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) ,
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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) < A(a, b)=M(a, b)=L(a, b) < Q(a, b)=M(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) 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) Gλ (a, b) αQ(a, b)+(–α)A(a, b) Qα (a, b)A(–α)(a, b) αC(a, b)+(–α)A(a, b) 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) αC(a, b)+(–α)A(a, b) αQ(a, b)+(–α)H(a, b) 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 , 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 . It follows from (.)and(.) together with the piecewise monotonicity of f (t)that f (t)> (.) √ for all t ∈ (, /). Zhao et al. Journal of Inequalities and Applications (2017) 2017:26 Page 6 of 10 √ Therefore, f (t) > for all t ∈ (, /) follows easily from (.), (.), (.), and (.). Subcase . λ∗ < p ≤ λ∗∗.Then(.)and(.)leadto √ f ≥ , (.) √ f <. (.) It follows from (.√)and(.) together with the piecewise monotonicity of f (t)that ∈ there exists t √(, /) such that f(t)isstrictlyincreasingon(,t]andstrictlyde- creasing on [t, /). Equation (.) and inequality (.) together with the piecewise monotonicity of f(t) lead to the conclusion that f(t)> (.) √ ∈ for all t (, /). √ Therefore, f (t) > for all t ∈ (, /) follows easily from (.)and(.)together with (.). Subcase . λ∗∗ < p ≤ λ.Then(.), (.), and (.)leadto √ f ≥ , (.) √ f <, (.) √ √ √ √ π ∗∗ f <E – K – λ √ √ √ π ∗ <E – K – λ =. (.) It follows from (.√)and(.) together with the piecewise monotonicity of f (t)that ∈ there exists t √(, /) such that f(t) is strictly increasing on (, t]andstrictlyde- creasing on [t, /). From (.), (.), and (.√) together with the piecewise monotonicity of f(t) we clearly ∈ see that there exists√t (, /) such that f (t) is strictly increasing on (, t]andstrictly decreasing on [t, /). √ Therefore, f (t) > for all t ∈ (, /) follows easily from (.)and(.)togetherwith the piecewise monotonicity of f (t). Case λ < p < /. Then (.), (.), (.), (.), and (.)leadto √ f < , (.) √ √ √ √ πλ∗∗ f < K – E – =, (.) √ √ √ √ ∗ πλ f <E – K – =, (.) Zhao et al. Journal of Inequalities and Applications (2017) 2017:26 Page 7 of 10 + f >, (.) √ √ √ √ ∗ f < E –K – πλ √ √ √ =– K – E <. (.) It follows√ from (.)and(.) together with the monotonicity of f (t) that there ex- ∈ ists√t (, /) such that f (t)isstrictlyincreasingon(,t] and strictly decreasing on [t, /). Equation (.) and inequality (.) together√ with the piecewise monotonicity of f (t) ∈ lead to the conclusion that there exists√t (, /) such that f(t) is strictly increasing on (, t] and strictly decreasing on [t, /). From (.),√ (.), (.), and the piecewise monotonicity of f(t) we clearly see that there ∈ exists√ t (, /) such that f (t) is strictly increasing on (, t] and strictly decreasing on [t, /). √ ∈ ∈ Therefore,√ there exists t (, /) such that f (t)>fort (, t)andf (t)<for t ∈ (t, /) follows from (.)and(.) together with the piecewise monotonicity of f (t). √ √ √ Lemma . Let r ∈ R, a, b >with r log –c c λ(r)= r (r =), λ = , (.) –c log c and /r U(r; a, b)= λ(r)ar + –λ(r) br (r =), U(; a, b)=aλ b–λ ,(.) respectively. Then the function r → U(r; a, b) is strictly decreasing on (–∞, ∞). √ Proof Let x = b/a ∈ (, ), r =,and V(r, x)= –λ(r) log x – log λ(r) .(.) Then from (.)-(.)onehas log U(r; a, b)=log a + log λ(r)+ –λ(r) xr , r ∂ log U(r; a, b) λ(r)( – xr)+(–λ(r))xr log x log(λ(r)+(–λ(r))xr) = – , ∂r r(λ(r)+(–λ(r))xr) r ∂ log U(r; a, b) =, (.) ∂r x= r r r r (c –)c log c –(c –)c log c λ (r)= , (cr –) r r r r | √ –c –c c λ(r)+ –λ(r) x x= = r + – r r = r , –c –c c c Zhao et al. Journal of Inequalities and Applications (2017) 2017:26 Page 8 of 10 λ(r)( – xr)+(–λ(r))xr log x c log r √ = , r(λ(r)+(–λ(r))x ) r c x= ∂ log U(r; a, b) √ =, (.) ∂r x= ∂ log U(r; a, b) λ(r)xr– = V(r, x), (.) ∂x ∂r (λ(r)+(–λ(r))xr) log log V(r,)= c – c <, (.) r r ( c ) – ( c ) – √ r log c log c V(r, ) = c r – r >, (.) c – c – r where inequalities (.)and(.)holdduetoc > c and the function t → log t/(t –) ∞ is strictly decreasing on (, ). √ ∈ → Note that λ(r) (, ) and the function x V(r, x)isstrictlyincreasingon(,√ ). Then ∈ (.)-(.) lead to the conclusion that there exists x (, ) such that the function√ x → ∂ log U(r; a, b)/∂r is strictly decreasing on (, x) and strictly increasing on (x, ). It follows from (.)and(.) together with√ the piecewise monotonicity of the func- tion x → ∂ log U(r; a, b)/∂r on the interval (, ) that ∂ log U(r; a, b) < (.) ∂r √ for all a, b >with 3 Main result √ √ Theorem . Let c =E( /)/π =...., c = / and λ(r) be defined by (.). Then 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 ≤ and a, b >with a = bifandonlyifα(r) ≥ / and β(r) ≤ λ(r), where r =is the limit value of r → . Proof We first prove that Theorem . holds for r =. Since A(a, b) √ A(a, b)=Q(a, b) –t, TD A(a, b), Q(a, b) = Q(a, b)E(t), π (.) pA(a, b)+(–p)Q(a, b)–TD A(a, b), Q(a, b) = Q(a, b)f (t), π where f (t) is defined as in Lemma .. Zhao et al. Journal of Inequalities and Applications (2017) 2017:26 Page 9 of 10 Therefore, Theorem . for r = follows easily from Lemma . and (.). Next, let r <anda, b >witha = b, then it follows from Theorem . for r =that A(a, b)+Q(a, b) < TD A(a, b), Q(a, b) < λ()A(a, b)+ –λ() Q(a, b). (.) Note that Q(a, b) √ < < , (.) A(a, b) TD[A(a, b), Q(a, b)] +/r E(t) = ,(.) Ar(a,b)+Qr(a,b) /r π [ + ( – t)r/]/r [ ] TD[A(a, b), Q(a, b)] E(t) = ,(.) [λ(r)Ar(a, b)+(–λ(r))Qr(a, b)]/r π [λ(r)( – t)r/ +–λ(r)]/r +/r E(t) E(t) lim lim r/ /r = √ r/ /r =. (.) t→+ π [ + ( – t ) ] t→ / π [λ(r)( – t ) +–λ(r)] Therefore, Theorem . for r < follows from (.)-(.) and Lemma . together with the monotonicity of the function r → [(ar + br)/]/r. Let r = . Then Theorems . leads to Corollary . immediately. √ √ Corollary . Let λ =(+ )[ – E( /)/π]. Then the double inequality π √ π π √ π –t + < E(t)< λ –t + ( – λ) √ holds for all t ∈ (, /). 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 1School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China. 2Icahn School of Medicine at Mount Sinai, Friedman Brain Institute, New York, 10033, United States. Acknowledgements The research was supported by the Natural Science Foundation of China under Grants 61673169, 11371125, 11401191, and 61374086. Received: 28 November 2016 Accepted: 17 January 2017 References 1. Toader, G: Some mean values related to the arithmetic-geometric mean. J. Math. Anal. Appl. 218(2), 358-368 (1998) 2. Neuman, E: Bounds for symmetric elliptic integrals. J. Approx. Theory 122(2), 249-259 (2003) 3. Kazi, H, Neuman, E: Inequalities and bounds for elliptic integrals. J. Approx. Theory 146(2), 212-226 (2007) 4. Kazi, H, Neuman, E: Inequalities and bounds for elliptic integrals II. In: Special Functions and Orthogonal Polynomials. Contemp. Math., vol. 471, pp. 127-138. Amer. Math. Soc., Providence, RI (2008) 5. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997) 6. Borwein, JM, Borwein, PB: Pi and the AGM. Wiley, New York (1987) 7. Vuorinen, M: Hypergeometric functions in geometric function theory. In: Special Functions and Differential Equations, Madras, 1977, pp. 119-126. Allied Publ., New Delhi (1998) 8. Qiu, S-L, Shen, J-M: On two problems concerning means. J. Hangzhou Inst. Electron. Eng. 17(3), 1-7 (1997) (in Chinese) Zhao et al. Journal of Inequalities and Applications (2017) 2017:26 Page 10 of 10 9. 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) 10. Alzer, H, Qiu, S-L: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172(2), 289-312 (2004) 11. 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) 12. Chu, Y-M, Wang, M-K: Inequalities between arithmetic-geometric, Gini, and Toader means. Abstr. Appl. Anal. 2012, Article ID 830585 (2012) 13. Chu, Y-M, Wang, M-K: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3-4), 223-229 (2012) 14. 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) 15. 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) 16. 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) 17. Song,Y-Q,Jiang,W-D,Chu,Y-M,Yan,D-D:OptimalboundsforToadermeanintermsofarithmeticand contraharmonic means. J. Math. Inequal. 7(4), 751-757 (2013) 18. Li, W-H, Zheng, M-M: Some inequalities for bounding Toader mean. J. Funct. Spaces Appl. 2013, Article ID 394194 (2013) 19. Hua, Y, Qi, F: The best bounds for Toader mean in terms of the centroidal and arithmetic means. Filomat 28(4), 775-780 (2014) 20. Sun, H, Chu, Y-M: Bounds for Toader mean by quadratic and harmonic means. Acta Math. Sci. 35A(1), 36-42 (2015) (in Chinese)