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JORDAN-TYPE INEQUALITIES FOR GENERALIZED BESSEL FUNCTIONS

ÁRPÁD BARICZ

BABE ¸S-BOLYAIUNIVERSITY, FACULTY OFECONOMICS

RO-400591 CLUJ-NAPOCA, ROMANIA

bariczocsi@yahoo.com

Received 8 December, 2007; accepted 02 May, 2008 Communicated by A. Laforgia

Dedicated to my son Koppány.

ABSTRACT. In this note our aim is to present some Jordan-type inequalities for generalized Bessel functions in order to extend some recent results concerning generalized and sharp versions of the well-known Jordan’s inequality.

Key words and phrases: Bessel functions, modified Bessel functions, Jordan’s inequality.

2000 Mathematics Subject Classification. 33C10, 26D05.

1. INTRODUCTION ANDPRELIMINARIES

The following inequality is known in the literature as Jordan’s inequality [8, p. 33]

2

π ≤ sinx

x <1, 0< x≤ π 2.

This inequality plays an important role in many areas of mathematics and it has been studied by several mathematicians. Recently many authors including, for example A. McD. Mercer, U.

Abel and D. Caccia [7], F. Yuefeng [17], F. Qi and Q.D. Hao [10], L. Debnath and C.J. Zhao [5], S.H. Wu [15], J. Sándor [12], X. Zhang, G. Wang and Y. Chu [18], L. Zhu [19, 20], A.Y. Özban [9], S. Wu and L. Debnath [16], W.D. Jiang and H. Yun [6] (see also the references therein) have improved Jordan’s inequality. For the history of this the interested reader is referred to the survey articles of J. Sándor [13] and F. Qi [11].

In a recent work [2] we pointed out that the improvements of Jordan’s inequality can be con- fined as particular cases of some inequalities concerning Bessel and modified Bessel functions.

Research partially supported by the Institute of Mathematics, University of Debrecen, Hungary. The author is grateful to Prof. Lokenath Debnath for a copy of paper [16].

363-07

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Our aim in this paper is to continue our investigation related to extensions of Jordan’s inequal- ity. The main motivation to write this note is the publication of S. Wu and L. Debnath [16], which we wish to complement. For this let us recall some basic facts about generalized Bessel functions.

The generalized Bessel function of the first kindvp is defined [4] as a particular solution of the generalized Bessel differential equation

x2y00(x) +bxy0(x) +

cx2 −p2+ (1−b)p

y(x) = 0, whereb, p, c∈R,andvp has the infinite series representation

vp(x) =X

n≥0

(−1)ncn

n!Γ p+n+ b+12 ·x 2

2n+p

for allx∈R.

This function permits us to study the classical Bessel functionJp [14, p. 40] and the modified Bessel function Ip [14, p. 77] together. For c = 1 (c = −1 respectively) and b = 1 the functionvp reduces to the function Jp (Ip respectively). Now the generalized and normalized (with conditionsup(0) = 1 andu0p(0) = −c/(4κ)) Bessel function of the first kind is defined [4] as follows

up(x) = 2pΓ (κ)·x−p/2vp(x1/2) =X

n≥0

(−c/4)n (κ)n

xn

n! for allx∈R,

whereκ:=p+(b+1)/26= 0,−1,−2, . . . ,and(a)n= Γ(a+n)/Γ(a), a6= 0,−1,−2, . . . is the well-known Pochhammer (or Appell) symbol defined in terms of Euler’s gamma function. This function is related in fact to an obvious transform of the well-known hypergeometric function

0F1,i.e.up(x) = 0F1(κ,−cx/4),and satisfies the following differential equation xy00(x) +κy0(x) + (c/4)y(x) = 0.

Now let us consider the functionλp :R→R,defined by λp(x) := up(x2) =X

n≥0

(−c/4)n (κ)n

x2n n! .

It is worth mentioning that ifc = b = 1,then λp reduces to the function Jp : R → (−∞,1], defined by

Jp(x) = 2pΓ(p+ 1)x−pJp(x).

Moreover, ifc=−1andb= 1,thenλp becomesIp :R→[1,∞),defined by Ip(x) = 2pΓ(p+ 1)x−pIp(x).

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For later use we note that in particular (forp= 1/2, p= 3/2respectively) the functionsJpand Ipreduce to some elementary functions, like [14, p. 54]

J1

2(x) = r π

2x ·J1

2(x) = sinx x , (1.1)

J3

2(x) = 3 x

r π 2x ·J3

2(x) = 3

sinx

x3 −cosx x2

,

I1

2(x) = r π

2x ·I1

2(x) = sinhx x , (1.2)

I3

2(x) = 3 x

rπ 2x ·I3

2(x) = −3

sinhx

x3 − coshx x2

.

2. EXTENSIONS OF JORDANSINEQUALITY TOBESSEL FUNCTIONS

The following theorems are extensions of Theorem 1 due to S. Wu and L. Debnath [16] to generalized Bessel functions of the first kind.

Theorem 2.1. Ifκ >0andc∈[0,1],then for all0< x≤r≤π/2we have λp(r) + c

2κx(r−x)λp+1(r) +

1−λp(r) r2

(r−x)2

≤λp(x)≤λp(r) + c

4κ(r2−x2p+1(r)− c

4κr(r−x)2λ0p+1(r).

Moreover, if κ > 0 and c ≤ 0, then the above inequalities hold for all 0 < x ≤ r, and equality holds if and only ifx=r.

Proof. Whenx=r, clearly we have equality. Assume thatx6=rand fixr.Let us consider the functionsϕ1, ϕ2, ϕ3, ϕ4 : (0, r)→R,defined by

ϕ1(x) := λp(x)−λp(r)− c

4κ(r2−x2p+1(r), ϕ2(x) :=

1− x r

2

, ϕ3(x) :=λp+1(r)−λp+1(x) and ϕ4(x) := 1− r

x. Then we have

ϕ01(x) ϕ02(x) = cr2

4κ · ϕ3(x)−ϕ3(r)

ϕ4(x)−ϕ4(r) and ϕ03(x)

ϕ04(x) = x2ϕ03(x)

r .

Here we applied the derivative formula

(2.1) λ0p(x) =−cx

2κλp+1(x),

which follows immediately from the series representation of λp. Suppose that c ∈ [0,1]. It is known [3] that the function Jp is decreasing and concave on [0, π/2] when p ≥ −1/2. On the other hand, λp(x) = Jκ−1(x√

c)and thus λp is decreasing and concave on[0, π/2]when κ ≥ 1/2. From this we obtain thatϕ3 is increasing and convex when κ > 0. Thus ϕ0304 is increasing too as a product of two positive and increasing functions. Using the monotone form

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of the l’Hospital rule due to G.D. Anderson, M.K. Vamanamurthy and M. Vuorinen [1, Lemma 2.2] we obtain thatϕ0102is also increasing on(0, r).

Now assume that c ≤ 0. Then clearly ϕ3 is decreasing and concave, since all coefficients of the corresponding power series are negative. Consequentlyϕ0304 is decreasing and hence ϕ0102is increasing on(0, r).Using again the monotone form of the l’Hospital rule [1, Lemma 2.2] this implies that the functionφ1 : (0, r)→R,defined by

φ1(x) := ϕ1(x)−ϕ1(r) ϕ2(x)−ϕ2(r), is increasing. Moreover from the l’Hospital rule we obtain that

φ1(0+) = 1−λp(r)− cr2

4κλp+1(r) and φ1(r) = −cr3

4κλ0p+1(r).

Hence for all κ > 0, c ∈ [0,1] and 0 < x ≤ r ≤ π/2 we have the following inequality:

φ1(0+)≤φ1(x)≤ φ1(r).Moreover, whenκ > 0, c ≤0and0< x≤r,the above inequality

also holds, hence the required result follows.

Theorem 2.2. Ifκ >0andc∈[0,1],then for all0< x≤r≤π/2we have λp(r) + c

4κ(r2−x2p+1(r)− c

16κr(r2−x2)2λ0p+1(r)

≤λp(x)≤λp(r) + c 4κ

x2

r2(r2−x2p+1(r) +

1−λp(r) r4

(r2 −x2)2. Moreover, ifκ > 0 andc ≤ 0,then the above inequalities are reversed for all0 < x ≤ r, and equality holds if and only ifx=r.

Proof. The proof of this theorem is similar to the proof of Theorem 2.1, so we sketch the proof.

Let us consider the functionsϕ5, ϕ6 : (0, r)→R,defined by ϕ5(x) :=

1−x2

r2 2

and ϕ6(x) :=x2−r2. In view of (2.1), easy computations show that

ϕ01(x) ϕ05(x) = cr4

8κ · ϕ3(x)−ϕ3(r) ϕ6(x)−ϕ6(r) and ϕ03(x)

ϕ06(x) =−λ0p+1(x)

2x = c

4(κ+ 1)λp+2(x).

Suppose that c ∈ [0,1]. Sinceλp is decreasing on[0, π/2]whenκ ≥ 1/2,we get thatϕ0306 is decreasing on(0, r), whenκ > 0.Thus from the monotone form of the l’Hospital rule [1, Lemma 2.2],ϕ0105is also decreasing on(0, r).

Now assume that c ≤ 0. Then clearly ϕ0306 is decreasing and consequently ϕ0105 is in- creasing on(0, r).Now consider the functionφ2 : (0, r)→R,defined by

φ2(x) := ϕ1(x)−ϕ1(r) ϕ5(x)−ϕ5(r).

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Then from the monotone form of the l’Hospital rule [1, Lemma 2.2], we conclude that φ2 is decreasing whenκ > 0andc ∈ [0,1], and is increasing whenκ > 0and c ≤ 0. In addition, from the usual l’Hospital rule we have thatφ2(0+) = φ1(0+)and4φ2(r) = φ1(r).Now for allκ >0, c ∈[0,1]and0< x≤r ≤π/2,using the inequalityφ2(0+)≥φ2(x)≥φ2(r),the asserted result follows. Finally, whenκ > 0, c ≤ 0and0 < x ≤ r the above inequalities are

reversed. Thus the proof is complete.

Remark 1. First note that taking c = b = 1 and p = 1/2 in Theorems 2.1 and 2.2 and using (1.1), we reobtain the results of S. Wu and L. Debnath [16, Theorem 1], but just for 0< x≤r ≤π/2.Moreover, the inequalities in Theorem 2.2 are improvements of inequalities established in [2, Theorem 5.14]. More precisely in [2, Theorem 5.14] we proved that ifκ >0, c∈[0,1]and0< x≤r≤π/2,then

(2.2) λp(r) + h c

λp+1(r) i

(r2 −x2)≤λp(x)≤λp(r) +

1−λp(r) r2

(r2−x2).

Easy computations show that

− c

16κr(r2−x2)2λ0p+1(r)≥0, c

4κ x2

r2(r2 −x20p+1(r) +

1−λp(r) r4

(r2−x2)2

1−λp(r) r2

(r2−x2),

whereκ >0, c∈ [0,1]and0< x ≤r ≤π/2.Thus Theorem 2.2 provides an improvement of (2.2).

Finally taking c = −1, b = 1 and p = 1/2 in Theorems 2.1 and 2.2 and using (1.2), we obtain the hyperbolic counterpart of Theorem 1 due to S. Wu and L. Debnath [16].

Corollary 2.3. If0< x≤r,then sinhr

r + 1 2

sinhr

r −coshr 1− x2 r2

− 3 2

−1

3rsinhr+ coshr− sinhr r

1− x

r 2

≥ sinhx

x ≥ sinhr r +1

2

sinhr

r −coshr 1− x2 r2

+3

2 2

3 +coshr

3 − sinhr r

1− x

r 2

, where the equality holds if and only ifx=rand the values

3 2

2

3 +coshr

3 −sinhr r

and 3 2

−1

3rsinhr+ coshr− sinhr r

are the best constants. Moreover,

sinhr r +1

2

sinhr

r −coshr 1− x2 r2

−3 8

−1

3rsinhr+ coshr− sinhr

r 1− x2 r2

2

≥ sinhx

x ≥ sinhr r +1

2

sinhr

r −coshr 1− x2 r2

+3

2 2

3 +coshr

3 − sinhr

r 1− x2 r2

2

, where equality holds if and only ifx=rand the values

−3 8

−1

3rsinhr+ coshr− sinhr r

and 3 2

2

3+ coshr

3 − sinhr r

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are the best constants.

REFERENCES

[1] G.D. ANDERSON, M.K. VAMANAMURTHYAND M. VUORINEN, Inequalities for quasicon- formal mappings in space, Pacific J. Math., 160(1) (1993), 1–18.

[2] Á. BARICZ, Some inequalities involving generalized Bessel functions, Math. Inequal. Appl., 10(4) (2007), 827–842.

[3] Á. BARICZ, Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math., (2008), doi: 10.1016/j.exmath.2008.01.001.

[4] Á. BARICZ, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen. in press.

[5] L. DEBNATHANDC.J. ZHAO, New strengthened Jordan’s inequality and its applications, Appl.

Math. Lett., 16 (2003), 557–560.

[6] W.D. JIANGANDH. YUN, Sharpening of Jordan’s inequality and its applications, J. Inequal. Pure Appl. Math., 7(3) (2006), Art. 102.

[7] A.McD. MERCER, U. ABELANDD. CACCIA, A sharpening of Jordan’s inequality, Amer. Math.

Monthly, 93 (1986), 568–569.

[8] D.S. MITRINOVI ´C, Analytic Inequalities, Springer–Verlag, Berlin, 1970.

[9] A.Y. ÖZBAN, A new refined form of Jordan’s inequality and its applications, Appl. Math. Lett., 19 (2006), 155–160.

[10] F. QIANDQ.D. HAO, Refinements and sharpenings of Jordan’s and Kober’s inequality, Mathemat- ics and Informatics Quarterly, 8(3) (1998), 116–120.

[11] F. QI, Jordan’s inequality. Refinements, generalizations, applications and related problems, RGMIA Research Report Collection, 9(3) (2006), Art. 12. Bùdˇengshì Y¯anji¯u T¯ongxùn (Communications in Studies on Inequalities) 13(3) (2006), 243–259.

[12] J. SÁNDOR, On the concavity ofsinx/x,Octogon Math. Mag., 13(1) (2005), 406–407.

[13] J. SÁNDOR, A note on certain Jordan type inequalities, RGMIA Research Report Collection, 10(1) (2007), Art. 1.

[14] G.N. WATSON, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cam- bridge, 1944.

[15] S.H. WU, On generalizations and refinements of Jordan type inequality, Octogon Math. Mag., 12(1) (2004), 267–272.

[16] S. WU AND L. DEBNATH, A new generalized and sharp version of Jordan’s inequality and its applications to the improvement of the Yang Le inequality, Appl. Math. Lett., 19 (2006), 1378–

1384.

[17] F. YUEFENG, Jordan’s inequality, Math. Mag., 69 (1996), 126–127.

[18] X. ZHANG, G. WANGANDY. CHU, Extensions and sharpenings of Jordan’s and Kober’s inequal- ities, J. Inequal. Pure Appl. Math., 7(2) (2006), Art. 63. [ONLINE:http://jipam.vu.edu.

au/article.php?sid=680].

[19] L. ZHU, Sharpening Jordan’s inequality and Yang Le inequality, Appl. Math. Lett., 19 (2006), 240–243.

[20] L. ZHU, Sharpening Jordan’s inequality and Yang Le inequality II, Appl. Math. Lett., 19 (2006), 990–994.

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