PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 99(113) (2016), 237–242 DOI: 10.2298/PIM141026009J

A DOUBLE INEQUALITY FOR THE COMBINATION OF TOADER MEAN AND THE ARITHMETIC MEAN IN TERMS OF THE CONTRAHARMONIC MEAN

Wei-Dong Jiang and Feng Qi

Abstract. We find the greatest value λ and the least value μ such that the double inequality

C(λa + (1 − λ)b, λb + (1 − λ)a)  < αA(a, b) + (1 − α)T (a, b)
< C(μa + (1 − μ)b, μb + (1 − μ)a)

holds for all α ∈ (0, 1) and a, b > 0 with a ≠ b, where C(a, b), A(a, b), and T (a, b) denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers a and b.

1. Introduction

For p ∈ ℝ and a, b > 0, the contraharmonic mean C(a, b), the p-th power mean M_{p}(a, b), and Toader mean T (a, b) are respectively defined by

C(a, b) = \frac{a^{2} + b^{2}}{a + b} , M_{p}(a, b) = \{ \begin{matrix}((a^{p} + b^{p})/2)^{1/p}, & p ≠ 0, \\ \sqrt{ab}, & p = 0, \end{matrix}

T (a, b) = \frac{2}{π} ∫_{0}^{π/2} \sqrt{a^{2} cos^{2} θ + b^{2} sin^{2} θ} dθ = \{ \begin{matrix}\frac{2a}{π} E(\sqrt{1 − (b/a)^{2}} ), & a > b, \\ \frac{2b}{π}E(\sqrt{1 − (a/b)^{2}} ), & a < b, \\ a, & a = b, \end{matrix}

where E = E(r) = ∫_{0}^{π/2} \sqrt{1 − r^{2} sin^{2} θ} dθ for r ∈ [0, 1] is the complete elliptic integral of the second kind. For more information on complete elliptic integrals, see [11, 13–15] and plenty of references therein.

Recently, the Toader mean has attracted attention of several researchers. In particular, many remarkable inequalities for T (a, b) can be found in the literature [6, 7, 9, 10, 18]. It was conjectured in [17] that

(1.1)  M_{3/2} (a, b) < T (a, b)

2010 Mathematics Subject Classification: Primary 26E60; Secondary 33E05.

Key words and phrases: bound; contraharmonic mean; arithmetic mean; Toader mean; complete elliptic integrals.

The second author was partially supported by the NNSF of China under Grant No. 11361038.

Communicated by Gradimir V. Milovanović.

for all a, b > 0 with a ≠ b. This conjecture was proved in [3,16] respectively. In [1], the best possible upper bound for the Toader mean was presented by

T (a, b) < M_{ln 2/ ln(π/2)} (a, b)

for all a, b > 0 with a ≠ b.

It is not difficult to verify that

(1.2)  C(a, b) > M_{2}(a, b) = \sqrt{(a^{2} + b^{2})/2}

for all a, b > 0 with a ≠ b. From (1.1) to (1.2) one has A(a, b) < T (a, b) < C(a, b) for all a, b > 0 with a ≠ b.

For positive numbers a, b > 0 with a ≠ b, let

J(x) = C(xa + (1 − x)b, xb + (1 − x)a)

on [ \frac{1}{2}, 1]. It is not difficult to verify that J(x) is continuous and strictly increasing on [ \frac{1}{2}, 1]. Note that J( \frac{1}{2}) = A(a, b) < T (a, b) and J(1) = C(a, b) > T (a, b).

In [8] it was proved that the double inequality

C(αa + (1 − α)b, αb + (1 − α)a) < T (a, b) < C(βa + (1 − β)b, βb + (1 − β)a)

holds for all a, b > 0 with a ≠ b if and only if α ⩽ \frac{3}{4} and β ⩾ \frac{1}{2} + \frac{\sqrt{4π−π^{2}}}{2π} .

The main purpose of the paper is to find the greatest value λ and the least value μ such that the double inequality

C(λa + (1 − λ)b, λb + (1 − λ)a) < αA(a, b) + (1 − α)T (a, b)
< C(μa + (1 − μ)b, μb + (1 − μ)a)

holds for all α ∈ (0, 1) and a, b > 0 with a ≠ b. As applications, we also present new bounds for the complete elliptic integral of the second kind.

2. Preliminaries and lemmas

In order to establish our main result, we need several formulas and lemmas below.

For 0 < r < 1 and r′ = \sqrt{1 − r^{2}} , Legendre’s complete elliptic integrals of the first and second kinds are defined in [4, 5] respectively by

\{ \begin{matrix}K = K(r) = ∫_{0}^{π/2}\frac{dθ}{(1 − r^{2} sin^{2} θ)^{1/2}} , \\ K′ = K′(r) = K(r′), \\ K(0) = \frac{π}{2}, K(1) = ∞ \end{matrix} and \{ \begin{matrix}E = E(r) = ∫_{0}^{π/2}(1 − r^{2} sin^{2} θ)^{1/2} dθ, \\ E′ = E′(r) = E(r′), \\ E(0) = \frac{π}{2}, E(1) = 1. \end{matrix}

For 0 < r < 1, the formulas

\frac{dK}{dr} = \frac{E − (r′)^{2}K}{r(r′)^{2}} , \frac{dE}{dr} = \frac{E − K}{r} , \frac{d(E − (r′)^{2}K)}{dr} = rK,

\frac{d(K − E)}{dr} = \frac{rE}{(r′)^{2}} , E( \frac{2\sqrt{r}}{1 + r} ) = \frac{2E − (r′)^{2}K}{1 + r}

were presented in [2, Appendix E, pp. 474–475].

AN INEQUALITY FOR COMBINATIONS OF TOADER AND ARITHMETIC MEAN 239

Lemma 2.1. [2, Theorem 3.21(1) and 3.43 Exercise 13(a)] The function \frac{E−(r′)^{2} K}{r^{2}} is strictly increasing from (0, 1) onto ( \frac{π}{4} , 1) and the function 2E − (r′)^{2}K is increasing from (0, 1) onto ( \frac{π}{2} , 2).

Lemma 2.2. Let u, α ∈ (0, 1) and

f_{u,α} (r) = ur^{2} − (1 − α)\{ \frac{2}{π}[2E(r) − (1 − r^{2})K(r)] − 1\}.

Then f_{u,α} > 0 for all r ∈ (0, 1) if and only if u ⩾ (1 − α)( \frac{4}{π} − 1) and f_{u,α} < 0 for all r ∈ (0, 1) if and only if u ⩽ \frac{1−α}{4} .

Proof. It is clear that

(2.1)  f_{u,α} (0^{+} ) = 0,
(2.2)  f_{u,α} (1^{−} ) = u − (1 − α)(4/π − 1),
(2.3)  f′_{u,α} (r) = 2r[u − (1 − α)g(r)],

where g(r) = \frac{1}{π} \frac{E−(r′)^{2} K}{r^{2}} .

When u ⩾ \frac{1−α}{π} , from (2.3) and Lemma 2.1 and by the monotonicity of g(r), it follows that f_{u,α} (r) is strictly increasing on (0, 1). Therefore, f_{u,α} (r) > 0 for all r ∈ (0, 1).

When u ⩽ \frac{1−α}{4} , from (2.3) and Lemma 2.1 and by the monotonicity of g(r), we obtain that f_{u,α} (r) is strictly decreasing on (0, 1). Therefore, f_{u,α} (r) < 0 for all r ∈ (0, 1).

When \frac{1−α}{4} < u ⩽ (1−α)( \frac{4}{π} −1), from (2.2) and (2.3) and by the monotonicity of g(r), we see that there exists λ ∈ (0, 1) such that f_{u,α} (r) is strictly increasing in (0, λ] and strictly decreasing in [λ, 1) and

(2.4)  f_{u,α} (1^{−} ) ⩽ 0.

Therefore, making use of equation (2.1), inequality (2.4), and the piecewise monotonicity of f_{u,α} (r) lead to the conclusion that there exists 0 < λ < η < 1 such that f_{u,α} (r) > 0 for r ∈ (0, η) and f_{u,α} (r) < 0 for r ∈ (η, 1).

When (1 − α)( \frac{4}{π} − 1) ⩽ u < \frac{1−α}{π} , by (2.2), it follows that

(2.5)  f_{u,α} (1^{−} ) ⩾ 0.

From (2.2) and (2.3) and by the monotonicity of g(r), we see that there exists λ ∈ (0, 1) such that f_{u,α} (r) is strictly increasing in (0, λ] and strictly decreasing in [λ, 1). Therefore, f_{u,α} (r) > 0 for r ∈ (0, 1) follows from (2.1) and (2.5) together with the piecewise monotonicity of f_{u,α} (r). □

3. Main results

Now we are in a position to state and prove our main results.

Theorem 3.1. If α ∈ (0, 1) and λ, μ ∈ ( \frac{1}{2}, 1), then the double inequality

C(λa + (1 − λ)b, λb + (1 − λ)a) < αA(a, b) + (1 − α)T (a, b)
< C(μa + (1 − μ)b, μb + (1 − μ)a)

holds for all a, b > 0 with a ≠ b if and only if

λ ⩽ \frac{1}{2} + \frac{\sqrt{1 − α}}{4} and μ ⩾ \frac{1}{2}[1 + \sqrt{(1 − α)(4/π − 1)} ].

Proof. Since A(a, b), T (a, b), and C(a, b) are symmetric and homogeneous of degree one, without loss of generality, assume that a > b. Let p ∈ ( \frac{1}{2}, 1), t = \frac{b}{a} ∈ (0, 1), and r = \frac{1−t}{1+t} . Then

C(pa+ (1 − p)b, pb + (1 − p)a) − αA(a, b) − (1 − α)T (a, b)

= a\frac{[p + (1 − p)b/a]^{2} + (pb/a + 1 − p)^{2}}{1 + b/a} − αa\frac{1 + b/a}{2}

− (1 − α)\frac{2a}{π}E( \sqrt{1 − (b/a)^{2}} )

= a\{ \frac{[p + (1 − p)t]^{2} + (pt + 1 − p)^{2}}{1 + t} − α\frac{1 + t}{2} − (1 − α)\frac{2}{π}E( \sqrt{1 − t^{2}} ) \}

= a\{ \frac{(1 − 2p)^{2}r^{2} + 1}{1 + r} − α\frac{1}{1 + r} − (1 − α)\frac{2}{π} \frac{2E − (r′)^{2}K}{1 + r} \}

= \frac{a}{1 + r} [(1 − 2p)^{2}r^{2} + 1 − α − (1 − α)\frac{2}{π}(2E − (r′)^{2}K)].

From this and Lemma 2.2, Theorem 3.1 follows. □

Corollary 3.1. For r ∈ (0, 1) and r′ = \sqrt{1 − r^{2}} , we have
(3.1)  \frac{π}{2}[ \frac{17 + 30r′+ 17(r′)^{2}}{8(1 + r′)} − \frac{3(1 + r′)}{2} ] < E(r) < π[ \frac{r′+ 2(1 − r′)^{2}/π}{1 + r′} ].

Proof. This follows from letting α = \frac{3}{4}, λ = \frac{5}{8}, and μ = \frac{1}{2}( 1 + ^{\frac{\sqrt{4/π−1}}{2}} ) in Theorem 3.1. □

4. Remarks

Remark 4.1. Recently, the complete elliptic integrals have attracted attention of numerous mathematicians. In [9], it was established that

(4.1) \frac{π}{2}[ \frac{1}{2}\sqrt{\frac{1 + (r′)^{2}}{2}} + \frac{1 + r′}{4} ] < E(r)

< \frac{π}{2}[ \frac{4 − π}{(\sqrt{2} − 1)π} \sqrt{\frac{1 + (r′)^{2}}{2}} + \frac{(\sqrt{2}π − 4)(1 + r′)}{2(\sqrt{2} − 1)π} ],

for all r ∈ (0, 1). In [11] it was proved that

(4.2)  \frac{π}{2} − \frac{1}{2} log \frac{(1 + r)^{1−r}}{(1 − r)^{1+r}} < E(r) < \frac{π − 1}{2} + \frac{1 − r^{2}}{4r} log \frac{1 + r}{1 − r} ,

for all r ∈ (0, 1). In [19] it was presented that

(4.3)  \frac{π}{2} \frac{\sqrt{6 + 2\sqrt{1 − r^{2}} − 3r^{2}}}{2\sqrt{2}} ⩽ E(r) ⩽ \frac{π}{2} \frac{\sqrt{10 − 2\sqrt{1 − r^{2}} − 5r^{2}}}{2\sqrt{2}}

for all r ∈ (0, 1). In [9] it was pointed out that the bounds in (4.1) for E(r) are better than the bounds in (4.2) for some r ∈ (0, 1).

Remark 4.2. The lower bound in (3.1) for E(r) is better than the lower bound in (4.1). Indeed,

\frac{17 + 30x + 17x^{2}}{8(1 + x)} − \frac{3(1 + x)}{2} − [ \frac{1}{2}\sqrt{\frac{1 + x^{2}}{2}} + \frac{1 + x}{4} ]

= \frac{3x^{2} + 2x + 3 − 2\sqrt{2(1 + x^{2})} (1 + x)}{8(1 + x)}

and

(3x^{2} + 2x + 3)^{2} − [2\sqrt{2(1 + x^{2})} (1 + x)]^{2} = (1 − x)^{4} > 0

for all x ∈ (0, 1).

Remark 4.3. The following equivalence relations show that the lower bound in (3.1) for E(r) is better than the lower bound in (4.3):

\frac{17 + 30x + 17x^{2}}{8(1 + x)} − \frac{3(1 + x)}{2} > \frac{\sqrt{6 + 2x − 3(1 − x^{2})}}{2\sqrt{2}}

⇔ (5x^{2} + 6x + 5)^{2} > 8(x + 1)^{2}(3x^{2} + 2x + 3)  ⇔ (x − 1)^{4} > 0,

where x ∈ (0, 1).

Remark 4.4. This paper is a slightly revised version of the preprint [12].

References

1. H. Alzer, S.-L. Qiu, Monotonicity theorems and inequalities for complete elliptic integrals, J. Comput. Appl. Math. 172(2) (2004), 289–312; DOI: 10.1016/j.cam.2004.02.009.

2. G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, Wiley, New York, 1997.

3. R. W. Barnard, K. Pearce, K. C. Richards, An inequality involving the generalized hypergeometric function and the arc length of an ellipse, SIAM J. Math. Anal. 31(3) (2000), 693–699; DOI: 10.1137/S0036141098341575.

4. F. Bowman, Introduction to Elliptic Functions with Applications, 2nd ed., Dover, New York, 1961.

5. P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., Springer-Verlag, New York, 1971.

6. Y.-M. Chu, M.-K. Wang, Inequalities between arithmetic-geometric, Gini, and Toader means, Abstr. Appl. Anal. 2012, Article ID 830585, 11 pages; DOI: 10.1155/2012/830585.

7. , Optimal Lehmer mean bounds for the Toader mean, Results Math. 61(3–4) (2012), 223–229; DOI: 10.1007/s00025-010-0090-9.

8. Y.-M. Chu, M.-K. Wang, X.-Y. Ma, Sharp bounds for Toader mean in terms of contraharmonic mean with applications, J. Math. Inequal. 7(2) (2013), 161–166; DOI: 10.7153/jmi-07-15.

9. Y.-M. Chu, M.-K. Wang, S.-L. Qiu, Optimal combinations bounds of root-square and arithmetic means for Toader mean, Proc. Indian Acad. Sci. Math. Sci. 122(1) (2012), 41–51; DOI: 10.1007/s12044-012-0062-y.

10. Y.-M. Chu, M.-K. Wang, S.-L. Qiu, Y.-F. Qiu, Sharp generalized Seiffert mean bounds for Toader mean, Abstr. Appl. Anal. 2011 (2011), Article ID 605259, 8 pages; DOI: 10.1155/2011/605259.

11. B.-N. Guo, F. Qi, Some bounds for the complete elliptic integrals of the first and second kind, Math. Inequal. Appl. 14(2) (2011), 323–334; DOI: 10.7153/mia-14-26.

12. W.-D. Jiang, F. Qi, Bounds for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean, arXiv:1402.4561.

13. F. Qi, L.-H. Cui, S.-L. Xu, Some inequalities constructed by Tchebysheff’s integral inequality, Math. Inequal. Appl. 2(4) (1999), 517–528; DOI: 10.7153/mia-02-42.

14. F. Qi, Z. Huang, Inequalities for complete elliptic integrals, Tamkang J. Math. 29(3) (1998), 165–169; DOI: 10.5556/j.tkjm.29.1998.165-169.

15. F. Qi, D.-W. Niu, B.-N. Guo, Refinements, generalizations, and applications of Jordan’s inequality and related problems, J. Inequal. Appl. 2009 (2009), Article ID 271923, 52 pages; DOI: 10.1155/2009/271923.

16. S.-L. Qiu, J.-M. Shen, On two problems concerning means, J. Hangzhou Insit. Electron. Engineering 17(3) (1997), 1–7 (in Chinese)

17. M. Vuorinen, Hypergeometric functions in geometric function theory, in: Special Functions and Differential Equations, Proc. Workshop at Inst. Math. Sci., Madras, India, January 13–24, 1997, Allied Publ., New Delhi, 1998, 119–126.

18. M.-K. Wang, Y.-M. Chu, S.-L. Qiu, Y.-P. Jiang, Bounds for the perimeter of an ellipse, J. Approx. Theory 164(7) (2012), 928–937; DOI: 10.1016/j.jat.2012.03.011.

19. L. Yin, F. Qi, Some inequalities for complete elliptic integrals, Appl. Math. E-Notes 14 (2014), 192–199; arXiv: 1301.4385.

Department of Information Engineering (Received 05 06 2014) Weihai Vocational College Weihai City, Shandong Province China

jackjwd@163.com
jackjwd@hotmail.com

Department of Mathematics College of Science Tianjin Polytechnic University Tianjin City China College of Mathematics Inner Mongolia University for Nationalities Tongliao City Inner Mongolia Autonomous Region China Institute of Mathematics Henan Polytechnic University Jiaozuo City Henan Province China

qifeng618@gmail.com
qifeng618@hotmail.com
qifeng618@qq.com