JJ II

(1)

volume 7, issue 2, article 63, 2006.

Received 15 September, 2005;

accepted 24 February, 2006.

Communicated by:S.S. Dragomir

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

EXTENSIONS AND SHARPENINGS OF JORDAN’S AND KOBER’S INEQUALITIES

XIAOHUI ZHANG, GENDI WANG AND YUMING CHU

57 Department of Mathematics Huzhou University

Huzhou 313000, P.R. China.

EMail:xhzhang@hutc.zj.cn

c

2000Victoria University ISSN (electronic): 1443-5756 274-05

(2)

Extensions and Sharpenings of Jordan’s and Kober’s

Inequalities

Xiaohui Zhang, Gendi Wang and Yuming Chu

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of7

J. Ineq. Pure and Appl. Math. 7(2) Art. 63, 2006

Abstract

In this paper the authors discuss some monotonicity properties of functions involving sine and cosine, and obtain some sharp inequalities for them. These inequalities are extensions and sharpenings of the well-known Jordan’s and Kober’s inequalities.

2000 Mathematics Subject Classification:Primary 26D05

Key words: Monotonicity; Jordan’s inequality; Kober’s inequality; Extension and sharpening.

The research is partly supported by N.S.Foundation of China under grant 10471039 and N.S.Foundation of Zhejiang Province under grant M103087.

1. Introduction

The well-known inequalities

(1.1) 2

πx≤sinx≤x, x∈h 0,π

2 i

and

(1.2) cosx≥1− 2

πx, x∈h 0,π

2 i

are called Jordan’s and Kober’s inequality, respectively. In fact, Jordan’s and Kober’s inequalities are dual in the sense that they follow from each other via the transformationT : x → π/2−x. Some different extensions and sharpenings of these inequalities have been obtained by many authors (see [1] – [4]).

In this note, we will extend and sharpen Jordan’s and Kober’s inequalities by using the monotone form of l’Hôpital’s Rule (cf. [5, Theorem 1.25]) and obtain the following results:

Theorem 1.1. Forx∈[0, π/2],

(1.3) 2

πx+π−2

π² x(π−2x)≤sinx≤ 2 πx+ 2

π²x(π−2x),

(1.4) 2

πx+ 1

π³x(π²−4x²)≤sinx≤ 2

πx+π−2

π³ x(π²−4x²), and

(1.5) 1− 2

πx+π−2

π² x(π−2x)≤cosx≤1− 2 πx+ 2

π²x(π−2x), where the coefficients are all best possible.

(4)

Inequalities

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of7

2. Proof of Theorem 1.1

The following monotone form of l’Hôpital’s Rule, which is put forward in [5, Theorem 1.25], is extremely useful in our proof.

Lemma 2.1 (The Monotone Form of l’Hôpital’s Rule). For−∞ < a < b <

∞, letf,g : [a, b] →Rbe continuous on[a, b], and differentiable on(a, b), let g⁰(x) 6= 0on(a, b). Iff⁰(x)/g⁰(x)is increasing (decreasing) on(a, b), then so

are f(x)−f(a)

g(x)−g(a) and f(x)−f(b) g(x)−g(b).

If f⁰(x)/g⁰(x) is strictly monotone, then the monotonicity in the conclusion is also strict.

We next prove the inequalities (1.3) – (1.5) by making use of the monotone form of l’Hôpital’s Rule.

Proof of Inequality (1.3). Let f(x) = ^sinx_x − _π²

/ ^π₂ −x

. Write f1(x) =

sinx

x − ²_π, andf₂(x) = ^π₂ −x. Thenf₁(π/2) =f₂(π/2) = 0and

(2.1) f₁⁰(x)

f₂⁰(x) = sinx−xcosx

x² = f₃(x) f₄(x),

wheref₃(x) = sinx−xcosxandf₄(x) =x². Thenf₃(0) =f₄(0) = 0and

(2.2) f₃⁰(x)

f₄⁰(x) = sinx 2 ,

(5)

Inequalities

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of7

which is strictly increasing on [0, π/2]. By (2.1), (2.2) and the monotone form of l’Hôpital’s rule,f(x)is strictly increasing on[0, π/2].

The limiting valuef(0) = _π²(1− _π²)is clear. By (2.1) and l’Hôpital’s Rule, we havef(π/2) = _π⁴2.

The inequality (1.3) follows from the monotonicity and the limiting values off(x).

Proof of Inequality (1.4). Letg(x) =g₁(x)/g₂(x), whereg₁(x) = ^sinx_x −²_π and g₂(x) = ^π₄² −x². Theng₁(π/2) =g₂(π/2) = 0. By differentiation, we have

(2.3) g⁰₁(x)

g⁰₂(x) = sinx−xcosx

2x³ = g3(x) g₄(x),

whereg₃(x) = sinx−xcosxandg₄(x) = 2x³. Theng₃(0) =g₄(0) = 0and

(2.4) g₃⁰(x)

g₄⁰(x) = sinx 6x ,

which is strictly decreasing on[0, π/2]. Hence, by the monotone form of l’Hôpital’s rule,g(x)is also strictly decreasing on[0, π/2].

The limiting valueg(0) = _π⁴2(1− ²_π)is clear. By (2.3) and l’Hôpital’s Rule, g(π/2) = _π⁴3.

The inequality (1.4) follows from the monotonicity and the limiting values ofg(x).

Proof of Inequality (1.5). Leth(x) = ^1−cos_x ^x − ²_π

/ ^π₂ −x

. Simple calculat- ing similar to proofs of inequalities (1.3) and (1.4) will yield the monotonicity and limiting values ofh(x), and the inequality (1.5) follow.

(6)

Inequalities

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of7

Remark 1.

1. The inequalities (1.3) and (1.5) areT−dual to each other.

2. Like the proof of inequality (1.4), we can construct a function

m(x) =

1−cosx

x − 2

π

π² 4 −x²

and obtain the following inequality:

(2.5) 1− 2

πx+π−2

2π³ x(π²−4x²)≤cosx≤1− 2 πx+ 2

π³x(π²−4x²).

But the inequalities (1.4) and (2.5) are not T−dual. Comparing the in- equality (1.5) with (2.5), we can find the inequality (1.5) is stronger than (2.5). Whereas the inequalities (1.3) and (1.4) cannot be compared on the whole interval[0, π/2].

3. Straightforward simplifications of the inequalities (1.3) – (1.5) yield that forx∈[0, π/2],

(2.6) x− 2(π−2)

π² x² ≤sinx≤ 4x π − 4

π²x²,

(2.7) 3

πx− 4

π³x³ ≤sinx≤x− 4(π−2) π³ x³, and

(2.8) 1− 4−π

π x− 2(π−2)

π² x² ≤cosx≤1− 4 π²x².

(7)

Inequalities

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of7

References

[1] G.H. HARDY, J.E. LITTLEWOOD AND G. PÓLYA, Inequalities, Second Edition, Cambridge, 1952.

[2] D.S. MITRINOVIC, Analytic Inequalities, Springer-Verlag, 1970.

[3] G. KLAMBAUER, Problems and Properties in Analysis, Marcel Dekker, Inc., New York and Basel, 1979.

[4] U. ABEL AND D. CACCIA, A sharpening of Jordan’s inequality, Amer.

Math. Monthly, 93 (1986), 568.

[5] G.D. ANDERSON, M.K. VAMANAMURTHY AND M. VUORINEN, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wi- ley & Sons, New York, 1997.