f(1,1))1/p,ifp≠0,q=0,ℋf(0,q;a,b)=(f(aq,bq)f(1,1))1/q,ifp=0,q≠0,ℋf(0,0;a,b)=afx(1,1)/f(1,1)bfy(1,1)/f(1,1),ifp=q=0,
and ℋf(p,q;a,a)=a.Remark 1.2.
Witkowski [17] proved that if the function (x,y)→f(x,y) is a symmetric and first-order homogeneous function, then for all p,qℋf(p,q;a,b) is a mean of positive numbers a and b if and only if f is increasing in both variables on ℝ+. In fact, it is easy to see that the condition “f(x,y) is symmetric” can be removed.
Ifℋf(p,q;a,b) is a mean of positive numbers a and b, then it is called two-parameter homogeneous mean generated by f.For simpleness,ℋf(p,q;a,b) is also denoted by ℋf(p,q) or ℋf(a,b).The two-parameter homogeneous functionℋf(p,q;a,b) generated by f is very important because it can generates many well-known means. For example, substituting L=L(x,y)=(x-y)/(lnx-lny) if x,y>0 with x≠y and L(x,x)=x for f yields Stolarsky means ℋL(p,q;a,b)=Sp,q(a,b) defined by
(1.8)Sp,q(a,b)={(qpap-bpaq-bq)1/(p-q),ifpq(p-q)≠0,L1/p(ap,bp),ifp≠0,q=0,L1/q(aq,bq),ifq≠0,p=0,I1/p(ap,bp),ifp=q≠0,ab,ifp=q=0,
where I(x,y)=e-1(xx/yy)1/(x-y) if x,y>0, with x≠y, and I(x,x)=x is the identric (exponential) mean (see [18]). Substituting A=A(x,y)=(x+y)/2 for f yields Gini means ℋA(p,q;a,b)=Gp,q(a,b) defined by
(1.9)Gp,q(a,b)={(ap+bpaq+bq)1/(p-q),ifp≠q,Z1/p(ap,bp),ifp=q,
where Z(a,b)=aa/(a+b)bb/(a+b) (see [19]).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

In [15, 16, 20], two decision functions play an important role, that are,
(2.1)ℐ=ℐ(x,y)=∂2lnf(x,y)∂x∂y=(lnf(x,y))xy=(lnf)xy,𝒥=𝒥(x,y)=(x-y)∂(xℐ)∂x=(x-y)(xℐ)x.
In [14], it is important to another key decision function defined by
(2.2)𝒯3(x,y):=-xy(xℐ)xln3(xy),whereℐ=(lnf)xy,x=at,y=bt.
Note that the function T defined by
(2.3)T(t):=lnf(at,bt),t≠0
has well properties (see [15, 16]). And it has shown in [14, (3.4)], [16, Lemma 4] the relation among T′′′(t), 𝒥(x,y) and 𝒯3(x,y):(2.4)T′′′(t)=t-3𝒯3(x,y),wherex=at,y=bt,(2.5)T′′′(t)=-Ct-3𝒥(x,y),whereC=xy(x-y)-1(lnx-lny)3>0.
Moreover, it has revealed in [14, (3.5)] that
(2.6)𝒯3(x,y)=𝒯3(xy,1)=𝒯3(1,yx).Now, we observe the monotonicities of ratio of certain mixed means defined by.Theorem 2.1.
Suppose thatf: ℝ+×ℝ+→ℝ+ is a symmetric, first-order homogenous, and three-time differentiable function, and 𝒯3(1,u) strictly increase (decrease) with u>1 and decrease (increase) with 0<u<1. Then, for any a,b,c,d>0 with b/a>d/c≥1 and fixed q≥0,  k≥0, but q,k are not equal to zero at the same time, Q1f is strictly increasing (decreasing) in p on (k,∞) and decreasing (increasing) on (-∞,k).
The monotonicity ofQ1f is converse if q≤0,  k≤0, but q,k are not equal to zero at the same time.Proof.
Sincef(x,y)>0 for (x,y)∈ℝ+×ℝ+, so T′(t) is continuous on [p,q] or [q,p] for p,q∈ℝ, then (2.13) in [13] holds. Thus we have
(2.7)lng1f(p)=12lnℋf(p,q)+12lnℋf(2k-p,q)=12∫01‍T′(t11)dt+12∫01‍T′(t12)dt,
where
(2.8)t12=tp+(1-t)q,t11=t(2k-p)+(1-t)q.
Partial derivative leads to
(2.9)(lng1f(p))′=12∫01‍tT′′(t12)dt-12∫01‍tT′′(t11)dt=12∫01‍tT′′(|t12|)dt-12∫01‍tT′′(|t11|)dt(by[13],(2.7))=12∫01‍t∫|t11||t12|‍T′′′(v)dvdt,
and then
(2.10)(lnQ1f(p))′=(lng1f(p;a,b))′-(lng1f(p;c,d))′=12∫01‍t∫|t11||t12|‍T′′′(v)dvdt-12∫01‍t∫|t11||t12|‍T′′′(v;c,d)dvdt=∫01‍t(|t12|-|t11|)∫|t11||t12|‍(T′′′(v;a,b)-T′′′(v;c,d))dv|t12|-|t11|dt:=∫01‍t(|t12|-|t11|)h(|t11|,|t12|)dt,
where
(2.11)h(x,y):={∫xy‍(T′′′(v;a,b)-T′′′(v;c,d))dvy-x,ifx≠y,T′′′(x;a,b)-T′′′(x;c,d),ifx=y.
Since 𝒯3(1,u) strictly increase (decrease) with u>1 and decrease (increase) with 0<u<1, (2.4) and (2.6) together with b/a>d/c≥1 yield
(2.12)T′′′(v;a,b)-T′′′(v;c,d)=v-3(𝒯3(av,bv)-𝒯3(cv,dv))=v-3(𝒯3(1,(ba)v)-𝒯3(1,(dc)v))>(<)0,forv>0,
and therefore h(x,y)>(<)0 for x,y>0. Thus, in order to prove desired result, it suffices to determine the sign of (|t12|-|t11|). In fact, if q≥0,  k≥0, then for t∈[0,1](2.13)|t12|-|t11|=t122-t112|t12|+|t11|=4tq(1-t)+ktt12+t11(p-k)={>0,ifp>k,<0,ifp<k.
It follows that
(2.14)(lnQ1f(p))′={>(<)0,ifp>k,<(>)0,ifp<k.
Clearly, the monotonicity of Q1f is converse if q≤0,  k≤0.
This completes the proof.Theorem 2.2.
The conditions are the same as those of Theorem2.1. Then, for any a,b,c,d>0 with b/a>d/c≥1 and fixed m,k with k≥0, k+m≥0, but m,k are not equal to zero at the same time, Q2f is strictly increasing (decreasing) in p on (k,∞) and decreasing (increasing) on (-∞,k).
The monotonicity ofQ2f is converse if k≤0 and k+m≤0, but m,k are not equal to zero at the same time.Proof.
By (2.13) in [13] we have
(2.15)lng2f(p)=12lnℋf(p,p+m)+12lnℋf(2k-p,2k-p+m)=12∫01‍T′(t22)dt+12∫01‍T′(t21)dt,
where
(2.16)t22=tp+(1-t)(p+m),t21=t(2k-p)+(1-t)(2k-p+m).
Direct calculation leads to
(2.17)(lng2f(p))′=12∫01‍T′′(t22)dt-12∫01‍T′′(t21)dt=12∫01‍∫|t21||t22|‍T′′′(v)dvdt,
and then
(2.18)(lnQ2f(p))′=(lng2f(p;a,b))′-(lng2f(p;c,d))′=12∫01‍∫|t21||t22|‍T′′′(v;a,b)dvdt-12∫01‍∫|t21||t22|‍T′′′(v;c,d)dvdt=12∫01‍(|t22|-|t21|)h(|t21|,|t22|)dt,
where h(x,y) is defined by (2.11). As shown previously, h(x,y)>(<)0 for x,y>0 if 𝒯3(1,u) strictly increase (decrease) with u>1 and decrease (increase) with 0<u<1; it remains to determine the sign of (|t22|-|t21|). It is easy to verify that if k≥0 and k+m≥0, then
(2.19)|t22|-|t21|=t222-t212|t22|+|t21|=4k+m(1-t)|t22|+|t21|(p-k)={>0,ifp>k,<0,ifp<k.
Thus, we have(2.20)(lnQ2f(p))′={>(<)0,ifp>k,<(>)0,ifp<k.
Clearly, the monotonicity of Q2f is converse if k≤0 and k+m≤0.
The proof ends.Theorem 2.3.
The conditions are the same as those of Theorem2.1. Then, for any a,b,c,d>0 with b/a>d/c≥1 and fixed m>0, 0≤k≤2m, Q3f is strictly increasing (decreasing) in p on (k,∞) and decreasing (increasing) on (-∞,k).
The monotonicity ofQ2f is converse if m<0, 2m≤k≤0.Proof.
From (2.13) in [13], it is derived that
(2.21)lng3f(p)=12lnℋf(p,2m-p)+12lnℋf(2k-p,2m-2k+p)=12∫01‍T′(t32)dt+12∫01‍T′(t31)dt,
where
(2.22)t32=(tp+(1-t)(2m-p)),t31=(t(2k-p)+(1-t)(2m-2k+p)).
Simple calculation yields
(2.23)(lng3f(p))′=12∫01‍(2t-1)(T′′(t32)-T′′(t31))dt=12∫01‍(2t-1)∫|t31||t32|‍T′′′(v;a,b)dvdt.
Hence,
(2.24)(lnQ3f(p))′=(lng3f(p;a,b))′-(lng3f(p;c,d))′=12∫01‍(2t-1)∫|t31||t32|‍(T′′′(v;a,b)-T′′′(v;c,d))dvdt=12∫01‍(2t-1)(|t32|-|t31|)h(|t31|,|t32|)dt,
where h(x,y) is defined by (2.11). It has shown that h(x,y)>(<)0 for x,y>0 if 𝒯3(1,u) strictly increase (decrease) with u>1 and decrease (increase) with 0<u<1, and we have also to check the sign of (2t-1)(|t32|-|t31|). Easy calculation reveals that if m>0, 0≤k≤2m, then
(2.25)(2t-1)(|t32|-|t31|)=(2t-1)(t322-t312)|t32|+|t31|=4(2t-1)2tk+(1-t)(2m-k)|t32|+|t31|(p-k)={>0,ifp>k,<0,ifp<k,
which yields
(2.26)(lnQ3f(p))′={>(<)0,ifp>k,<(>)0,ifp<k.
It is evident that the monotonicity of Q3f is converse if m<0, 2m≤k≤0.
Thus the proof is complete.Theorem 2.4.
The conditions are the same as those of Theorem2.1. Then, for any a,b,c,d>0 with b/a>d/c≥1 and fixed k,r,s∈ℝ with r+s≠0, Q4f is strictly increasing (decreasing) in p on (k,∞) and decreasing (increasing) on (-∞,k) if k(r+s)>0.
The monotonicity ofQ4f is converse if k(r+s)<0.Proof.
By (2.13) in [13], lnℋf(pr,ps) can be expressed in integral form
(2.27)lnℋf(pr,ps)={1r-s∫sr‍T′(pt)dt,ifr≠s,T′(pr),ifr=s.
The case r=s≠0 has no interest since it can come down to the case of m=0 in Theorem 2.2. Therefore, we may assume that r≠s. We have
(2.28)lng4f(p)=lnℋf(pr,ps)ℋf((2k-p)r,(2k-p)s)=121r-s∫sr‍T′(pt)dt+121r-s∫sr‍T′((2k-p)t)dt,
and then
(2.29)(lng4f(p))′=121r-s∫sr‍tT′′(pt)dt-121r-s∫sr‍tT′′((2k-p)t)dt=121r-s∫sr‍t(T′′(pt)-T′′((2k-p)t)).
Note that T′′(t) is even (see [13, (2.7)]) and so t(T′′(pt)-T′′((2k-p)t)) is odd, then make use of Lemma  3.3 in [13], (lng4f(p))′ can be expressed as
(2.30)(lng4f(p))′=12r+s|r|-|s|∫|s||r|‍t(T′′(|pt|)-T′′(|(2k-p)t|))dt=12r+s|r|-|s|∫|s||r|‍t∫|t41||t42|‍T′′′(v)dvdt,
where
(2.31)t42=pt,t41=(2k-p)t.
Hence,
(2.32)(lnQ4f(p))′=(lng4f(p;a,b))′-(lng4f(p;c,d))′=12r+s|r|-|s|∫|s||r|‍t∫|t41||t42|‍(T′′′(v;a,b)-T′′′(v;c,d))dvdt=12r+s|r|-|s|∫|s||r|‍t(|t42|-|t41|)h(|t41|,|t42|)dt,
where h(x,y) is defined by (2.11). We have shown that h(x,y)>(<)0 for x,y>0 if 𝒯3(1,u) strictly increase (decrease) with u>1 and decrease (increase) with 0<u<1, and we also have
(2.33)sgn(|t42|-|t41|)=sgn(t422-t412)=sgn(k)sgn(p-k).
It follows that
(2.34)sgnQ4f′(p)=sgn(r+s)sgn(k)sgn(p-k)sgnh(|t41|,|t42|)={>(<)0,ifk(r+s)>0,p>k,<(>)0,ifk(r+s)>0,p<k,<(>)0,ifk(r+s)<0,p>k,>(<)0,ifk(r+s)<0,p<k.
This proof is accomplished.


## 3. Applications

As shown previously,Sp,q(a,b)=ℋL(p,q;a,b), where L=L(x,y) is the logarithmic mean. Also, it has been proven in [14] that 𝒯3′(1,u)<0 if u>1 and 𝒯3′(1,u)>0 if 0<u<1. From the applications of Theorems 2.1–2.4, we have the following.Corollary 3.1.
Leta,b,c,d>0 with b/a>d/c≥1. Then, the following four functions are all strictly decreasing (increasing) on (k,∞) and increasing (decreasing) on (-∞,k):(i)
Q1L is defined by
(3.1)Q1L(p)=Sp,q(a,b)S2k-p,q(a,b)Sp,q(c,d)S2k-p,q(c,d),
for fixed q≥(≤)0,  k≥(≤)0, but q,k are not equal to zero at the same time,(ii)
Q2L is defined by
(3.2)Q2L(p)=Sp,p+m(a,b)S2k-p,2k-p+m(a,b)Sp,p+m(c,d)S2k-p,2k-p+m(c,d),
for fixed m,k with k≥(≤)0 and k+m≥(≤)0, but m,k are not equal to zero at the same time,(iii)
Q3L is defined by
(3.3)Q3L(p)=Sp,2m-p(a,b)S2k-p,2m-2k+p(a,b)Sp,2m-p(c,d)S2k-p,2m-2k+p(c,d),
for fixed m>(<)0, k∈[0,2m]([2m,0]).(iv)
Q4L is defined by
(3.4)Q4L(p)=Spr,ps(a,b)S(2k-p)r,(2k-p)s(a,b)Spr,ps(c,d)S(2k-p)r,(2k-p)s(c,d),
for fixed k,r,s∈ℝ with k(r+s)>(<)0.Remark 3.2.
Letting in the first result of Corollary3.1,q=k yields Theorem  3.4 in [13] since Sp,kS2k-p,k=Sp,2k-p. Letting q=1,  k=0 yields
(3.5)G(a,b)G(c,d)=Q1L(∞)<Sp,1(a,b)S-p,1(a,b)Sp,1(c,d)S-p,1(c,d)<Q1L(0)=L(a,b)L(c,d).
Inequalities (3.5) in the case of d=c were proved by Alzer in [21]. By letting q=1,  k=1/2 from Q1L(1/2)>Q1L(1)>Q1L(2), we have
(3.6)A(a,b)+G(a,b)A(c,d)+G(c,d)>L(a,b)I(a,b)L(c,d)I(c,d)>A(a,b)G(a,b)A(c,d)G(c,d).
Inequalities (3.6) in the case of d=c are due to Alzer [22].Remark 3.3.
Letting in the second result of Corollary3.1,m=1,  k=0 yields Cheung and Qi’s result (see [23, Theorem 2]). And we have
(3.7)G(a,b)G(c,d)=Q2L(∞)<Sp,p+1(a,b)S-p,-p+1(a,b)Sp,p+1(c,d)S-p,-p+1(c,d)<Q2L(0)=L(a,b)L(c,d).
When d=c, inequalities (3.7) are changed as Alzer’s ones given in [24].Remark 3.4.
In the third result of Corollary3.1, letting k=m also leads to Theorem  3.4 in [13]. Put m=1/2,k=1/4. Then from Q3L(1/4)>Q3L(1/2), we obtain a new inequality
(3.8)He1/2(a,b)He1/2(c,d)>L(a,b)I1/2(a,b)L(c,d)I1/2(c,d).
Putting m=1/2,k=1/3 leads to another new inequality
(3.9)A1/3(a,b)A1/3(c,d)>S1/6,5/6(a,b)I1/2(a,b)S1/6,5/6(c,d)I1/2(c,d).Remark 3.5.
Letting in the third result of Corollary3.1,k=1/2 and (r,s)=(1,0), (1,1), (2,1), and we deduce that all the following three functions
(3.10)p⟶Lp(a,b)L1-p(a,b)Lp(c,d)L1-p(c,d),p⟶Ip(a,b)I1-p(a,b)Ip(c,d)I1-p(c,d),p⟶Ap(a,b)A1-p(a,b)Ap(c,d)A1-p(c,d),
are strictly decreasing on (1/2,∞) and increasing on (-∞,1/2), where Lp=L1/p(ap,bp), Ip=I1/p(ap,bp), and Ap=A1/p(ap,bp) are the p-order logarithmic, identric (exponential), and power mean, respectively, particularly, so are the functions LpL1-p, IpI1-p, ApA1-p.


## 4. Other Results

Letd=c in Theorems 2.1–2.4. Then, ℋf(p,q;c,d)=c and T′′′(t;c,c)=0. From the their proofs, it is seen that the condition “𝒯3(1,u) strictly increases (decreases) with u>1 and decreases (increases) with 0<u<1” can be reduce to “T′′′(v)>(<)0 for v>0”, which is equivalent with 𝒥=(x-y)(xℐ)x<(>)0, where ℐ=(lnf)xy, by (2.4). Thus, we obtain critical theorems for the monotonicities of gif, i=1-4, defined as (1.2)–(1.5).Theorem 4.1.
Suppose thatf: ℝ+×ℝ+→ℝ+ is a symmetric, first-order homogenous, and three-time differentiable function and 𝒥=(x-y)(xℐ)x<(>)0, where ℐ=(lnf)xy. Then, for a,b>0 with a≠b, the following four functions are strictly increasing (decreasing) in p on (k,∞) and decreasing (increasing) on (-∞,k):(i)
g1f is defined by (1.2), for fixed q,k≥0, but q,k are not equal to zero at the same time;(ii)
g2f is defined by (1.3), for fixed m,k with k≥0 and k+m≥0, but m,k are not equal to zero at the same time;(iii)
g3f is defined by (1.4), for fixed m>0 and 0≤k≤2m;(iv)
g4f is defined by (1.5), for fixed k,r,s∈ℝ with k(r+s)>0.Iff is defined on ℝ+2∖{(x,x),x∈ℝ+}, then T′(t) may be not continuous at t=0, and (2.13) in [13] may not hold for p,q∈ℝ but must be hold for p,q∈ℝ+. And then, we easily derive the following from the proofs of Theorems 2.1–2.4.Theorem 4.2.
Suppose thatf: ℝ+2∖{(x,x),x∈ℝ+}→ℝ+ is a symmetric, first-order homogenous and three-time differentiable function and 𝒥=(x-y)(xℐ)x<(>)0, where ℐ=(lnf)xy. Then for a,b>0 with a≠b the following four functions are strictly increasing (decreasing) in p on (k,2k) and decreasing (increasing) on (0,k):(i)
g1f is defined by (1.2), for fixed q,k>0;(ii)
g2f is defined by (1.3), for fixed m,k with k>0 and k+m>0;(iii)
g3f is defined by (1.4), for fixed m>0 and 0≤k≤2m;(iv)
g4f is defined by (1.5), for fixed k,r,s>0.If we substituteL, A, and I for f, where L, A, and I denote the logarithmic, arithmetic, and identric (exponential) mean, respectively, then from Theorem 4.1, we will deduce some known and new inequalities for means. Similarly, letting in Theorem 4.2f(x,y)=D(x,y)=|x-y|, K(x,y)=(x+y)|ln(x/y)|, where x,y>0 with x≠y, we will obtain certain companion ones of those known and new ones. Here no longer list them.



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*Source: 540710-2012-10-24.xml*