Zhao et al. Journal of Inequalities and Applications (2017)2017:26 DOI 10.1186/s13660-017-1300-8 R E S E A R C H Open Access Optimal inequalities for bounding Toader mean by arithmetic and quadratic means Tie-Hong Zhao1, Yu-Ming Chu1* and Wen Zhang2 *Correspondence: [email protected] Abstract 1School of Mathematics and Computation Sciences, Hunan City In this paper, we present the best possible parameters α(r)andβ(r) such that the University, Yiyang, 413000, China double inequality Full list of author information is available at the end of the article α(r)Ar(a, b)+(1–α(r))Qr(a, b) 1/r < TD A(a, b), Q(a, b) < β(r)Ar(a, b)+(1–β(r))Qr(a, b) 1/r ≤ holds for all r 1anda, b >0with a = b, and we provide new√ bounds for the complete elliptic integral E(r)= π/2(1 – r2 sin2 )1/2 d (r ∈ (0, 2/2)) of the second √ 0 θ θ 2 π/2 2 2 2 2 kind, where√ TD(a, b)= π 0 a cos θ + b sin θ dθ, A(a, b)=(a + b)/2 and Q(a, b)= (a2 + b2)/2 are the Toader, arithmetic, and quadratic means of a and b, respectively. MSC: 26E60 Keywords: arithmetic mean; Toader mean; quadratic mean; complete elliptic integral 1 Introduction For p ∈ [, ], q ∈ R and a, b >witha = b,thepth generalized Seiffert mean Sp(a, b), qth Gini mean Gq(a, b), qth power mean Mq(a, b), qth Lehmer mean Lq(a, b), harmonic mean H(a, b), geometric mean G(a, b), arithmetic mean A(a, b), quadratic mean Q(a, b), Toader mean TD(a, b)[], centroidal mean C(a, b), contraharmonic mean C(a, b)are,respectively, defined by ⎧ ⎨⎪ p(a – b) ,<p ≤ , S (a, b)= arctan[p(a – b)/(a + b)] p ⎩⎪ (a + b)/, p =, ⎧ ⎨ [(aq– + bq–)/(a + b)]/(q–), q =, Gq(a, b)=⎩ (aabb)/(a+b), q =, ⎧ ⎨ [(aq + bq)/]/q, q =, √ Mq(a, b)=⎩ ab, q =, © The Author(s) 2017. 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, pro- vided 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. Zhao et al. Journal of Inequalities and Applications (2017)2017:26 Page 2 of 10 aq+ + bq+ ab √ L (a, b)= , H(a, b)= , G(a, b)= ab, (.) q aq + bq a + b a + b a + b A(a, b)= , Q(a, b)= , π/ TD(a, b)= a cos θ + b sin θ dθ, π (a + ab + b) a + b C(a, b)= , C(a, b)= . (a + b) a + b It is well known that Sp(a, b), Gq(a, b), Mq(a, b), and Lq(a, b) are continuous and strictly increasing with respect to p ∈ [, ] and q ∈ R for fixed a, b >witha = b, and the inequal- ities H(a, b)=M–(a, b)=L–(a, b)<G(a, b)=M(a, b)=L–/(a, b) < A(a, b)=M(a, b)=L(a, b)<TD(a, b)<C(a, b) < Q(a, b)=M(a, b)<C(a, b)=L(a, b) hold for all a, b >witha = b. The Toader mean TD(a, b) has been well known in the mathematical literature for many years, it satisfies TD(a, b)=RE a , b , where ∞ [a(t + b)+b(t + a)]t R a b dt E( , )= / / π (t + a) (t + b) stands for the symmetric complete elliptic integral of the second kind (see [–]), therefore it cannot be expressed in terms of the elementary transcendental functions. ∈ K π/ –/ E π/ / Let r (, ), (r)= (–r sin θ) dθ and (r)= (–r sin θ) dθ be, respec- tively, the complete elliptic integrals of the first and second kind. Then K(+)=E(+)= π/, K(r), and E(r) satisfy the derivatives formulas (see [], Appendix E, p.-) dK(r) E(r)–(–r)K(r) dE(r) E(r)–K(r) = , = , dr r( – r) dr r d[K(r)–E(r)] rE(r) = , dr –r √ √ the values K( /) and E( /) can be expressed as (see [], Theorem .) √ √ (/) (/) + (/) K = √ =...., E = √ =...., π π Zhao et al. Journal of Inequalities and Applications (2017)2017:26 Page 3 of 10 ∞ x– –t where (x)= t e dt(Re x > ) is the Euler gamma function, and the Toader mean TD(a, b)canberewrittenas ⎧ ⎨ aE( –(b/a))/π, a ≥ b, TD(a, b)=⎩ (.) bE( –(a/b))/π, a < b. Recently, the Toader mean TD(a, b) has been the subject of intensive research. Vuorinen [] conjectured that the inequality TD(a, b)>M/(a, b) holds for all a, b >witha = b. This conjecture was proved by Qiu and Shen [], and Barnard, Pearce and Richards [], respectively. Alzer and Qiu []presentedabestpossibleupperpowermeanboundfortheToader mean as follows: TD(a, b)<Mlog /(log π–log )(a, b) for all a, b >witha = b. Neuman [], and Kazi and Neuman [] proved that the inequalities √ √ (a + b) ab – ab (a + b) ab +(a – b) < TD(a, b)< , AGM(a, b) AGM(a, b) √ √ √ √ TD(a, b)< ( + )a +(– )b + ( + )b +(– )a hold for all a, b >witha = b,whereAGM(a, b) is the arithmetic-geometric mean of a and b. In [–], the authors presented the best possible parameters λ, μ ∈ [, ] and ∈ R λ, μ, λ, μ such that the double inequalities Sλ (a, b)<TD(a, b)<Sμ (a, b), Gλ (a, b)<TD(a, b)<Gμ (a, b)andLλ (a, b)<TD(a, b)<Lμ (a, b) hold for all a, b >with a = b. Let λ, μ, α, β ∈ (/, ). Then Chu, Wang and Ma [], and Hua and Qi []provedthat the double inequalities C λa +(–λ)b, λb +(–λ)a < TD(a, b)<C μa +(–μ)b, μb +(–μ)a , C αa +(–α)b, αb +(–α)a < TD(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 – /. In [–], the authors proved that the double inequalities αQ(a, b)+(–α)A(a, b)<TD(a, b)<βQ(a, b)+(–β)A(a, b), Qα (a, b)A(–α)(a, b)<TD(a, b)<Qβ (a, b)A(–β)(a, b), αC(a, b)+(–α)A(a, b)<TD(a, b)<βC(a, b)+(–β)A(a, b), Zhao et al. Journal of Inequalities and Applications (2017)2017:26 Page 4 of 10 α –α β –β + < < + , A(a, b) C(a, b) TD(a, b) A(a, b) C(a, b) αC(a, b)+(–α)H(a, b)<TD(a, b)<βC(a, b)+(–β)H(a, b), α C(a, b)–H(a, b) + A(a, b)<TD(a, b)<β C(a, b)–H(a, b) + A(a, b), αC(a, b)+(–α)A(a, b)<TD(a, b)<βC(a, b)+(–β)A(a, b), α –α β –β + < < + , A(a, b) C(a, b) TD(a, b) A(a, b) C(a, b) αQ(a, b)+(–α)H(a, b)<TD(a, b)<βQ(a, b)+(–β)H(a, b), α –α β –β + < < + H(a, b) Q(a, b) TD(a, b) H(a, b) Q(a, b) √ hold for all a, b >witha = b if and only if α ≤ /, β ≥ ( – π)/[( –)π], α ≤ /, β ≥ –log π/ 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 α(r)andβ(r) such that the double inequality /r α(r)Ar(a, b)+ –α(r) Qr(a, b) < TD A(a, b), Q(a, b) /r < β(r)Ar(a, b)+ –β(r) Qr(a, b) holds for all r ≤ anda, b >witha = b. 2 Lemmas Inordertoproveourmainresultweneedtwolemmas,whichwepresentinthissection. √ √ √ Lemma . Let p ∈ (, ), t ∈ (, /), λ =(+ )[ – E( /)/π]=....and πp√ π f (t)= –t + ( – p)–E(t). (.) √ √ Then f (t)<for all t ∈ (, /) if and only if p ≥ / and f (t)>for all t ∈ (, /) if and only if p ≤ λ. Proof It follows from (.)that f + = , (.) √ √ π f = – (λ – p), (.) f (t) f (t)= √ ,(.) t –t where √ πp f (t)= –t K(t)–E(t) – t, f + =, (.) Zhao et al. Journal of Inequalities and Applications (2017)2017:26 Page 5 of 10 √ √ √ √ πp f = K – E – ,(.) E K t[ √(t)– (t)] f (t)= – πpt,(.) –t + f =, (.) √ √ √ √ πp f =E – K – ,(.) ( – t )E(t)–(–t )K(t) f (t)= – πp,(.) ( – t)/ f + = π – p ,(.) √ √ √ √ f = E –K – πp,(.) ( + t )[K(t)–E(t)] + t K(t) f (t)=– < (.) t( – t)/ √ ∈ for all t (, /). √ It follows from (.)thatf (t) is strictly decreasing on (, /). We divide the proof into three cases. Case p ≥ /. Then (.)leadsto + ≤ f . (.) From√ (.) and the monotonicity of f (t) we√ clearly see that f (t) is strictly decreasing on (, /). Therefore, f (t) < for all t ∈ (, /) follows easily from (.), (.), (.), (.), and the monotonicity of f (t). √ √ Case <p ≤ λ.Thenfrom(.)and(.)togetherwithE( /) – K( /) = –....weclearlyseethat √ f + >, f < . (.) √ ∈ It follows from (.) and the monotonicity of f (t) that there exists√t (, /) such that f (t) is√ strictly√ increasing√ on (, t] and strictly decreasing√ √ on [t,√ /). ∗ E K ∗∗ K E Let λ = π [ ( )– ( )]=.... and λ = π [ ( )– ( )]=..... We divide the proof into three subcases. Subcase . <p ≤ λ∗.Then(.)leadsto √ f ≥ . (.) It follows from (.)and(.) together with the piecewise monotonicity of f (t)that f (t)> (.) √ for all t ∈ (, /). Zhao et al.