THEOREM AMS WORDS

Internat. J. Math. & Math. Sci.

VOL. 20 NO. 3 (1997) 561-566

561

A NOTE

ON

A MONOTONICITY PROPERTY OF BESSEL FUNCTIONS

STAMATISKOUMANDOS

Department

ofPureMathematics TheUniversity ofAdelaide Adelaide5005,SOUTH AUSTRALIA

CurrentAddress:

Department

ofMathematics and Statistics University ofCyprus

P.O.Box537 1678Nicosia,CYPRUS

(ReceivedSeptember 18, 1995)

ABSTRACT. AtheoremofLorch,MuldoonandSzeg0^statesthat thesequence

3oJ, k=

isdecreasing fora

> 1/2,

where

Jo(t)

theBesselfunction ofthe first kind orderaandja,k itskth positiveroot. Thismonotonicity property impliesSzegO’sinequality

ot-Jo(t)t

>_ o,

whena

> a’

and

a’

istheuniquesolutionof

fo.2

^{t-oj(t)dt 0}

We givea new andsimplerproofof theseclassicalresults by expressing theabove Bessel function integralas anintegral involving elementaryfunctions.

KEY WORDS AND PHRASES: Besselfunctions, positive integral ofBeselfunctions, monotonicity propertyof Besselfunctions.

1991AMSSUBJECTCLASSIlCICATIONCODES: 33C10, 33C45.

1. INTRODUCTION

Let

Jo(t)

^{be the} Bessei function ofthe first kind and order a,ja,l,ja,9_, its positive roots in

increasing orderandJo,0 ⁰

In Lorch, MuldoonandSzegOderived,among other things,thefollowing.

THEOREM1. Fora

>

thesequenceof areas

k ^Jo,k k=

isdecreasing.

As it is shown in [1], thistheorem is aspecial caseofa moregeneral resultconcerning cylinder functions and itsproofis based onan application ofa Sturm-typeoscillationtheorem (formulatedby

Watson [2,p. 518], sharpenedandappliedin greater detailby^Makai [3])^{to a}certain linear differential equation ofsecond order.

Inaddition, astheauthorspointedoutin ],Theorem givesanotherproof ofaclassicalinequality ofSzegOcontainedintheNoteswhichheappendedto aposthumouspaper of Feldheim[4]inthecourse ofpreparingitfor publication.

SzegOprovedin[4]that

J’’=

t-’J, (t)dt >

O, k 2, 3 (1.2) wherea istheuniquerootofthetranscendentalequation

whosenumericalvalue is

a’

0.26938

As indicatedby Szeg6,(1.2)in combination with anapplication of the Sonineimegral(see [4, p.

279]or[2,p.373]),yieldtheinequality

Zt-aJa(t)dt

>

0 for all ^x

>

0, ^{whq’l c}

> t.

(1 4) SzegO’sproof of

(1.2)

isratherimricateas itrelies onvarious properties ofBessel functions as well asofcertain idemitiesinvolving theLommel functions.

It shouldbenoted that inequality(1.4)suggestsamuchstrongerinequality involving ultraspherical polynomialswhich wasrecentlyestablished in[5].

Over theyears, generalizations of(1.4) havebeen proved by several authors Inparticular, the inequality

t-Jo(t)dt

>

O, ^x

>

O,

<

a

+

I, (1.5) wasproved byMakai[6]^for

<

^a

< 1/2

^and

^{(c) _<}

^fi/<a

⁺

^1,where

’’ t-(’) J, (t)dt

O. ₍₆₎

Askeyand Steinig[7]proved(1.5)for 1

<

a

<

for thesamerange

of/.

^Fora

,

^(1.5)

rams

outtobeaclassicalinequality forcosineintegrals. Whena

> , ^(1.5,)

^{holds for <_/}

^<

^a

⁺

¹

and thisfollows fromaworkofGasper [8],inwhichanexplicit expression of the integralinquestionas a sumof squaresof theBesselfunctions withpositivecoefficients isproved. Seealso

[9]

^{for some}more recentresults onpositive integrals ofBessel functions.

Thepurpose ofthis note istoshow thatTheorem can beestablished in asimplermanner than that in andhence togivea newandmorestraightforwardproofofSzeg0’sinequalities(1.2)and(1.4)

We proceedbyobservingthatTheorem isequivalentto TREOREM2. Fora

> ,

^wedefine

f(x,a) t-J(t)dt,

^x

>

O. (17)

(1 8) Then,thelocalminimaof

f(x, a),

as afunctionof x, formanincreasingsequence,^i.e.

f(j,9_t,

a) <

f(ja,2t+2,

a),

^g 1,2, and itslocalmaximaformadecreasingsequence,i.e.

NOTEONAMONOTONICITY PROPERTYOFBESSEL FUNCTIONS 563

f(3o,9_-l,a) >

f(ja,2t+i,a),

e

1,2,.... (1.9)

It iswell known that thegraphof y

Jo(t), (a > 1)

consists ofwaves alternately aboveand below the axisoft, whose areas form asteadily decreasingsequence,t being positive. This classical result wasproved originallybyCookein 10]and 11]. Cooke’s proofisrathercomplicatedas itdepends on somedelicate estimatesinvolving the Lommelfunctionsand severalpropertiesofBesslfunctions In [3], ^Makai proved thisresult for

]al >

ina simpler way using a differential equation approach of Smrm-Liouville type. Aparticularly simple proofof Cooke’s Theoremhas been devisedby Steinigin [12].

Sincethesequence

2o.k+,

]Jo(t)lclt

"2o.k k

issteadily decreasing and for^ct

>

0, t is apositive decreasingfunctionoft,wehave

f ’<’’+’-lao()l, > f"’+’ (.<,.,<+ lao()l<i

2a.k+l 2o.k+l

ffhich

establishesTheorem foralla

>

0. Itis clearthatthe casea 0 reducestoCooke’s resultas well.

Inthenextsection wegiveasimple proofofTheorem 2forthe range

<

^a

<

0 Thisis, of course, the interestingcaseasthe criticalvalue

a’

forwhichtheSzegO’sinequalities(1.2)and(1 4)are valid,iscontained in this interval.

2. IROOFOFTIEOREM2FOR

<

a

<

0 Forthisproofweneedthe following elementarylemma.

LEMMA. Let 0

<

#

<

^1and

(t)=t(l_t)"

sin ^for 0<t<l.

Thenwehave

9’ ^{(t) <}

^{0for E}

(0,1). Moreover, 9" ^{(t) <}

^0for

^{e (0, 1),}

^when

1/2

^</

^<

¹

PROOF. Weobservethat

(1 t)"+’t/=

1 _t2

^(Tr

^{cos7r sin7r}

^t) + 2#

^sin7rt.

Toprovethenegativity of

g’ (t),

^{itsufficestoshowthat}

1 t

t (TrtcosTrt-

sinTrt)

+

^2sinTrt

<

0, orequivalemly

1 3t rtctg(Trt)

<

1 (2.1)

Nowtakinginto account the familiarformula

7rtctg(Trt) 1

+

^2t _t2 1_k2

k=l

(2.2) weseethat(2.1)^isequivalentto

Ek

1 ^>0

which isapparentlytruefor 0

<

t

<

1.

Hence, g’(t) <

0, for 0

< <

^1.

Supposethat

<

#

<

^1. Aroutine calculationshows that the negativity of

g"(t),

for 0

< <

1, follows fromtheinequality

7r2(1 t-) 2#[1 + (2# + 1)t 2] +

²

⁽¹ -}2)2

_(rtctgTrt-

_{1)- 4#(1} t2)(TrtctgTrt- 1) >

0.

t2 Inviewof(2.2),this isequivalemto

7r2t + (6# +

⁴

4# 27r2)t

²

+

^7r

2#

⁴

4(1 t2)(1 (2# + 1)t 2) E

k 1

k=2

>

0. (2

3)

Since

.2

1 3

--1< E

^k_t

^<

’

0<t<l,

k=2

inequality

(2.3)

follows easilybyanelementarycomputation.

Theproofof the lemmaiscomplete. El

Now,

in orderto proveTheorem2 weobserve thattheintegralin

(1.7)

^{thatdefinesthe}function

f (x, a)

coincides with anintegral ofcertainelementarythnctions.

Infact, byPoisson’sintegral(cf. [2,p.48])

fo’

Ja(z) (1- t2,-1/2cos,zt,dt,

^{for a}

^>

itfollows easily that

o ^{t-J(t)dt r()r(, 21-a + 1/2) foX sin(xt)}

forx

>

Oanda

> 1/2.

Since thezerosof

Jo(t)

areincreasingwitha[2,p.508]and

dt

J_(t)=

^cost,

J(t)= - ^sin},

wehave for

-

^<a<

^{1/2 (-1/2) <}

^jo.<,

Inaddition,Szeg6showedin

[13]

that, for

<

a

< ,

ja,v--j,v-l<r,

v=1,2,....

v=1,2,....

(2.4)

(2 5)

Combining this with(2 5)weget

jo.v+ jo.

<

2

<

j.v+3

J.,

^v=1,2, (2.6)

when <a<

1/2

Let

sin(zt)

(z, )

^dt

(1- t2)

NOTE ONA MONOTONICITY PROPERTYOFBESSEL FUNCTIONS 565 Takingimo consideration

(2.4)

^and

(2.6)

^{weseethatinorderto}prove (1.8)it sufficestoshow that

(z, a) (x

27r,

a) >

0, for ^2gTr

+

37r

-- ^<

^z

^<

^2eTr

⁺

^2r,

^e

^{1,2, (2 7)}

Similarly,

(1.9)

^{canbeobtained}by showing

(, )- (z-

2,

) <

0, for

2zr+<z<2gr+Tr,

^{t?=1,2 (2.8)}

Itisevidentthat(2.7)isequivalemto

sin()

^r

cos(yt)

t(l_t2) -dt>O

^for

2br+<y<2eTr+vr,

^e=1,2,...

which,in turn, isequivalentto

fo ^K(t)

^{eos t clt}

^>

^O,

2eTr+<y<2err+vr,

^{=1,2,... (2.9)}

whe

sn

-

O<t<y.

Ku(t)

t(Y t211/2_o,

Wehave

UKu(t)costdt ^Ku((2j 2)r +t)costdt + Ku(2eTr +

^t)costdt

3= JO

T(t)costdt ^{Ku((2e+ 1)r}

^t)costdt,

g+l)r-

(2 I0) where

T ^(t) E {K((2j- 2)7r + t) K((2j- 1)7r t)

j=l

Ku((2j- 1)Tr + t) + Ku(2jTr t)} + Ku(2eTr + t).

Weobserve thatthe function

Atu(t T ^{(t)- K((2e + 1)r- t)}

isdecreasing for

(2e + ^1)r

^y

^{< <} ,

^{since ithastheform}

zx(t) Q(t)

where

Qtu(t) Ku(t) + Z ^{{Ku(2jTr +} t)+ Ku(2jr- t)}.

Bythe lemma of this section it follows that

Qeu(t

^isadecreasing function oft,therefore

A(t)>A _g

^=0,

(e+)-<t<-, ^e=,.,....

^(2.11)

Fromthis itfollowsthat

(t)>0, 0<t<7, ^e=,,....

^(2.12)

Finally, by(2.10),

(2.11)

^and(2.12)we get

K(t)costdt ^> A(t)costdt ^>

^O,

t+l)’-/

whichgives

(2.9). By

a similarargumentweestablish

(2.8)

and complete theproofof Theorem2. [21 REFERENCES

[1]

LORCH,

L.,

MULDOON,

M.E. and

SZEGO, P,

Some monotonicity properties of Bessel functions,SIAMJ.Math. Anal.4(1973),385-392.

[2]

WATSON, G.N.,

A Treatise on the Theory

of

^{BesaelFunctions, 2nd} ed., Cambridge University

Press,

1944.

[3]

MAKAI, E.,

Ona monotonic propertyofcertain Sturm-Liouvillefunctions,ActaMath. AcadSct.

Hungar.

3(1952), 165-172.

[4]

FELDHEIM, E.,

Onthe positivity ofcertain sumsof ultraspherical polynomials,J.AnalyseMath.

11 (1963), 275-284(editedwith additional notesbyG. Szegt), alsoinG. Szeg0 collected papers vol. 3,BirkhauserBoston, 1982,821-830.

[5] BROWN, G., KOUMANDOS, S. and WANG,

K-Y.,

Positivity ofbasic sums of ultraspherical polynomials,submitted.

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ttungar.

25

(1974),387-390.

[7] ASKEY, R. and

STEINIG, J.,

Somepositive trigonometric sums, Trans. Amer. Math. Soc. 187 (1974),295-307.

[8] GASPER, G.,Positiveintegrals of Bessel functions,SIAMJ.Math. Anal. 6(1975),868-881.

[9] MISIEWlCZ, J.K. and RICHARDS, D.S.P.,Positivity of imegrals of Bessel functions, SlAMJ.

Math. Anal. 25--2(1994),596-601.

10]

COOKE,

R.G.,Gibbs’phenomenoninFourier-Bessel series andintegrals,Proc.London Math.Soc.

27(1927), 171-192.

[11] COOKE,

R.G,

A

monotonic propertyof Bessel functions,o London Math.Soc. 12(1937), ^180- 185.

[12] STE1NIG,J., Onamonotonicity property of Besselfunctions,Math.Z. 122(1971),363-365.

13]

SZEGO,

G.,Inequalitiesforthe zerosof Legendre polynomialsandrelated functions,Trans.Amer.

Math. Soc. 39(1936), ^{1-17,also in} G. Szeg0 collected papersvol. 2, BirldaauserBoston, 1982, 593-610.