optimal evaluation of a toader-type mean by power mean · 2020. 1. 23. ·...

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

R E S E A R C H Open Access

Optimal evaluation of a Toader-type meanby power meanYing-Qing Song1, Tie-Hong Zhao2, Yu-Ming Chu1* and Xiao-Hui Zhang1

*Correspondence:[email protected] of Mathematics andComputation Science, Hunan CityUniversity, Yiyang, 413000, ChinaFull list of author information isavailable at the end of the article

AbstractIn this paper, we present the best possible parameters p,q ∈R such that the doubleinequalityMp(a,b) < T [A(a,b),Q(a,b)] 0 with a �= b, and weget sharp bounds for the complete elliptic integral E (t) =

∫ π /20 (1 – t

2 sin2 θ )1/2 dθ ofthe second kind on the interval (0,

√2/2), where

T (a,b) = 2π

∫ π /20

√a2 cos2 θ + b2 sin2 θ dθ , A(a,b) = (a + b)/2, Q(a,b) =

√(a2 + b2)/2,

Mr(a,b) = [(ar + br)/2]1/r (r �= 0), andM0(a,b) =√ab are the Toader, arithmetic,

quadratic, and rth power means of a and b, respectively.

MSC: 33E05; 33C05; 26E60

Keywords: arithmetic mean; Toader mean; quadratic mean

1 IntroductionFor r ∈ R and a, b > , the Toader mean T(a, b) (see []) and rth power mean Mr(a, b) aredefined by

T(a, b) =π

∫ π/

√acos θ + bsin θ dθ (.)

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 fixed a, b > with a �= b. Many classical bivariate means are a special case of the powermean, 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 anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.

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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 Toadermean T(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 [–]), thereforeit 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 mean T(a, b) given in (.) can be expressed as

T(a, b) =

⎧⎨

⎩

aπE(

√ – ( ba )), a > b,

bπE(

√ – ( 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(

)

= . . . . ,

K(

)

= . . . . , E(

)

= . . . . .

Recently, the power mean Mr(a, b) and Toader mean T(a, b) have been the subject of in-tensive research. In particular, many remarkable inequalities for both means can be foundin 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 [], andBarnard et al. [], respectively.

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

T(a, b) < Mlog /(logπ–log )(a, b)

for all a, b > with a �= b.


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

(a + b)√

ab – abAGM(a, b)

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

√ab + (a – b)

AGM(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 aand b.

Let λ,μ,α,β ∈ (/, ). Then Chu et al. [], and Hua and Qi [] proved that the doubleinequalities

C[λa + ( – λ)b,λb + ( – λ)a

]< T(a, b) < C

[μa + ( – μ)b,μb + ( – μ)a

],

C[αa + ( – α)b,αb + ( – α)a

]< T(a, b) < C

[βa + ( – β)b,βb + ( – β)a

]

hold for all a, b > with a �= b if and only if λ ≤ /, μ ≥ / + √π ( – π )/(π ), α ≤ / +√/, and β ≥ / + √/π – /, 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

αQ(a, b) + ( – α)A(a, b) < T(a, b) < βQ(a, b) + ( – β)A(a, b),

Qα (a, b)A(–α)(a, b) < T(a, b) < Qβ (a, b)A(–β)(a, b),

αC(a, b) + ( – α)A(a, b) < T(a, b) < βC(a, b) + ( – β)A(a, b),

α

A(a, b)+

– αC(a, b)

<

T(a, b)<

β

A(a, b)+

– βC(a, b)

,

αC(a, b) + ( – α)H(a, b) < T(a, b) < βC(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),

αC(a, b) + ( – α)A(a, b) < T(a, b) < βC(a, b) + ( – β)A(a, b),

α

A(a, b)+

– αC(a, b)

<

T(a, b)<

β

A(a, b)+

– βC(a, b)

,

αQ(a, b) + ( – α)H(a, b) < T(a, b) < βQ(a, b) + ( – β)H(a, b),

α

H(a, b)+

– αQ(a, b)

<

T(a, b)<

β

H(a, b)+

– βQ(a, b)

hold for all a, b > with a �= b if and only if α ≤ /, 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 ≤ , and β ≥ /.The main purpose of this paper is to present the best possible parameters p, q ∈R such

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.


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

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

p ≤ C(q) := infr∈(,)

{rE(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 Let u = ( – t)/. Then u ∈ (/ √, ), t = – u, and the first inequality of (.) isequivalent to

u + u +

<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 suffices to prove that

ρ(v) :=[(v +

√ – v)/ + (v –

√ – v)/]

–

(v + )

=

[

v – v +(v –

)/ –(v + )

]

> (.)

for all v ∈ (√/, ).We claim that

(v –

)/ > – v + v + v (.)

for all v ∈ (√/, ).


Indeed, if v ∈ (√/, (√ – )/], then we clearly see that the function – v + v + vis 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) > 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)

t( – t), (.)


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 increasingon (, t] and strictly decreasing on [t, /).

Therefore, Lemma . follows easily from (.) and (.) together with the piecewisemonotonicity of f (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 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 λ = log /[ logπ – log – logE(√

/)] = . . . . and

g(t) =π

√ + tE

(t√

+ t

)

–[

( + t)λ + ( – t)λ

]/λ.

Then g(t) > for all t ∈ (, /).


Proof It follows from t/√

+ t ∈ (, /), λ < /, Lemma ., Lemma . and the mono-tonicity of Mr( + t, – t) with respect to r ∈ R that

g(t) >π

√ + t × π

[

–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 conclu-sion that there exists t ∈ (, /) such that g(t) is strictly increasing on (, t] and strictlydecreasing on [t, /).

Therefore, Lemma . follows from (.)-(.) and the piecewise monotonicity ofg(t). �


/)] = . . . . . Then the functiont–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) + Eλ(y)

)/λ(.)

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


It follows from the monotonicity of the function E(t) and the power mean Mp(x, y) withrespect to p ∈R together with λ > that

E(

x + y

)

= E(M(x, y)

)> E

(Mλ(x, y)

)(.)

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

Eλ(

x + y

)

>Eλ(x) + Eλ(y)

for all x, y ∈ (, ) with x �= y, which implies that the function Eλ(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 of Eλ(t) on (, ).


/)] = . . . .

h(t) =+λ

πλEλ(t) – ( + t)

λ + ( – t)λ

λ

and

h(t) =+λ

πλEλ(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+

πλt–Eλ–(t)

(E(t) – K(t)

)+ ( – t)λ– – ( + t)λ–

]

, (.)

h′(t) = λ[(

π

)λt–Eλ–(t)

(E(t) – K(t)

)+ (

√ – t)λ–

]

, (.)

h′

(

)

= –. . . . < , 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 [/,√

/). �


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

πE(t) < ( – t

)/ + ( – t)/ +

(.)

holds for all t ∈ (, ).

3 Main resultsTheorem . Let λ = 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 ≤ λ 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 lossof 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)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 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

[πE(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 (.).


Case r ≤ λ. 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)] – 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 ∈ (, /). Let u = t/√ – t. Then u ∈ (, /) and (.) leads to

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

]– Mr(a, b)

> A(a, b)[

π

√ + uE

(u√

+ u

)

–(

( + u)λ + ( – u)λ

)/λ]

. (.)

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) =+λ

πλEλ(t) –

(√ – t + t

)λ –(√

– t – t)λ. (.)

It is easy to verify that

√ – t ≤ – t

and

√ – t <

√ – t (.)

for all t ∈ (,√/).Equation (.) and inequality (.) lead to

h(t) >+λ

πλEλ(t) – ( + t)

λ + ( – t)λ

λ(.)

and

h(t) >+λ

πλEλ(t) – (

√ – t)λ – λ/. (.)

Therefore,

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

]> Mr(a, b)

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


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

limx→+

[log T

[A(, x), Q(, x)

]– log Mr(, x)

]

= log[√E(

√

)π

]

+log

r

= –(r – λ) log

λr< . (.)

Inequality (.) implies that there exists δ > such that

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

]< Mr(a, b)

for all a, b > with a/b ∈ (, δ).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)

=π

√

– x +x

E(

x

√

– x + x

)

–[

+ ( – x)r

]/r

= –x

+x

–

[

–x

+(

– – r

)

x]

+ o(x

)

= – r

x + o

(x

). (.)

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

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

]> Mr(a, b)

for all a, b > with a/b ∈ ( – δ, ). �

From Theorem . we get Corollary . immediately.

Corollary . Let λ = log /[ logπ – log – logE(√

/)] = . . . . . Then the doubleinequality

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 ≤ λ and q ≥ /.

Competing interestsThe authors declare that they have no competing interests.

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

Author details1School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China. 2Department ofMathematics, Hangzhou Normal University, Hangzhou, 310036, China.


AcknowledgementsThis 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 ScienceFoundation of Hunan Province under Grant 12C0577.

Received: 23 September 2015 Accepted: 1 December 2015

References1. Toader, G: Some mean values related to the arithmetic-geometric mean. J. Math. Anal. Appl. 218(2), 358-368 (1998)2. Bullen, PS, Mitrinović, DS, Vasić, PM: Means and Their Inequalities. Reidel, Dordrecht (1988)3. Neuman, E: Bounds for symmetric elliptic integrals. J. Approx. Theory 122(2), 249-259 (2003)4. Kazi, H, Neuman, E: Inequalities and bounds for elliptic integrals. J. Approx. Theory 146(2), 212-226 (2007)5. Kazi, H, Neuman, E: Inequalities and bounds for elliptic integrals II. In: Special Functions and Orthogonal Polynomials.

Contemp. Math., vol. 471, pp. 127-138. Am. Math. Soc., Providence, RI (2008)6. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley,

New York (1997)7. Lin, TP: The power mean and the logarithmic mean. Am. Math. Mon. 81, 879-883 (1974)8. Pittenger, AO: Inequalities between arithmetic and logarithmic means. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.

678-715, 15-18 (1980)9. Pittenger, AO: The symmetric, logarithmic and power means. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 678-715,

19-23 (1980)10. Stolarsky, KB: The power and generalized logarithmic means. Am. Math. Mon. 87(7), 545-548 (1980)11. Alzer, H: Ungleichungen für (e/a)a(b/e)b . Elem. Math. 40, 120-123 (1985)12. Alzer, H: Ungleichungen für Mittelwerte. Arch. Math. 47(5), 422-426 (1986)13. Burk, F: The geometric, logarithmic, and arithmetic mean inequality. Am. Math. Mon. 94(6), 527-528 (1987)14. Alzer, H, Qiu, S-L: Inequalities for means in two variables. Arch. Math. 80(2), 201-215 (2003)15. Costin, I, Toader, G: Optimal evaluations of some Seiffert-type means by power mean. Appl. Math. Comput. 219(9),

4745-4754 (2013)16. Chu, Y-M, Wang, M-K, Qiu, S-L, Qiu, Y-F: Sharp generalized Seiffert mean bounds for Toader mean. Abstr. Appl. Anal.

2011, Article ID 605259 (2011)17. Chu, Y-M, Wang, M-K: Inequalities between arithmetic-geometric, Gini, and Toader means. Abstr. Appl. Anal. 2012,

Article ID 830585 (2012)18. Chu, Y-M, Wang, M-K: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3-4), 223-229 (2012)19. Vuorinen, M: Hypergeometric functions in geometric function theory. In: Special Functions and Differential Equations

(Madras, 1977), pp. 119-126. Allied Publ., New Delhi (1998)20. Qiu, S-L, Shen, J-M: On two problems concerning means. J. Hongzhou Inst. Electron. Eng. 17(3), 1-7 (1997)

(in Chinese)21. Barnard, RW, Pearce, K, Richards, KC: An inequality involving the generalized hypergeometric function and the arc

length of an ellipse. SIAM J. Math. Anal. 31(3), 693-699 (2000)22. Alzer, H, Qiu, S-L: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math.

172(2), 289-312 (2004)23. Chu, Y-M, Wang, M-K, Ma, X-Y: Sharp bounds for Toader mean in terms of contraharmonic mean with applications.

J. Math. Inequal. 7(2), 161-166 (2012)24. Hua, Y, Qi, F: A double inequality for bounding Toader mean by the centroidal mean. Proc. Indian Acad. Sci. Math. Sci.

124(4), 527-531 (2014)25. Chu, Y-M, Wang, M-K, Qiu, S-L: Optimal combination bounds of root-square and arithmetic means for Toader mean.

Proc. Indian Acad. Sci. Math. Sci. 122(1), 41-51 (2012)26. Song, Y-Q, Jiang, W-D, Chu, Y-M, Yan, D-D: Optimal bounds for Toader mean in terms of arithmetic and

contraharmonic means. J. Math. Inequal. 7(4), 751-757 (2013)27. Li, W-H, Zheng, M-M: Some inequalities for bounding Toader mean. J. Funct. Spaces Appl. 2013, Article ID 394194

(2013)28. Hua, Y, Qi, F: The best bounds for Toader mean in terms of the centroidal and arithmetic means. Filomat 28(4),

775-780 (2014)29. Sun, H, Chu, Y-M: Bounds for Toader mean by quadratic and harmonic means. Acta Math. Sci. 35A(1), 36-42 (2015)

(in Chinese)30. Chu, Y-M, Wang, M-K, Jiang, Y-P, Qiu, S-L: Concavity of the complete elliptic integrals of the second kind with respect

to Hölder means. J. Math. Anal. Appl. 395(2), 637-642 (2012)

Optimal evaluation of a Toader-type mean by power meanAbstractMSCKeywords

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