SOME REFINEMENTS OF WELL-KNOWN INEQUALITIES INVOLVING TRIGONOMETRIC FUNCTIONS

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SOME REFINEMENTS OF WELL-KNOWN INEQUALITIES INVOLVING TRIGONOMETRIC

FUNCTIONS

Raouf Chouikha, Christophe Chesneau, Yogesh Bagul

To cite this version:

Raouf Chouikha, Christophe Chesneau, Yogesh Bagul. SOME REFINEMENTS OF WELL-KNOWN

INEQUALITIES INVOLVING TRIGONOMETRIC FUNCTIONS. 2020. �hal-02538194�

INVOLVING TRIGONOMETRIC FUNCTIONS

ABD RAOUF CHOUIKHA, CHRISTOPHE CHESNEAU, AND YOGESH J. BAGUL

Abstract. In this paper, we determine new and sharp inequalities involving trigonometric functions. More specifically, a new general result on the lower bound for log(1−uv),u, v∈(0,1) is proved, allowing to determine sharp lower and upper bounds for the so-called sinc function, i.e., sin(x)/x, lower bounds for cos(x) and upper bounds for (cos(x/3))³. The obtained bounds improve some well-established results. The findings are supported by graphical analy- ses.

2010 Mathematics Subject Classification:26D05, 26D07, 26D20, 33B10.

Key words and phrases:Cusa-Huygens inequality; sinc function; series expan- sion; positive and increasing function.

1. Introduction

Over the last two decades, efforts were made to bound special trigonometric and hyperbolic functions as sharp as possible, with a focus on the sinc function sin(x)/x.

The resulting inequalities find applications in many applied fields, allowing quick evaluations of complex functions involving these trigonometric functions. The lit- erature on the subject is vast and growing fast. We may refer the reader to [1], [2], [3], [4], [5], [6], [7], [8], [10], [11], [12], [13], [14], [15], [16], [17] and [18], and the references therein.

This paper contributes to the subject in the following way. First of all, we prove a general and sharp lower bound result for log(1−uv), u, v∈(0,1). Then, we apply this result to determine polynomial-exponential lower bounds for sin(x)/x and cos(x), also with the use of infinite product series. We prove that they are sharp, improving some recent results of the literature. Also, as intermediate results, some new polynomial-exponential inequalities are set. As an alternative approach, we use these results to conjecture upper bounds for sin(x)/xand (cos(x/3))³. Proofs are given in details by the means of Taylor developments. Again, some recent results in the fields are refined, including the famous Cusa-Huygens inequality[9]. All the findings are supported by the visual checks of appropriate functions.

The rest of the paper is planned as follows. Section 2 investigate the lower bounds. Section 3 is devoted to the upper bounds for sin(x)/x and (cos(x/3))³, with discussions.

2. Lower bounds

This section is devoted to the proof of new lower bounds, involving sharp lower bounds for sin(x)/xand cos(x) as applications. Some graphics support the findings.

1

2.1. Some new general results. The result below proposes a new lower bound for log(1−uv) with u, v ∈(0,1), which will be at the center of the proofs of new sharp lower bounds for sin(x)/xand cos(x).

Proposition 2.1. For any u, v∈(0,1), the following inequality holds:

log(1−uv)> uv(u−1)h

u+ 1 + uv 2

i

+u³log(1−v).

Proof. By virtue of the logarithmic series expansion and, for k≥3, u^k < u³, after some algebraic manipulations, we get

log(1−uv) =−

+∞

X

k=1

u^kv^k

k =−uv−u²v²

2 −

+∞

X

k=3

u^kv^k k

>−uv−u²v² 2 −u³

+∞

X

k=3

v^k

k =−uv−u²v² 2 +u³

"

−

+∞

X

k=1

v^k

k +v+v² 2

#

=−uv−u²v² 2 +u³

log(1−v) +v+v² 2

=uv(u−1)h

u+ 1 +uv 2

i

+u³log(1−v).

This ends the proof of Proposition 2.1.

Proposition 2.2. Foru, v∈(0,1) the following inequalities hold:

1−uv >(1−v)^u3e^uv(u−1)[^u+1+uv2]>(1−v)^u2e^uv(u−1)>(1−v)^u. Proof. The first inequality is Proposition 2.1, by taking the exponential trans- formation. The second one takes the first steps of the proof of Proposition 2.1 in the following sense: Sinceu³< u², we have

uv(u−1)h

u+ 1 + uv 2

i

+u³log(1−v) =−uv−u²v² 2 −u³

+∞

X

k=3

v^k k

>−uv−u²v² 2 −u²

+∞

X

k=3

v^k

k =−uv−u²v² 2 +u²

"

−

+∞

X

k=1

v^k

k +v+v² 2

#

=uv(u−1) +u²log(1−v).

The desired inequality follows by taking the exponential transformation. The last inequality is follows from [7, Theorem 1-1], showing thatuv(u−1) +u²log(1−v)>

ulog(1−v). This ends the proof of Proposition 2.2.

2.2. Lower bounds for sin(x)/x. The result below presents a new sharp bound for sin(x)/xinvolving the exponential function.

Proposition 2.3. Forx∈(0, π), we have the following inequalities:

sin(x) x >

1−x²

π^{2 π}

6 945

e^x

2h_π4

945−¹₆+^x₂²_π2 945−₉₀¹i

>

1−x²

π^{2 π}

4 90

e^x

2

π2 90−¹₆

>

1−x²

π^{2 π}

2 6

.

Proof. By using the infinite product expression of sin(x)/x, takingu= 1/k²and v=x²/π²in Proposition 2.2, and using the following well-known results on the zeta function, i.e., ζ(k) =P+∞

n=11/n^k: ζ(2) =π²/6, ζ(4) = π⁴/90 andζ(6) =π⁶/945, we get

sin(x)

x =

+∞

Y

k=1

1− x²

π²k²

>

+∞

Y

k=1

"

1−x²

π² _k¹6

e ^x

2

π2k2(_k¹2−1)^h_k¹2+1+ ^x2

2k2π2

i#

=

1−x² π²

^P+∞_k=1k¹6

e

x2 π2

P+∞

k=1

h₁

k6−¹

k2+^x2

2π2(_k¹6−¹

k4)ⁱ

=

1−x² π²

^π

6 945

e^x2

hπ4

945−¹₆+^x₂²

π2 945−₉₀¹i

.

The second inequality is due to Proposition 2.2, with similar lines of proof. The last inequality follows from [7, Proposition 2-4]. This ends the proof of Proposition

2.3.

Figure 1 illustrates the two first inequalities of Proposition 2.3 by plotting the two following functions forx∈(0, π):

A(x) =sin(x)

x −

1−x²

π^{2 π}

6 945

e^x2

hπ4

945−¹₆+^x₂²

π2 945−₉₀¹i

and

B(x) =

1−x² π²

^π

6 945

e^x

2h

π4

945−¹₆+^x₂²

π2 945−₉₀¹i

−

1−x² π²

^π

4 90

e^x

2

π2 90−¹₆

.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0e+002e−044e−046e−04

A(x)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0000.0010.0020.003

B(x)

, Figure 1. Plots for A(x) andB(x) forx∈(0, π), respectively.

As expected, we see thatA(x) andB(x) are positive, with very small variations, attesting the sharpness of the obtained bounds.

2.3. Lower bounds forcos(x). The result below presents a new sharp bound for cos(x) involving the exponential function.

Proposition 2.4. Forx∈(0, π/2), we have the following inequalities:

cos(x)>

1−4x²

π^{2 π}

6 960

e^x2

hπ4

240−¹₂+x²

π2 120−₁₂¹i

>

1−4x²

π^{2 π}

4 96

e^x

2

π2 24−¹₂

>

1−4x²

π^{2 π}

2 8

.

Proof. By using the infinite product expression of cos(x), takingu= 1/(2k−1)² and v = 4x²/π² in Proposition 2.2, and using the following well-known results:

P+∞

k=11/(2k−1)² = π²/8, P+∞

k=11/(2k−1)⁴ = π⁴/96 and P+∞

k=11/(2k−1)⁶ = π⁶/960 we get

cos(x) =

+∞

Y

k=1

1− 4x² π²(2k−1)²

>

+∞

Y

k=1

"

1−4x²

π²

_(2k−1)6¹ e

4x2 π2 (2k−1)2

₁

(2k−1)2−1h ₁

(2k−1)2+1+_{π2 (2k−1)2}^2x2 i#

=

1−4x² π²

P+∞

k=1 1 (2k−1)6

e

4x2 π2

P+∞

k=1

h ₁

(2k−1)6−_(2k−1)2¹ +^2x2

π2

₁

(2k−1)6−_(2k−1)4¹ i

=

1−4x² π²

^π

6 960

e^x2

hπ4 240−¹₂+x²

π2 120−₁₂¹i

.

The second inequality is also derived to Proposition 2.2, with similar mathematical arguments. The last one is a consequence of [7, Proposition 2-4]. This ends the

proof of Proposition 2.4.

As a matter of fact, Proposition 2.4 improves [2, Proposition 4], i.e., for any x∈(0, π/2),

cos(x)>

1−4x²

π^{2 π}

2 8

. (2.1)

Figure 2 illustrates the two first inequalities of Proposition 2.4 by plotting the two following functions forx∈(0, π/2):

C(x) = cos(x)−

1−4x² π²

^π

6 960

e^x2

hπ4

240−¹₂+x²

π2 120−₁₂¹i

and

D(x) =

1−4x² π²

^π

6 960

e^4x

2h

π4

960−¹₈+2x²

π2 960−₉₆¹i

−

1−4x² π²

^π

4 96

e^x

2

π2 24−¹₂

.

0.0 0.5 1.0 1.5

0e+004e−058e−05

C(x)

0.0 0.5 1.0 1.5

0e+004e−048e−04

D(x)

, Figure 2. Plots forC(x) andD(x) forx∈(0, π/2), respectively.

We observe thatC(x) andD(x) are positive, with very small variations, attesting the sharpness of the obtained bounds.

3. Upper bounds

Here, we derive some new sharps upper bounds for sin(x)/x and (cos(x/3))³, which naturally appear in many inequality involving trigonometric functions.

3.1. Upper bounds for sin(x)/x. In [12], the authors proved the inequality of Cusa-Huygens: forx∈(0, π/2),

sin(x)

x < 2 + cos(x)

3 .

(3.1)

That is, is natural to address the following question: Based on (3.1), can we use lower bounds for cos(x) to derive upper bounds for sin(x)/x?

As a first remark, by using the simple but improvable lower bounds in (2.1), we provide a first answer to the question: Is the following inequality true ? For any x∈(0, π/2),

00 2 + cos(x) 3 > 2

3+1 3

1−4x²

π^{2 π}

2 8

>sin(x) x

00.

The answer is negative because the functionh(x) = 3 sin(x)/x−2− 1−4x²/π²π²/8

has not a constant sign. Indeed, as countered example, we haveh(0.3)≈0.0001>0 andh(0.8)≈ −0.00034<0.

However, based on Proposition 2.4, the following chain of inequalities is true, providing a new upper bound for sin(x)/x.

Proposition 3.1. Forx∈(0, π/2), we have 2 + cos(x)

3 > 2 3+1

3

1−4x² π²

^π

6 960

e^x

2h

π4

240−¹₂+x²

π2 120−₁₂¹i

> 2 3+1

3

1−4x² π²

^π

4 96

e^x

2_π2 24−¹₂

>sin(x) x .

Proof. We only need to prove the last right inequalities, the others follow from Proposition 2.4. The proof is based on the analytical study of the following function:

k(x) = 3sin(x) x −2−

1−4x²

π^{2 π}

4 96

e^x

2

π2 24−¹₂

. (3.2)

The desired inequality comes by proving that k(x) is non positive. The Taylor development of sin(x) gives

sin(x) =x−x³ 3! +x⁵

5! −x⁷

7! +. . .+ (−1)^k−1 x^2k−1

(2k−1)!+ (−1)^k cos(θx) (2k+ 1)!x^2k+1, where θ ∈ (0,1). As a direct application, the following inequalities hold forx ∈ (0, π/2):

1−1

6x²+ 1

120x⁴− 1

5040x⁶< sin(x)

x < `(x), (3.3)

where

`(x) = 1−1

6x²+ 1

120x⁴− 1

5040x⁶+ 1 362880x⁸

≈1−0.166667x²+ 0.0083333x⁴−0.00019841x⁶+ 0.0000027x⁸. Similarly, we have

e^x

2_π2 24−¹₂

> m(x), (3.4)

where

m(x) = 1 + π²

24−1 2

x²+1

2 π²

24−1 2

² x⁴+1

6 π²

24−1 2

³ x⁶

≈1−0.088766x²+ 0.0039392x⁴−0.00011655x⁶.

Applying again the Taylor decomposition technique, since 4x²/π²<1 andπ⁴/96>

1, we have

1−4x²

π^{2 π}

4 96

> n(x), (3.5)

where

n(x) = 1− 1

24π²x²+ 1

1152π⁴− 1 12

x⁴− 4 3π²

π⁴ 1152− 1

12 π⁴ 96 −2

x⁶ + 4

3π⁴ π⁴

1152− 1 12

π⁴

96 −2 π⁴ 96−3

x⁸

≈1−0.411246x²+ 0.001225x⁴+ 0.00016305x⁶+ 0.000032799x⁸.

By putting (3.2), (3.3), (3.4) and (3.5) together, we obtain

k(x)<3 1−0.166667x²+ 0.0083333x⁴−0.00019841x⁶+ 0.0000027x⁸

−2

− 1−0.41124x²+ 0.001225x⁴+ 0.000163x⁶+ 0.0000328x⁸

× 1−0.08876x²+ 0.0039392x⁴−0.00011655x⁶ .

After development and a acceptable approximation (with 5 digits), we arrive at k(x)<−0.0000001x¹²+ 0.000003x¹⁰−0.0000654x⁸+ 0.0010868x⁶−0.01667x⁴. This last polynomial is non positive because it has no root in the interval (0, π/2).

This ends the proof of Proposition 3.1.

Thanks to Proposition 3.1, we then obtain a better bound for the Cusa-Huygens inequality.

Also, we may derive from Propositions 2.1 and 3.1 the following new frame for sin(x)/x:

1−4x²

π^{2 π}

6 960

e^x

2h

π4

240−¹₂+x²

π2 120−₁₂¹i

< sin(x) x

<2 3 +1

3

1−4x² π²

^π

6 960

e^x

2h

π4

240−¹₂+x²

π2 120−₁₂¹i

.

Figure 3 provides a graphical illustration of the main finding of Proposition 3.1 by displaying the following function forx∈(0, π/2):

E(x) = sin(x)

x −





 2 3 +1

3

1−4x² π²

^π

4 96

e^x

2

π2 24−¹₂





 .

0.0 0.5 1.0 1.5

0.0000.0100.0200.030

E(x)

, Figure 3. Plots forE(x) forx∈(0, π/2).

We see thatE(x) is positive, as proved analytically.

3.2. Upper bounds for (cos(x/3))³. In [15], the following chain of inequalities is proved. Forx∈(0, π/2),

sin(x) x <

1 3

2 cosx 2

+ 1²

<

cosx 3

3

< 2 + cos(x)

3 .

The following result proposes a refinement of this results, by the use of Propo- sition 2.4.

Proposition 3.2. Forx∈(0, π/2) we have

cosx 3

3

<2 3 +1

3

1−4x² π²

^π

6 960

e^x

2h

π4

240−¹₂+x²

π2 120−₁₂¹i

<2 3 +1

3

1−4x² π²

^π

4 96

e^x

2_π2 24−¹₂

< 2 + cos(x)

3 .

Proof. We follow the lines of the proof of Proposition 3.1, by proving the first left inequality; the others follows from Proposition 2.4. Firstly, the classic Taylor development of cos(x) gives

cosx= 1−x² 2! +x⁴

4! −x⁶

6! +. . .+ (−1)^kx^2k

2k! + (−1)^k+1 cos(θx) (2k+ 2)!x^2k+2, whereθ∈(0,1). Among others, this implies that

cos(x)<1−x² 2! +x⁴

4! −x⁶ 6! +x⁸

8!. That is, we have

cosx

3 3

< q(x), (3.6)

where, doing a standard development and a acceptable approximation, q(x) =

1−(x/3)²

2! +(x/3)⁴

4! −(x/3)⁶

6! +(x/3)⁸ 8!

³

≈1−1

6x²+ 7

648x⁴− 61

174960x⁶+ 547 88179840x⁸

≈1−0.166667x²+ 0.010802x⁴−0.0003486x⁶+ 0.000006x⁸. It follows from (3.4) that

e^x

2

π2 24−¹₂

> r(x), (3.7)

where

r(x) = 1 + π²

24−1 2

x²+1

2 π²

24−1 2

² x⁴+1

6 π²

24−1 2

³ x⁶

≈1−0.088766x²+ 0.0039392x⁴−0.00011655x⁶. We have also

1−4x²

π^{2 π}

4 96

> s(x), (3.8)

where

s(x) = 1− 1

24π²x²+ 1

1152π⁴− 1 12

x⁴− 4 3π²

π⁴ 1152− 1

12 π⁴ 96 −2

x⁶ + 4

3π⁴ π⁴

1152− 1 12

π⁴

96 −2 π⁴ 96−3

x⁸

≈1−0.411246x²+ 0.001225x⁴+ 0.0001631x⁶+ 0.0000328x⁸.

The relations (3.6), (3.7) and (3.8) allow us to derive an estimate of the difference

t(x) = 3 cosx

3 3

−2−

1−4x² π²

^π

4 96

e^x2(π

4 24−¹₂). Indeed, we have

t(x)<3

1−1/6x²+ 7

648x⁴− 61

174960x⁶+ 547 88179840x⁸

−2

− 1 + π²

24−1 2

x²+1

2 π²

24 −1 2

² x⁴+1

6 π²

24 −1 2

³ x⁶

!

×

1− 1

24π²x²+ 1

1152π⁴− 1 12

x⁴− 4 3π²

π⁴ 1152− 1

12 π⁴ 96−2

x⁶ + 4

3π⁴ π⁴

1152− 1 12

π⁴

96−2 π⁴ 96−3

x⁸

, implying that

t(x)<3 1.0−0.166667x²+ 0.010802x⁴−0.0003486x⁶+ 0.000006x⁸

−2

− 1−0.088766x²+ 0.0039392x⁴−0.0001166x⁶

× 1−0.411246x²+ 0.001225x⁴+ 0.0001631x⁶+ 0.0000328x⁸ .

That is, after development and a suitable approximation (with 5 digits), we arrive at

t(x)<−0.00925926x⁴+ 0.000636532x⁶−0.00005245x⁸+ 0.00000241x¹⁰. We verify that this polynomial is non positive because it has no root in the interval (0, π/2). This proved the first left inequality, ending the proof of Proposition 3.2.

As illustration of the main result in Proposition 3.2, Figure 4 shows the curve of the following function forx∈(0, π/2):

F(x) = 2 3+1

3

1−4x² π²

^π

6 960

e^x

2h

π4

240−¹₂+x²

π2 120−₁₂¹i

− cosx

3 3

.

0.0 0.5 1.0 1.5

0.0000.0050.0100.015

F(x)

, Figure 4. Plots forF(x) forx∈(0, π/2).

As expected, we see thatF(x) is positive.

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4, Cour des Quesblais 35430 Saint-Pere Email address:chouikha@math.univ-paris13.fr

LMNO, University of Caen-Normandie, Caen, France Email address:christophe.chesneau@unicaen.fr

Department of Mathematics, K. K. M. College, Manwath, Dist : Parbhani(M.S.) - 431505, India

Email address:yjbagul@gmail.com