Optimal bounds for arithmetic geometric and Toader means in terms of generalized logarithmic mean

R E S E A R C H

Open Access

Optimal bounds for arithmetic-geometric

and Toader means in terms of generalized

logarithmic mean

Qing Ding

_{and Tiehong Zhao}

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

Hangzhou Normal University, Hangzhou, 311121, China Full list of author information is available at the end of the article

Abstract

In this paper, we find the greatest values

α

α

β

β

that the double inequalitiesLα1(a,b) < AG(a,b) <Lβ1(a,b) and

Lα2(a,b) <T(a,b) <Lβ2(a,b) hold for alla,b> 0 witha=b, where AG(a,b),T(a,b) and

Lp(a,b) are the arithmetic-geometric, Toader and generalized logarithmic means of

two positive numbersaandb, respectively.

MSC: 26E60

Keywords: arithmetic-geometric mean; Toader mean; logarithmic mean

1 Introduction

Forp∈R, thepth generalized logarithmic meanLp(a,b) [] of two positive numbersaand

bis defined by

Lp(a,b) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

[b_(pp_+)(+–a_bp_–+_a)]/p_{, a=b,p= –,p= ,} 

e( bb

aa)/b–a, a=b,p= ,

b–a

logb–loga, a=b,p= –,

a, a=b.

(.)

It is well known thatLp(a,b) is continuous and strictly increasing with respect top∈R for fixeda,b>  witha=b. Many remarkable inequalities for the generalized logarithmic mean can be found in the literature [–].

The classical arithmetic-geometric meanAG(a,b) of two positive numbersaandbis defined by starting witha=a,b=band then iterating

an+=

an+bn

 , bn+=

anbn (.)

forn∈Nuntil two sequences{an}and{bn}converge to the same number. The well-known Gauss identity [] shows that

AG(,r)K√ –r₌π

 (.)

forr∈(, ), whereK(r) = _π/( –r_sin_t)–/_{dt,r∈[, ), is the complete elliptic integral} of the first kind.

In [], the Toader meanT(a,b) of two positive numbersaandbwas given by

T(a,b) = 

π

π/



a_cos_θ+b_sin_θdθ

= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

aE(√–(b/a)₎

π , a>b,

bE(√–(a/b)₎

π , a<b,

a, a=b,

(.)

whereE(r) = _π/( –r_sin_θ)/_{dθ,r∈[, ] is the complete elliptic integral of the second} kind.

Recently, the bounds for the arithmetic-geometric mean AG(a,b) and Toader mean

T(a,b) have attracted the attention of many mathematicians. The double inequality

L–(a,b) =L(a,b) <AG(a,b) <L/a/,b/ (.)

holds for alla,b>  witha=b. The left inequality of (.) was first proposed by Carlson and Vuorinen [] and also was proved by different methods in [–]. Vamanamurthy and Vuorinen [] proved thatAG(a,b) < (π/)L(a,b) for alla,b>  witha=b. The second inequality of (.) was proved by Borwein and Borwein [] and Yang [].

Vuorinen [] conjectured that

M/(a,b) <T(a,b) (.)

for alla,b>  witha=b, whereMp(a,b) = [(ap+bp)/]/p (p= ) andM(a,b) = √

abis the power mean of orderp. This conjecture was proved by Qiu and Shen [] and Barnard et al. [].

In [], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows:

T(a,b) <Mlog/log(π/)(a,b) (.)

for alla,b>  witha=b.

In [–], the authors proved that

L(a,b) <T(a,b) <L/(a,b), (.) S√_/(a,b) <T(a,b) <S/(a,b) (.)

for alla,b>  witha=b, whereLp(a,b) = (ap++bp+)/(ap+bp) denotes thepth Lehmer mean andSp(a,b) is the generalized Seiffert mean given bySp(a,b) =p(a–b)/arctan[p(a–

b)/(a+b)] ( <p≤,a=b),S(a,b) = (a+b)/(a=b) andSp(a,a) =a. Very recently, Chu and Wang [] proved that

Sp(a,b) <T(a,b) <Sq(a,b) (.)

for alla,b>  witha=bif and only ifp≤/,q≥ andp≤,q≥/. Here thepth Gini mean of two positive numbersaandbis defined by

Sp(a,b) = ⎧ ⎨ ⎩

(ap–_a++b_bp–)/(p–), p= ,

(aa_bb₎/(a+b)_{, p= .} (.)

The main purpose of this paper is to find the greatest valuesα,αand the smallest val-uesβ,βsuch that the double inequalitiesLα(a,b) <AG(a,b) <Lβ(a,b) andLα(a,b) <

T(a,b) <Lβ(a,b) hold for alla,b>  witha=band give some new bounds for the com-plete elliptic integrals.

2 Basic knowledge and lemmas

In order to prove our main results, we need several formulas and lemmas, which we present in this section.

Forr∈(, ) andr=√ –r_{, the well-known complete elliptic integrals of the first and} second kinds are defined by

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

K=K(r) = _π/( –r_sin_θ)–/_dθ, K_{=K(r) =K(r),}

K() =π/, K() = +∞,

and

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

E=E(r) = _π/( –r_sin_θ)/_dθ, E_{=E(r) =E(r)}

E() =π/, E() = ,

respectively, and the following formulas were presented in [], Appendix E, pp.-:

dK dr =

E–r_K

rr ,

dE dr =

E–K

r ,

d(E–r_K)

dr =rK,

d(K–E)

dr =

rE r,

(.)

E

√r

 +r

=E–r

_K

 +r . (.)

In what follows, four special valuesE(√/),K(√/) andE(.),K(.) will be used. By numerical computations, these are given by

Lemma .(See [], Theorem .) For–∞<a<b<∞,let f,g: [a,b]→Rbe continu-ous on[a,b]and be differentiable on(a,b),let g(x)=  on(a,b).If f(x)/g(x)is increasing

(decreasing)on(a,b),then so are

f(x) –f(a)

g(x) –g(a) and

f(x) –f(b)

g(x) –g(b).

If f(x)/g(x)is strictly monotone,then the monotonicity in the conclusion is also strict.

Lemma . ()The function r→(E–rK)/ris strictly increasing from(, )onto(π/, ); ()The function r→E–r_{Kis increasing and log-convex from(, )onto(π/, );} ()The functionK/log(e_{/r)is strictly increasing from(, )onto(π/, );}

()The function(K–E)/ris strictly increasing on(, );in particular,K–E> (π/)r for all r∈(, ).

Proof Parts () and () follow from [], Theorem .() and Exercise .().

Lemma . The equation

( +p)/p=π 

has a unique solution p=p= .· · ·.

Proof Let

ϕ(p) = ⎧ ⎨ ⎩

( +p)/p_{–π/, p∈(–, )∪(, +∞),}

e–π/, p= .

It is easy to verify that the functionϕis continuous and strictly decreasing from (–, +∞)

onto (, +∞). Therefore, Lemma . easily follows from the continuity and monotonicity ofϕtogether with the facts thatϕ(.) = .×–andϕ(.) = –.×

–_.

Lemma . The function

f(r) =(E–r

_{K)/π–  –r}_/

r

is strictly increasing from(, )onto(/, /π– /).

Proof Letf(r) = (E–r_{K)/π–  –r}_{/ andf(r) =r}_{, thenf}

() =f() =  andf(r) =

f(r)/f(r).

A simple calculation yields

f(r) f_(r)=

(E–rK) –πr

πr

f(r)

f(r), (.)

f_(r) f_(r)=

K–π

πr

f(r)

f(r), (.)

f() =f(), (.)

f_(r) f_(r)=

 π ·

E–r_K

r ·



r. (.)

Following from Lemma .() and (.) together with the monotonicity of /r_, we clearly see that f(r)/f(r) is strictly increasing on (, ). Equations (.)-(.) and Lemma . lead to the conclusion thatf(r) is strictly increasing on (, ).

Therefore, Lemma . follows from the monotonicity off(r) together with the facts that

f(+_{) = / andf(}–_{) = /π– /.}

The following double inequalities can be obtained from Lemma . immediately.

Corollary . Inequalities

 +r 

 +

r <



π

E–rK<  +r 

 +



π –

 

r

hold for <r< .

Lemma . The inequality

( +r)/_{– ( –r)}/ r

/ <  +r



 (.)

holds for <r< .

Proof In order to prove inequality (.), it suffices to prove that

g(r) =( +r)/– ( –r)/– r

 +r



 

= ( +r)_{+ ( –r)}_{– r}

 +r 

 

–  –r/

=g(r) –g(r) <  (.)

for  <r< , where

g(r) = ( +r)+ ( –r)– r

 +r



 

,

g(r) =  –r/.

Observe that

g_(r) = – r 

 – r+ ,r+ ,r+ r+ r< , (.)

we conclude, from (.) and (.), that there existsr∈(., .) such thatg(r) >  forr∈(,r) andg(r) <  forr∈(r, ).

In order to prove (.), we divide it into two cases.

CaseAr∈[r, ). In this case, we clearly see thatg(r)≤ andg(r) > . This implies thatg(r) =g(r) –g(r) < .

CaseBr∈(,r). In this case,g(r) > . Letg(r) =  – r+_r– r, the difference betweeng(r) andg(r) yields

g(r) –g(r) = –r

_{(, + ,r}_{+ r}_{+ r}₎

, < . (.)

We know from (.) thatg(r) >g(r) > . Moreover,

g_(r) –g_(r) = –r 



 – r +r+r

r– 

+ 

< ,

this in conjunction withg(r) >  implies that

g(r) –g(r) < . (.)

Therefore, we clearly see thatg(r) = [g(r) –g(r)] + [g(r) –g(r)] <  from (.) and

(.).

Lemma . Letη(r) = [( +r)p+_{– ( –r)}p+_{]/r andω(r) = [( –r)}p_{( +p}

r) – ( +r)p( –

pr)]/r,then the functionsη(r)andω(r)both are strictly increasing on(, ).

Proof We assume that

η(r) = ( +r)p+_{– ( –r)}p+_{, η(r) =r,}

ω(r) = ( –r)p_{( +p}

r) – ( +r)p( –pr), ω(r) =r,

thenη(r) =η(r)/η(r) andω(r) =ω(r)/ω(r). A simple calculation yields

η() =η() =ω() =ω() = , (.)

η_(r)

η_(r)= ( +p)

( +r)p_{– ( –r)}p_{, (.)}

ω_(r)

ω_(r)=

p(p+ )[( +r)p–– ( –r)p–]

 . (.)

Lemma . and (.)-(.) lead to the conclusion thatη(r) andω(r) are strictly

increas-ing on (, ).

Lemma . Let

φp(r) = 

π

E–rK–

( +r)p+– ( –r)p+ (p+ )r

/p ,

Proof It is well known thatLp(a,b) is strictly increasing with respect top∈Rfor fixed

a,b>  witha=b, thenφp(r) is strictly decreasing with respect top∈R. In order to prove Lemma ., we divide it into three cases.

Casep= /.

From Corollary . and Lemma ., we clearly see that

φ/(r) = 

π

E–rK–

( +r)/– ( –r)/ r

/

>  +r 

 +

r –

( +r)/_{– ( –r)}/ r

/

> r 

> 

for  <r< .

Casep=p.

We divide it into two subcases.

SubcaseAφp(r) <  forr∈(, .).

Sinceφp(r) is strictly decreasing with respect top∈R, we clearly see thatφp(r) <φ(r). It suffices to prove thatφ(r) <  forr∈(, .).

Forr∈(,√/], it follows from Corollary . that

φ(r) = 

π

E–rK–

( +r)_{– ( –r)} r

/

<  +r   +  π –  

r–

 +r 

 –

r 



= –r    –  π –   r

≤–r    –   π –  

= –(π– )r 

π < ,

where the first inequality easily follows from

( +r)_{– ( –r)}

r –

 +r

  – r   = r  

 +  –r+r> .

Forr∈(√/, .), taking the derivative ofφ(r) yields

φ_(r) = (E–r

_K)

πr –

r

( +r₎/ =μ(r) +μ(r), (.)

where

μ(r) =(E–r

_K)

πr –

r

, μ(r) =

r

– r

From Lemma .(), we clearly see that

dμ(r)

dr =



πr

K–E–π r



>  (.)

forr∈(, ) and

dμ(r)

dr =

( +r₎/_{–  + r} ( +r₎/ >

[ + (/)r_{] –  + r} ( +r₎/

= r _{– }

( +r₎/ >  (.)

forr∈(√/, .). Equations (.)-(.) lead to the conclusion thatφ_(r) is strictly in-creasing on (√/, .). This in conjunction with (.) implies that

φ_(r) >φ_(√/) = .· · ·>  (.)

forr∈(√/, /). Therefore, from (.) we clearly see thatφ(r) is strictly increasing on (√/, .). This in conjunction with (.) yieldsφ(r) <φ(.) = –.· · ·<  forr∈(√/, .).

SubcaseBφp(r) <  forr∈[., ).

For .≤r< , taking the derivation ofφp(r) yields

φ_p

(r) =

(E–r_K)

πr –

ω(r)

p(p+ )/pη(r)–/p

, (.)

whereω(r) andη(r) are defined as in Lemma .. From Lemma .(), we clearly see that (E–r_{K)/ris strictly increasing on (, ). Lemma . and (.), (.) lead to the} conclu-sion that

φ_p(r)≥[E(.) – ( – .

_)K(.)]

.π –

ω()

p(p+ )/pη(.)–/p = .· · ·– .· · ·= .· · ·> 

for .≤r< .

Therefore, it follows from the monotonicity ofφ_p(r) on (/, ) thatφp(r) <φp() = [/π– /( +p)/p_{] =  for  <r< .}

Case / <p<p.

Taking the Taylor series ofφp(r) atr=  yields

φp(r) =

 –

p



r+( – p+ p

_{+ p}_)r

, +o

r. (.)

From (.) we clearly see that there exists a sufficiently smallδ>  such thatφp(r) <  for

3 Main results

Theorem . Inequality L–(a,b) <AG(a,b) <L–/(a,b)holds for all a,b> with a=b,

where L–(a,b)and L–/(a,b)are the best possible lower and upper generalized logarithmic mean bounds for the arithmetic-geometric meanAG(a,b),respectively.

Proof Firstly, from (.) we clearly see thatL–(a,b) <AG(a,b) for alla,b>  witha=b. Next, we prove thatAG(a,b) <L–/(a,b) for alla,b>  witha=b. SinceAG(a,b) and

Lp(a,b) are symmetric and homogeneous of degree , without loss of generality, it suffices to give an assumption thata=  >b. Lett=b∈(, ),r= ( –t)/( +t), then (.) and (.) lead to

AG(a,b) –L–/(a,b) =

π

K(√ –t₎–

 –t

( –√t) 

=   +r

π

K –

r

√

 +r–√ –r



= h(r)

( +r)K(r), (.)

where

h(r) =π– +rK(r).

We can rewriteh(r) as

h(r) =π–λr· K

log(e_/r), (.)

whereλ(r) = ( +r)log(e_/r). A simple calculation yields

λr=  – 

r –logr

_{, (.)}

λ() = , (.)

λr= –r

r > . (.)

Equations (.)-(.) lead to the conclusion thatλ(r) is strictly decreasing on (, ) with respect tor. Moreover, the functionr=√ –r _{is strictly decreasing on (, ). Hence} the functionλ(r) is strictly increasing on (, ) with respect tor. It follows from (.) and Lemma .() thath(r) is strictly decreasing on (, ). This implies thath(r) <  for  <r<  together withh() = .

Therefore,AG(a,b) <L–/(a,b) for alla,b>  witha=bfollows from (.) andh(r) < . Finally, we prove thatL–(a,b) andL–/(a,b) are the best possible lower and upper gen-eralized logarithmic mean bounds for the arithmetic-geometric meanAG(a,b).

For any  <ε< / and  <x< , it follows from (.) and (.) that

lim

x→

AG(,x) –L–+ε(,x)

=lim

x→

π

K(√ –x₎–

 –xε ε( –x)



ε–

and making use of the Taylor expansion asx→, one has

AG(,  –x) –L–/–ε(,  –x)

= π

K(√x–x₎–

 – ( –x)/–ε

(/ –ε)x

–  +ε

=

 –x –

x

+o

x–

 –x –

 + ε

 x

_+ox

= ε x

_+ox_{. (.)}

Equations (.) and (.) imply that for any  <ε< / there exist δ =δ(ε)∈(, ) and δ =δ(ε)∈(, ) such that AG(,x) <L–+ε(,x) for x∈(,δ) and AG(,  –x) >

L–/–ε(,  –x) forx∈(,δ).

Theorem . Inequality L/(a,b) <T(a,b) <Lp(a,b)holds for all a,b>  with a=b,

where pis defined as in Lemma.and L/(a,b),Lp(a,b)are the best possible lower and

upper generalized logarithmic mean bounds for the Toader mean T(a,b),respectively.

Proof From (.) and (.) we clearly see that bothT(a,b) andLp(a,b) are symmetric and homogeneous of degree . Without loss of generality, we assume thata=  >b. Lett=b∈

(, ),r= ( –t)/( +t), then from (.) and (.) together with (.) we have

T(a,b) –Lp(a,b) = 

πE

√  –t_–

 –tp+ (p+ )( –t)

/p

= 

πE

√r

 +r

– 

 +r

( +r)p+_{– ( –r)}p+ (p+ )r

/p

=   +r



π

E–rK–

( +r)p+– ( –r)p+ (p+ )r

/p

= φp(r)

( +r), (.)

whereφp(r) is defined as in Lemma ..

Therefore, Theorem . follows from (.) and Lemma ..

4 Corollaries and remarks

From Theorem . we get a lower bound for the complete elliptic integral of the first kind K(r) as follows.

Corollary . Inequality

K(r) >π[ + √

 –r_{– ( –r}₎/_]

( –√ –r₎ (.)

holds for all r∈(, ).

Table 1 Comparison ofK(r) withH(r) for somer∈(0, 1)

r K(r) H(r)

0.1 1.5747455615· · · 1.5747455614· · · 0.2 1.5868678474· · · 1.5868678471· · · 0.3 1.608048619· · · 1.608048612· · · 0.4 1.63999986· · · 1.63999977· · · 0.5 1.68575035_{· · ·} 1.68574965_{· · ·} 0.6 1.75075380· · · 1.75074958· · · 0.7 1.84569400· · · 1.84567106· · · 0.8 1.99530278· · · 1.99517293· · ·

Theorem . enables us to give new bounds for the complete elliptic integrals of the second kindE(r).

Corollary . Inequality

π



( – ( –r₎/₎ ( –√ –r₎

/

<E(r) <π 

 – ( –r₎(p+)/ (p+ )( –

√  –r₎

/p

(.)

holds for all r∈(, ),where p= .· · · is defined as in Lemma..

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1_{College of Mathematics and Statistics, Hunan University of Finance and Economics, Changsha, 410205, China.} 2_{Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China.}

Acknowledgements

The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions. This research was supported by the National Natural Science Foundation of China (Grant no. 11301127), the general project Foundation of the Department of Education of Hunan Province (Grant no. 16C0265) and the scientific research Foundation for Young teachers in Hunan University of Finance and Economics (Grant no. Q201501).

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 15 February 2017 Accepted: 13 April 2017

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