Optimal evaluation of a Toader-type mean by power mean

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

In this paper, we present the best possible parameters ...... such that the double inequality ...... holds for all ...... with ......, and we get sharp bounds for the complete elliptic integral ...... of the second kind on the interval ......, where ......, ......, ......, ...... (......), and ...... are the Toader, arithmetic, quadratic, and rth power means of a and b, respectively.

Song et al. Journal of Inequalities and Applications (2015) 2015:408 DOI 10.1186/s13660-015-0927-6

Optimal evaluation of a Toader-type mean by power mean

http://crossmark.crossref.org/dialog/?doi=10.1186/s13660-015-0927-6&domain=pdf

Web End = Ying-Qing Song1, Tie-Hong Zhao2, Yu-Ming Chu1* and Xiao-Hui Zhang1

*Correspondence: mailto:[email protected]

Web End [email protected]

1School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China Full list of author information is available at the end of the article

1 Introduction

For r

R and a, b > , the Toader mean T(a, b) (see []) and rth power mean Mr(a, b) are

dened by

T(a, b) =

/

acos + bsin d (.)

and

Mr(a, b) =

(ar+br )/r, r = , ab, r = ,

(.)

respectively.

It is well known that Mr(a, b) is continuous and strictly increasing with respect to r

R

for xed a, b > with a = 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) (.)

2015 Song et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Song et al. Journal of Inequalities and Applications (2015) 2015:408 Page 2 of 12

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

T(a, b) = RE a, b ,

where

RE(a, b) =

[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) =

/ ( r sin )/ d, and E(r) =

/ ( r sin )/ d be, respectively, the complete elliptic integrals of the rst and second kind. Then K(+) =

E(+) = /, the Toader mean T(a, b) given in (.) can be expressed as

T(a, b) =

a

E(

(ba)), a > b,

b

E(

(ab)), a < b,

(.)

and K(r) and E(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

= . . . . ,

= . . . . , E

= . . . . .

Recently, the power mean Mr(a, b) and Toader mean T(a, b) have been the subject of intensive 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 all a, b > with a = b. This conjecture was proved by Qiu and Shen [], and Barnard et al. [], respectively.

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

T(a, b) < Mlog/(loglog)(a, b)

for all a, b > with a = b.

K

Song et al. Journal of Inequalities and Applications (2015) 2015:408 Page 3 of 12

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

(a + b)ab abAGM(a, b) < T(a, b) <

( + )a + ( )b + ( + )b + ( )a

hold for all a, b > with a = b, where AGM(a, b) is the arithmetic-geometric mean of a and b.

Let , , , (/, ). Then Chu et al. [], and Hua and Qi [] proved that the double inequalities

C a + ( )b, b + ( )a < T(a, b) < C a + ( )b, b + ( )a ,

C a + ( )b, b + ( )a < T(a, b) < C a + ( )b, b + ( )a

hold for all a, b > with a = b if and only if /, / + ( )/(), / +

/, and / + / /, where C(a, b) = (a + b)/(a + b) and C(a, b) = (a + ab + b)/[(a + b)] are, respectively, the contraharmonic and centroidal means of a and b.

In [], the authors proved that the double inequalities

Q(a, b) + ( )A(a, b) < T(a, b) < Q(a, b) + ( )A(a, b),

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

C(a, b) + ( )A(a, b) < T(a, b) < C(a, b) + ( )A(a, b),

A(a, b) +

A(a, b) +

H(a, b) +

hold for all a, b > with a = b if and only if /, ( )/[( )], /,

log / log , /, / , / , /, /, /,

/, / /, /, / , , /, /, /, , and /.

The main purpose of this paper is to present the best possible parameters p, q

that the double inequality

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

holds for all a, b > with a = b.

(a + b)ab + (a b) AGM(a, b) ,

T(a, b) <

C(a, b),

C(a, b) + ( )H(a, b) < T(a, b) < C(a, b) + ( )H(a, b),

C(a, b) H(a, b) + A(a, b) < T(a, b) < C(a, b) H(a, b) + A(a, b),

C(a, b) + ( )A(a, b) < T(a, b) < C(a, b) + ( )A(a, b),

A(a, b) +

C(a, b) <

T(a, b) <

C(a, b) <

T(a, b) <

A(a, b) +

C(a, b),

Q(a, b) + ( )H(a, b) < T(a, b) < Q(a, b) + ( )H(a, b),

H(a, b) +

Q(a, b) <

T(a, b) <

Q(a, b)

R such

Song et al. Journal of Inequalities and Applications (2015) 2015:408 Page 4 of 12

2 Lemmas

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

Lemma . (See [], Theorem .) The inequality E[Mp(x, y)] > Mq[E(x), E(y)] holds for all x, y (, ) if and only if

p C(q) := inf

r(,)

rE(r)( r)[K(r) E(r)]

where q C(q) is a continuous function which satises C(q) = for all q / and C(q) < for all q > /.

Lemma . The double inequality

( t)/ +

( t)/ + <

holds for all t (, /).

Proof Let u = ( t)/. Then u (/ , ), t = u, and the rst inequality of (.) is equivalent to

u + u + <

v)/ + (v v)/]

v v + v / (v + )

for all v (/, ).

We claim that

v / > v + v + v (.)

for all v (/, ).

+ ( q)[K(r) E(r)]E(r)

,

/

t

<

( t + t)/ + ( t t)/

(.)

u +

(.)

for all u (/ , ).

We clearly see that (.) follows from

(u + ) u + u + = (u + ) u + ( u) (u ) + u + u + u >

for all u (/ , ).

For the second inequality of (.), let v = t (/, ), then it suces to prove that

(v) := [(v +

(v + )

> (.)

=

Song et al. Journal of Inequalities and Applications (2015) 2015:408 Page 5 of 12

Indeed, if v (/, ( )/], then we clearly see that the function v + v + v is strictly increasing on (/, ( )/], and (.) follows from

v + v + v

+

+

< .

If v (( )/, ), 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 all v (/, ).

Lemma . The inequality

E(t) >

t

holds for all t (, /).

Proof Let

f (t) = E(t)

. (.)

Then simple computations lead to

f + = , f

t

= . . . . > , (.)

f (t) = tf(t), (.)

where

f(t) = E(t) K(t)

t +

,

f + =

> , f

= . . . . < , (.)

f (t) = f(t)

t( t), (.)

Song et al. Journal of Inequalities and Applications (2015) 2015:408 Page 6 of 12

where

f(t) = t K(t) t E(t),f + = , (.)

f (t) = t K(t) E(t) < (.)

for t (, /).

From (.)-(.) we clearly see that f(t) is strictly decreasing on (, /). Then (.) and (.) lead to the conclusion that there exists t (, /) such that f (t) is strictly increasing on (, t] and strictly decreasing on [t, /).

Therefore, Lemma . follows easily from (.) and (.) together with the piecewise monotonicity of f (t).

Lemma . The inequality

+ t + t

/

> + t

holds for all t (, /).

Proof It suces to prove that the inequalities

+ t + t > +

t

(.)

and

+ t

/

> + t

(.)

hold for all t (, /).

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

+ t + t + t = t( t)( + t)

and

+ t

+ t

= t

+

t

+

,t , +

,t ,, +

t , +

t

,,

.

Lemma . Let = log /[ log log log E(/)] = . . . . and

g(t) =

+ tE

t

/

.

+ t

( + t) + ( t)

Then g(t) > for all t (, /).

Song et al. Journal of Inequalities and Applications (2015) 2015:408 Page 7 of 12

Proof It follows from t/ + t (, /), < /, Lemma ., Lemma . and the monotonicity of Mr( + t, t) with respect to r

R that

g(t) >

+ t

t

( + t)

( + t)/ + ( t)/

/

/

= + t

+ t

( + t)/ + ( t)/

> + t

/

( + t)/ + ( t)/

/

(.)

for t (, /). 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)/

< (.)

for t (, /).

From (.) and (.) we know that there exists t (, /) such that g (t) is strictly increasing on (, t] and strictly decreasing on [t, /). Then (.) leads to the conclusion that there exists t (, /) such that g(t) is strictly increasing on (, t] and strictly decreasing on [t, /).

Therefore, Lemma . follows from (.)-(.) and the piecewise monotonicity of g(t).

Lemma . Let = log /[ log log log E(/)] = . . . . . Then the functiontE(t)[E(t) K(t)] is strictly decreasing on (, ).

Proof From Lemma . we clearly see that the inequality

E M

(x, y) > M E(x), E(y) =

E(x) + E(y)

/

(.)

holds for all x, y (, ) with x = y.

Song et al. Journal of Inequalities and Applications (2015) 2015:408 Page 8 of 12

It follows from the monotonicity of the function E(t) and the power mean Mp(x, y) with

respect to p

R together with > that

E

x + y

= E M(x, y) > E M

(x, y) (.)

for all x, y (, ) with x = y.Inequalities (.) and (.) lead to

E

x + y

> E(x) + E(y)

for all x, y (, ) with x = y, which implies that the function E(t) is strictly concave on

(, ).Note that

tE(t) E(t) K(t) =

dt . (.)

Therefore, Lemma . follows easily from (.) and the concavity of E(t) on (, ).

Lemma . Let = log /[ log log log E(/)] = . . . .

h(t) = +

E(t)

( + t) + ( t)

dE(t)

and

h(t) = +

E(t) (

t) /.

Then h(t) > for t [/, /) and h(t) > for t [/, /).

Proof Simple computations lead to

h

= . . . . > , h

= , (.)

h (t) =

+

tE(t) E(t) K(t) + ( t) ( + t)

, (.)

h (t) =

tE(t) E(t) K(t) + (

t) , (.)

h

= . . . . < . (.)

From (.) and (.) together with Lemma . we clearly see that both h (t) and h (t) are strictly decreasing on (, /). Then (.) leads to the conclusion that h(t) is strictly decreasing on [/, /] and h(t) is strictly decreasing on [/, /).

Therefore, Lemma . follows from (.) and the monotonicity of h(t) on [/, /] and h(t) on [/, /).

= . . . . < , h

Song et al. Journal of Inequalities and Applications (2015) 2015:408 Page 9 of 12

Lemma . (See [], Corollary .) The inequality

E(t) <

( t)/ +

( t)/ + (.)

holds for all t (, ).

3 Main results

Theorem . Let = log /[ log log log E(/)] = . . . . . Then the doubleinequality

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 and q /.

Proof Since the arithmetic mean A(a, b), quadratic mean Q(a, b), Toader mean T(a, b), and rth power mean Mr(a, b) are symmetric and homogeneous of degree , without loss of generality, we assume that a > b. Let t = (a b)/ (a + b). Then t (, /) and equations (.)-(.) lead to

Mr(a, b) = A(a, b)

t

( t + t)r + ( t t)r

/r

, (.)

T A(a, b), Q(a, b) = A(a, b)E(t)

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 to r that

T A(a, b), Q(a, b) Mr(a, b)

T A(a, b), Q(a, b) M/(a, b)

= A(a, b)

t

E(t)

( t)/ +

( t)/ +

+ A(a, b)

t

( t)/ +

( t)/ +

( t + t)/ + ( t t)/

/ . (.)

Therefore,

T A(a, b), Q(a, b) < Mr(a, b)

for all a, b > with a = b follows from Lemmas . and . together with (.).

Song et al. Journal of Inequalities and Applications (2015) 2015:408 Page 10 of 12

Case r . Then equations (.) and (.) together with the monotonicity of Mr(a, b) with respect to r lead to

T A(a, b), Q(a, b) Mr(a, b)

T A(a, b), Q(a, b) M(a, b)

= A(a, b)

t

We divide the proof into two subcases.Subcase . t (, /). Let u = t/ t. Then u (, /) and (.) leads to

T A(a, b), Q(a, b) Mr(a, b)

> A(a, b)

+ uE

u

+ u

Therefore,

T A(a, b), Q(a, b) > Mr(a, b)

for < |a b|/ (a + b) < / with a = b follows from Lemma . and (.).

Subcase . t [/, /). Let

h(t) = +

E(t)

It is easy to verify that

t

for all t (, /).

Equation (.) and inequality (.) lead to

h(t) > +

E(t)

and

E(t) ( t) /. (.)

Therefore,

T A(a, b), Q(a, b) > Mr(a, b)

for / |a b|/ (a + b) with a = b follows from Lemma ., (.), (.), (.), and (.).

E(t)

( t + t) + ( t t)

/ . (.)

( + u) + ( u)

/ . (.)

t + t t t . (.)

t

and

t < t (.)

( + t) + ( t)

(.)

h(t) > +

Song et al. Journal of Inequalities and Applications (2015) 2015:408 Page 11 of 12

Case < r < /. On the one hand, equations (.) and (.) lead to

lim

x

+

log T A(, x), Q(, x) log Mr(, x)

= log

E(

)

+

log r

r < . (.)

Inequality (.) implies that there exists > such that

T A(a, b), Q(a, b) < Mr(a, b)

for all a, b > with a/b (, ).

On the other hand, by the Taylor expansion and let x > and x , then equations (.) and (.) lead to

T A(, x), Q(, x) Mr(, x)

=

= (r

) log

x + x

x

/r

E

x + x

+ ( x)r

= x

+

x

x

+

r

x + o x

x . (.)

Equation (.) implies there exists (, ) such that

T A(a, b), Q(a, b) > Mr(a, b)

for all a, b > with a/b ( , ).

From Theorem . we get Corollary . immediately.

Corollary . Let = log /[ log log log E(/)] = . . . . . Then the double

inequality

( t + t)p + ( t t)p

= r

x + o

( t + t)q + ( t t)q

/q

/p

< E(t) <

holds for all t (, /) if and only if p and 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 nal manuscript.

Author details

1School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China. 2Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China.

Song et al. Journal of Inequalities and Applications (2015) 2015:408 Page 12 of 12

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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