Optimal evaluation of a Toader type mean by power mean

R E S E A R C H

Open Access

Optimal evaluation of a Toader-type mean

by power mean

Ying-Qing Song

_{, Tie-Hong Zhao}

_{, Yu-Ming Chu}

_{and Xiao-Hui Zhang}

*_{Correspondence:}

[email protected]

1_{School of Mathematics and}

Computation Science, Hunan City University, Yiyang, 413000, China Full list of author information is available at the end of the article

Abstract

In this paper, we present the best possible parametersp,q∈Rsuch that the double inequalityMp(a,b) <T[A(a,b),Q(a,b)] <Mq(a,b) holds for alla,b> 0 witha=b, and we

get sharp bounds for the complete elliptic integralE(t) =₀π/2(1 –t2_sin2

_θ

the second kind on the interval (0,√2/2), where T(a,b) = 2

π

π/2 0

√

a2_cos2

_θ

_θ

Mr(a,b) = [(ar+br)/2]1/r(r= 0), andM0(a,b) =

√

abare the Toader, arithmetic, quadratic, andrth power means ofaandb, respectively.

MSC: 33E05; 33C05; 26E60

Keywords: arithmetic mean; Toader mean; quadratic mean

1 Introduction

Forr∈Randa,b> , the Toader meanT(a,b) (see []) andrth power meanMr(a,b) are

defined by

T(a,b) = 

π

π/



a_cos_θ+b_sin_{θdθ (.)}

and

Mr(a,b) =

⎧ ⎨ ⎩

(ar+_br)/r_{, r= ,}

√

ab, r= ,

(.)

respectively.

It is well known thatMr(a,b) is continuous and strictly increasing with respect tor∈R

for fixeda,b>  witha=b. Many classical bivariate means are a special case of the power mean, for example,H(a,b) = ab/(a+b) =M–(a,b) is the harmonic mean,G(a,b) =

√ ab= M(a,b) is the geometric mean,

A(a,b) = (a+b)/ =M(a,b) (.)

is the arithmetic mean, and

Q(a,b) = a_+b_{/ =M}

(a,b) (.)

is the quadratic mean. The main properties of the power mean are given in []. The Toader meanT(a,b) has been well known in the mathematical literature for many years, it satisfies

T(a,b) =RE a,b

,

where

RE(a,b) =



π

∞



[a(t+b) +b(t+a)]t (t+a)/_(t+b)/ dt

stands for the symmetric complete elliptic integral of the second kind (see [–]), therefore it cannot be expressed in terms of the elementary transcendental functions.

Let r∈ (, ),K(r) =_π/( –r_sin_θ)–/_{dθ, and E(r) =}π/

 ( –rsin

_θ)/_{dθ be,}

respectively, the complete elliptic integrals of the first and second kind. Then K(+) =

E(+_{) =π/, the Toader meanT(a,b) given in (.) can be expressed as}

T(a,b) = ⎧ ⎨ ⎩

a

πE(

 – (b

a)), a>b,

b

πE(

 – (a_b)_{), a<b,} (.)

andK(r) andE(r) satisfy the derivatives formulas (see [], Appendix E, p. -)

dK(r) dr =

E(r) – ( –r_)K(r)

r( –r₎ ,

dE(r) dr =

E(r) –K(r)

r ,

d(K(r) –E(r))

dr =

rE(r)  –r.

Numerical computations show that

E

√  

= . . . . , K

 

= . . . . , E

 

= . . . . ,

K

 

= . . . . , E

 

= . . . . .

Recently, the power meanMr(a,b) and Toader meanT(a,b) have been the subject of

in-tensive research. In particular, many remarkable inequalities for both means can be found in the literature [–].

Vuorinen [] conjectured that the inequality

M/(a,b) <T(a,b)

holds for alla,b>  witha=b. This conjecture was proved by Qiu and Shen [], and Barnardet al.[], respectively.

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

T(a,b) <Mlog/(logπ–log)(a,b)

Neuman [], and Kazi and Neuman [] proved that the inequalities

(a+b)√ab–ab

AGM(a,b) <T(a,b) <

(a+b)√ab+ (a–b)

AGM(a,b) ,

T(a,b) <   ( +

√

)a_{+ ( –}√_)b₊

( +√)b_{+ ( –}√_)a

hold for alla,b>  witha=b, whereAGM(a,b) is the arithmetic-geometric mean ofa andb.

Letλ,μ,α,β∈(/, ). Then Chuet al.[], and Hua and Qi [] proved that the double inequalities

Cλa+ ( –λ)b,λb+ ( –λ)a<T(a,b) <Cμa+ ( –μ)b,μb+ ( –μ)a,

Cαa+ ( –α)b,αb+ ( –α)a<T(a,b) <Cβa+ ( –β)b,βb+ ( –β)a

hold for alla,b>  witha=bif and only ifλ≤/,μ≥/ +√π( –π)/(π),α≤/ + √

/, andβ≥/ +√/π– /, whereC(a,b) = (a_+b_{)/(a+b) andC(a,b) = (a}₊

ab+b_{)/[(a+b)] are, respectively, the contraharmonic and centroidal means ofaandb.}

In [–], the authors proved that the double inequalities

αQ(a,b) + ( –α)A(a,b) <T(a,b) <βQ(a,b) + ( –β)A(a,b),

Qα_(a,b)A(–α)_{(a,b) <T(a,b) <Q}β_(a,b)A(–β)_(a,b),

αC(a,b) + ( –α)A(a,b) <T(a,b) <βC(a,b) + ( –β)A(a,b), α

A(a,b)+  –α

C(a,b)<  T(a,b)<

β

A(a,b)+  –β

C(a,b),

αC(a,b) + ( –α)H(a,b) <T(a,b) <βC(a,b) + ( –β)H(a,b),

αC(a,b) –H(a,b)+A(a,b) <T(a,b) <βC(a,b) –H(a,b)+A(a,b),

αC(a,b) + ( –α)A(a,b) <T(a,b) <βC(a,b) + ( –β)A(a,b),

α

A(a,b)+  –α

C(a,b)<  T(a,b)<

β

A(a,b)+  –β

C(a,b),

αQ(a,b) + ( –α)H(a,b) <T(a,b) <βQ(a,b) + ( –β)H(a,b), α

H(a,b)+  –α

Q(a,b)<  T(a,b)<

β

H(a,b)+  –β

Q(a,b)

hold for alla,b>  witha=bif and only ifα≤/,β≥( –π)/[(√ – )π],α≤/,

β≥ – logπ/log,α≤/,β≥/π– ,α≤π/ – ,β≥/,α≤/,β≥/π,

α≤/,β≥/π– /,α≤/,β≥/π– ,α≤π– ,β≥/,α≤/,β≥

√/π,α≤, andβ≥/.

The main purpose of this paper is to present the best possible parametersp,q∈Rsuch that the double inequality

Mp(a,b) <T

A(a,b),Q(a,b)<Mq(a,b)

2 Lemmas

In order to prove our main results, we need several lemmas which we present in this sec-tion.

Lemma .(See [], Theorem .) The inequalityE[Mp(x,y)] >Mq[E(x),E(y)]holds for

all x,y∈(, )if and only if

p≤C(q) := inf

r∈(,)

r_E(r)

( –r_)[K_{(r) –}E_(r)]+

( –q)[K(r) –E(r)]

E(r)

,

where q→C(q)is a continuous function which satisfies C(q) = for all q≤/and C(q) < for all q> /.

Lemma . The double inequality

( –t₎/_{+ }

( –t₎/_{+ }<  –

t

 <

(√ –t_+t)/_{+ (}√_{ –t}_–t)/



/

(.)

holds for all t∈(,√/).

Proof Letu= ( –t)/. Thenu∈(/√, ),t=  –u, and the first inequality of (.) is equivalent to

u_{+ }

u+  < u_{+ }

 (.)

for allu∈(/√, ).

We clearly see that (.) follows from

(u+ ) u+ –  u+ = (u+ ) u+ ( –u)(u– ) + u+ u+u> 

for allu∈(/√, ).

For the second inequality of (.), letv=√ –t_∈(√_{/, ), then it suffices to prove that}

ρ(v) :=[(v+ √

 –v₎/_{+ (v–}√_{ –v}₎/_]

 –

(v+ ) 

= 

v– v+ v– /–(v

_{+ )}



>  (.)

for allv∈(√/, ). We claim that

v– />  – v+ v+ v (.)

Indeed, ifv∈(√/, (√ – )/], then we clearly see that the function  – v+ v_{+ v}

is strictly increasing on (√/, (√ – )/], and (.) follows from

 – v+ v+ v≤ – × √

 –   + ×

√  – 

 

+ × √

 –  



= –  √

  < .

Ifv∈((√ – )/, ), then (.) follows easily from

v– –  – v+ v+ v=  –v – + v+ v

>  –v– + × √

 –   + 

√  – 

 

= .

Therefore, inequality (.) follows from (.) and

v– v+ v– /–(v

_{+ )}

 > v– v

_{+  – v+ v}_{+ v}_–(v+ )



=( –v

_{)( + v+ v}_+v₎

 > 

for allv∈(√/, ).

Lemma . The inequality

E(t) >π 

 –t





holds for all t∈(, /).

Proof Let

f(t) =E(t) –π 

 –t





. (.)

Then simple computations lead to

f += , f

 

= . . . . > , (.)

f (t) =tf(t), (.)

where

f(t) = E

(t) –K(t) t +

π

,

f +

= π

> , f

 

= –. . . . < , (.)

f_(t) = f(t)

where

f(t) =   –t

K(t) –  –tE(t),

f +

= , (.)

f_(t) = –tK(t) –E(t)<  (.)

fort∈(, /).

From (.)-(.) we clearly see thatf(t) is strictly decreasing on (, /). Then (.) and

(.) lead to the conclusion that there existst∈(, /) such thatf(t) is strictly increasing

on (,t] and strictly decreasing on [t, /).

Therefore, Lemma . follows easily from (.) and (.) together with the piecewise monotonicity off(t).

Lemma . The inequality

 + t

√ +t

/

>  +t





holds for all t∈(, /).

Proof It suffices to prove that the inequalities

 + t

√ +t >  +

t

 (.)

and

 +t



 /

>  +t



 (.)

hold for allt∈(, /).

Indeed, inequalities (.) and (.) follow easily from the identities

 + t–  +t  + t=t( – t)( + t)

and

 +t



 

–

 +t



 

= t

 +

t

+

,t

,+

,t

,,+ t

,+ t

,,

.

Lemma . Letλ= log/[logπ–log – logE(√/)] = . . . .and

g(t) = 

π

√  +tE

t √

 +t

–

( +t)λ_{+ ( –t)}λ 

/λ .

Proof It follows fromt/√ +t∈_{(, /),λ< /, Lemma ., Lemma . and the}

mono-tonicity ofMr( +t,  –t) with respect tor∈Rthat

g(t) > 

π

√

 +t×π



 – t



( +t₎

–

( +t)/_{+ ( –t)}/



/

=  + t



√ +t –

( +t)/_{+ ( –t)}/



/

>

 +t



 /

–

( +t)/+ ( –t)/ 

/

(.)

fort∈(, /). Let

g(t) = 

 +t





–( +t)/+ ( –t)/. (.)

Then simple computations lead to

g() = , g

 

= . . . . > , (.)

g_(t) =  

t– ( +t)/+ ( –t)/,

g_() = , g_

 

= –. . . . < , (.)

g_ (t) =  

 –  ( +t)/–

 ( –t)/

,

g_ () = 

> , g



 

= –. . . . < , (.)

g_ (t) =  

 ( +t)/ –

 ( –t)/

<  (.)

fort∈(, /).

From (.) and (.) we know that there existst∈(, /) such thatg(t) is strictly

increasing on (,t] and strictly decreasing on [t, /). Then (.) leads to the

conclu-sion that there existst∈(, /) such thatg(t) is strictly increasing on (,t] and strictly

decreasing on [t, /).

Therefore, Lemma . follows from (.)-(.) and the piecewise monotonicity of

g(t).

Lemma . Letλ= log/[logπ–log – logE(√/)] = . . . . .Then the function t–_Eλ–_{(t)[E(t) –K(t)]is strictly decreasing on(, ).}

Proof From Lemma . we clearly see that the inequality

E Mλ(x,y)>Mλ E(x),E(y)

=

_Eλ_{(x) +E}λ_(y)



/λ

(.)

It follows from the monotonicity of the functionE(t) and the power meanMp(x,y) with

respect top∈Rtogether withλ>  that

E

x+y



=E M(x,y)

>E Mλ(x,y) (.)

for allx,y∈(, ) withx=y.

Inequalities (.) and (.) lead to

Eλ

x+y 

>E

λ_{(x) +E}λ_(y) 

for allx,y∈(, ) withx=y, which implies that the functionEλ_{(t) is strictly concave on} (, ).

Note that

t–Eλ–_{(t)E(t) –K(t)=} λ

dEλ_(t)

dt . (.)

Therefore, Lemma . follows easily from (.) and the concavity ofEλ_{(t) on (, ).}

Lemma . Letλ= log/[logπ–log – logE(√/)] = . . . .

h(t) =

+λ

πλ E λ

(t) –( +t)

λ_{+ ( – t)}λ λ

and

h(t) =

+λ

πλ E

λ_{(t) – (}√_{ – t)}λ_{– }λ/_.

Then h(t) > for t∈[/, /)and h(t) > for t∈[/,

√ /).

Proof Simple computations lead to

h

 

= . . . . > , h

√  

= , (.)

h_(t) = λ λ

λ+

πλ t

–_Eλ–_{(t)E(t) –K(t)+ ( – t)}λ–_{– ( +t)}λ–

, (.)

h_(t) = λ



π

λ

t–Eλ–(t) E(t) –K(t)+ (√ – t)λ–

, (.)

h_

 

= –. . . . < , h_

 

= –. . . . < . (.)

From (.) and (.) together with Lemma . we clearly see that bothh_(t) andh_(t) are strictly decreasing on (,√/). Then (.) leads to the conclusion thath(t) is strictly

decreasing on [/, /] andh(t) is strictly decreasing on [/,

√ /).

Therefore, Lemma . follows from (.) and the monotonicity ofh(t) on [/, /]

andh(t) on [/,

√

Lemma .(See [], Corollary .) The inequality



πE(t) <

( –t₎/_{+ }

( –t₎/_{+ } (.)

holds for all t∈(, ).

3 Main results

Theorem . Letλ= log/[logπ–log – logE(√/)] = . . . . .Then the double inequality

Mp(a,b) <T

A(a,b),Q(a,b)<Mq(a,b)

holds for all a,b> with a=b if and only if p≤λand q≥/.

Proof Since the arithmetic meanA(a,b), quadratic meanQ(a,b), Toader meanT(a,b), andrth power meanMr(a,b) are symmetric and homogeneous of degree , without loss

of generality, we assume thata>b. Lett= (a–b)/(a_+b_{). Thent}∈_(,√_{/) and}

equations (.)-(.) lead to

Mr(a,b) =

A(a,b) √

 –t

(√ –t_+t)r_{+ (}√_{ –t}_–t)r



/r

, (.)

TA(a,b),Q(a,b)=A(a,b)E(t)

π√ –t . (.)

We divide the proof into three cases.

Caser≥/. Then it follows from (.) and (.) together with the monotonicity of Mr(a,b) with respect torthat

TA(a,b),Q(a,b)–Mr(a,b)

≤TA(a,b),Q(a,b)–M/(a,b)

=√A(a,b)  –t



πE(t) –

( –t₎/_{+ }

( –t₎/_{+ }

+√A(a,b)  –t

( –t₎/_{+ }

( –t₎/_{+ }–

(√ –t_+t)/_{+ (}√_{ –t}_–t)/



/

. (.)

Therefore,

TA(a,b),Q(a,b)<Mr(a,b)

Caser≤λ. Then equations (.) and (.) together with the monotonicity ofMr(a,b)

with respect torlead to

TA(a,b),Q(a,b)–Mr(a,b)

≥TA(a,b),Q(a,b)–Mλ(a,b)

=√A(a,b)  –t



πE(t) –

(√ –t_+t)λ_{+ (}√_{ –t}_–t)λ 

/λ

. (.)

We divide the proof into two subcases.

Subcase.t∈(, /). Letu=t/√ –t_{. Thenu∈(, /) and (.) leads to}

TA(a,b),Q(a,b)–Mr(a,b)

>A(a,b)



π

√  +uE

u √

 +u

–

( +u)λ_{+ ( –u)}λ 

/λ

. (.)

Therefore,

TA(a,b),Q(a,b)>Mr(a,b)

for  <|a–b|/(a_+b_{) < / witha}=_{bfollows from Lemma . and (.).}

Subcase.t∈[/,√/). Let

h(t) =

+λ

πλ E λ

(t) – √ –t_+tλ_– √_{ –t}_–tλ_{. (.)}

It is easy to verify that

√

 –t≤ – t

 and √

 –t_<√_{ –t (.)}

for allt∈(,√/).

Equation (.) and inequality (.) lead to

h(t) >

+λ

πλ E λ

(t) –( +t)

λ_{+ ( – t)}λ

λ (.)

and

h(t) >

+λ

πλ E

λ_{(t) – (}√_{ – t)}λ_{– }λ/_{. (.)}

Therefore,

TA(a,b),Q(a,b)>Mr(a,b)

for /≤ |a–b|/(a_+b_{) witha=bfollows from Lemma ., (.), (.), (.), and}

Caseλ<r< /. On the one hand, equations (.) and (.) lead to

lim

x→+

logTA(,x),Q(,x)–logMr(,x)

=log

√ E(

√

  ) π

+log

r

= –(r–λ)log

λr < . (.)

Inequality (.) implies that there existsδ>  such that

TA(a,b),Q(a,b)<Mr(a,b)

for alla,b>  witha/b∈(,δ).

On the other hand, by the Taylor expansion and letx>  andx→, then equations (.) and (.) lead to

TA(,  –x),Q(,  –x)–Mr(,  –x)

= 

π

 –x+x



 E

x



 –x+x_

–

 + ( –x)r 

/r

=  –x +

x –

 –x

+

 –

 – r 

x

+o x

= – r  x

_{+o x}_{. (.)}

Equation (.) implies there existsδ∈(, ) such that

TA(a,b),Q(a,b)>Mr(a,b)

for alla,b>  witha/b∈( –δ, ).

From Theorem . we get Corollary . immediately.

Corollary . Letλ= log/[logπ–log – logE(√/)] = . . . . .Then the double inequality

π



(√ –t_+t)p_{+ (}√_{ –t}_–t)p



/p

<E(t) <π 

(√ –t_+t)q_{+ (}√_{ –t}_–t)q



/q

holds for all t∈(,√/)if and only if p≤λand q≥/.

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_{School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China.}2_{Department of}

Acknowledgements

This research was supported by the Natural Science Foundation of China under Grants 11301127, 11371125, 11401191, and 61374086, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004 and the Natural Science Foundation of Hunan Province under Grant 12C0577.

Received: 23 September 2015 Accepted: 1 December 2015

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