Some Monotonicity Properties of Bessel Functions

SIAM J. MATH. ANAL.Vol. 4, No. 2, May 1973

SOME MONOTONICITY PROPERTIES OF BESSEL FUNCTIONS*

LEE LORCH?, M. E. MULDOONf AND PETER SZEGO:

Abstract. It is proved that the sequence

rg(t dtk=

is decreasing for all v, for < < -, and for suitable u, where rgv(t is an arbitrary Bessel function oforder and Cvk its kth positive zero. This subsumes and unifies results obtained by G. Szeg6 and R. G.Cooke, extending and sharpening some. For one of his results Szeg0 used a Sturm comparison theoremwhich is shown here to permit the requisite generalization and to incorporate and extend other resultsoriginally proved by quite different methods. Auxiliary results are derived. Various remarks are col-lected in the final section.

1. Introduction and results. G. Szeg6 has proved I2, p. 104] that the sequenceof areas

(1) ICC(t)l atk

under the successive arches of an arbitrary Bessel (cylinder) function rgv(t), withkth positive zero Cvk, form a decreasing sequence when is selected properly(see Corollary 3 below) and, using a different approach, that [5, p. 281, (19)]

j=,2k

(2) t-J(t) dt > O, k 2, 3,4,...,0

where J(t) is the Bessel function of the first kind, Jk is its kth positive zero, andis the unique value satisfying

(3) t-J(t) dt O.

He showed that -1/2 < =< 0 and mentioned that D. R. Snow had computed eto be -0.2693885

Using the Sonin integral [5, p. 279, (12)], [11, p. 373, (1)] as indicated bySzeg6 [5, p. 280, (16) if.I, it can be shown that (2) remains valid when e is replacedby any larger value, and consequently that

(2’) t-vJ(t)dt > O, z > O, v > .The inequalities (2) and (2’), with accompanying discussion and inferences,

are found in the Notes which G. Szeg6 appended to a posthumous [5, p. 275, firstfootnote] paper of Ervin Feldheim in the course of preparing it for publication.

* Received by the editors February 8, 1972, and in revised form April 17, 1972. This research wassupported by the National Research Council of Canada.

" Department of Mathematics, York University, Downsview 463, Ontario, Canada.

:l: Research Department, Ampex Corporation, Redwood City, California 94063.

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386 LEE LORCH, M. E. MULDOON AND PETER SZEGO

Our purpose here is to connect the Szeg6 results with one another so thateach emerges from the method he used to establish the first; his own proof of (2)follows a quite different line of reasoning. This unification yields (Corollary 2)a sharper and more general formulation for (2’). Also, a lemma due to R. G. Cooke[3, p. 282] can be subsumed and generalized by this approach (Corollary 4).Remarks on these results are collected in 6; 2, 3, 4, 5 are devoted to proofs.

The Szeg6 and Cooke results are contained in the following.THEOREM 1. For all v, andfor -o < < -, the sequence

(4) if- llv(t)l dtk=

is decreasing, where is the smallest integer such that

1/212(7 9v2)/(3 27)] /, 0 < < 23-, Ivl(5) cv __> 2(v;,)(0 otherwise.

Thus, when Ivl >= 1/2 or when <= O.If O < < 1, then <= 2 and when Ivl >= 1/2. If cv(t Jr(t) (still with

/=< 1), then for any v.The final assertion in the theorem can be generalized. This is recorded

separately as Corollary 1 because its proof is rather more tedious (and is based onless elementary arguments) than the other proofs required here and we do notwish it to obscure the main line of argument. The extension is not required,for example, for our proof of Szeg6’s positivity results (2) and (2’), nor for ourgeneralization of Cooke’s lemma (Corollary 4).

COROLLARY 1. If? <= and c >= Yvl (for example, when cg(t) =- Yv(t)), thenin Theorem 1 even for the remaining range Iv[ < 1/2’.

Here y denotes the first positive zero of Y(t), the Bessel function of the.,

second kind.That this corollary does subsume the final sentence of the theorem follows

from the inequalityj > yx valid for v > -1/2 (see [7 (a), p. 364, (i) for v >__ 0, andCorollary, p. 366, for 0 > v >-1/2 (after replacing v by -v), wherebyJr1 > Y-,I > Yv], since y increases with v > -1/2 [11, p. 508, (3)]).

Upon choosing ,- 1- v, Theorem becomes (in view of (3))a sharperand more general version of (2’). The precise result is as follows.

COROLLARY 2. For v > --1/2, the sequence

(6) t-lcg(t)l at

is decreasing, where is the smallest integer such that

(7) [0 /y v>=1/4.Here <= 2 for all v > -1/2, and 1 when v >_ 1/4 or when cgv(t J(t). TheSzeg6 inequalities (2) and (2’)follow on choosing v a and c(t) J(t) in (6).

In Corollary 2, also for %(0 Y(t) (or, more generally, for any%(0 for which c1 >- yvl), at least when v =>/, where/ is defined by the equationy, 2(/; 1 -#). The root kt is unique, since y increases for v > -1/2 while

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MONOTONICITY PROPERTIES OF BESSEL FUNCTIONS 387

2(v;1 v), as defined by (7), decreases. Moreover, -1/2 </ < 0, since 2(-1/2;)x/q > .38 > y_ 1/3.1 (the last inequality is obtained from [11, p. 714]) and

2(0, 1)= 1/2x/} < .89 < Yol [11, p. 748].Using the York University computer facilities, Dr. Marian Shepherd has

calculated p to be -0.1866..., where there is uncertainty about the last digit.We thank her most cordially.

The Szeg6 result on (1) [2, p. 104] follows on choosing 7 in the theorem;we have added only the two last sentences in the next corollary.

COROLLARY 3. For all v, the sequence of areas (1) is decreasing, where is thesmallest integer such that

cv > 2(v;1) 1/212(1-- 9v2)] /2 if ,v,<1/2,8) 0 if Ivl _-> 1/2,Here < 2. When Ivl >_-1/2, or when (for Ivl < 1/2),ca > y (for example, ifcv(t) J(t) or c(t) Y(t)), then rc 1.

The following result reduces to the Cooke lemma I3, p. 282] in the specialcase (t) J(t); it is obtained from Theorem on putting 7 2 v. (Cf. 6 (xi)for a further generalization.)

COROLLARY 4. If v > 1/2, then

(9) - lCg(t)l dtk=

is a decreasing sequence.In view especially of Corollary 1, it is of some interest to characterize those

nontrivial Bessel functions

v(t) AJ(t) + B Y(t) Ivl < -3for which cx >= Yvl in the following way.THEOREM 2. For 0 =< v < l, c >= yv when and only when AB <0; for

-1/2 < v < 0, c >= y if and only if A 0 or B/A <= -tan vrr. (It is assumedthroughout that A and B are not both zero.)

In the latter case (-1/2 < v < 0), tan vrr < 0, so that c1 => y whenever Aand B have opposite signs, or one is zero, and also for certain values in which theyhave the same sign, such as J_ /,(t) + Y_ 1/4(t) which equals x/Jx/4(t). A changeof behavior occurs at v -3.

2. Proof of Theorem 1. This is based on an application of a Sturm-typelemma, formulated by G. N. Watson [11, p. 518], sharpened and applied ingreater detail by E. Makai [8], to the differential equation

(o) y" + o(x)y o,where q(x) is monotonic. In particular, this lemma states that the areas undersuccessive arches of the graph of y(x) decrease (increase) when q(x) increases(decreases).

Here, as in [2, p. 104], we choose

(11) o(x) 4x-(1 4f12v2 + 4flZxZt)x-2; y(x) x’/Zcgv(xt), fl > 1.

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Regardless of the value of v, qg(x) is an increasing function for all x > 0 forwhich

(12) x2 > 1/4(1 4f12v2)f1-2(fl 1)-

Hence, for such x, the areas bounded by successive arches of y(x) form adecreasing sequence. Now with xa,

fCv,kly(x)l dx - - l%(t)l dr,k Cvk

,,1/ is the kth positive zero ofwhere 7 3/(2/) and xWe note that 0 < 7 < , since oe >/ > 1. Conversely, given any 7, 0 < 7 < -,

there is a unique/ > 1, namely/ 3/(27), which can be used in (11). Substitutingthis value for/ in (12) yields (5) and the main part of the theorem is proved for0 < <-}.

If 7 <- 0, pick such that 0 _< 2(v; 6) < cvl, 0 < 6 < 23-, and write

fcTV,kt- (t) at e-{t- (t)} ,it,Cvk

k 1,2,.... Here 7- 3 < 0 so that - is a positive decreasing function.Hence the second mean value theorem shows that the integral on the left, beingequal to the one on the right, has the same sign as

- Ic(t) dt, k 1,2,

and the principal assertion of the theorem is proved for all 7 =< 0 as well as for0 < <-}.

The proof shows that z for all %(0 when 7 < 0, in conformity withdefinition (5) of 2(v 7).

To prove the assertions concerning the range 0 < 7 =< 1, it is sufficient toconsider 7 1, since 2(v; 7) -< 2(v; 1), 0 < 7 =< 1, for all v.

That z =< 2 for such 7 follows from the inequalities 2(v 7) =< 2(v; 1) N -,, <and c2 > [6, p. 1254, Remark (i)], val.id for all v.

When Ivl _>- , (5) requires 2(v; 1) 0 and so z 1 for such v and 0 < 7 -<_ 1.Thus, only the remaining range Ivl < 1/2 need be considered for the final case

in which cg(t) J,,(t), with, still, 0 < 7 -< 1. Here j > j_ /2, 1/2n > 1, sincejv increases with v for v >-1 [11, p. 508], so that jl > 2(v;1)>= 2(v;7),0<7_<1.

The theorem is proved.

3. Proof of Corollary 1. Without loss of generality it can be assumed that%(0 > 0 for 0 < < y, since c => yv. Using again the observation, alreadymade in the course of proving the parts of Theorem 1 concerned with the range0 < 7 =< 1, that 2(v; 7) -< 2(v 1) for 0 < 7 =< 1, it follows that n ify > 2(v 1).

Hence, it suffices to show that

(13) Yvl >= -}[2(1 9v2)] 1/2, --} < v < -.

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Now [11, p. 748], Yol 0.89... > 1/2x/ and Yvl is an increasing function of vfor v > -1/2 [11, pp. 508-509], so that (13) holds for 0 =< v < 1/2. Since Yvl increaseswith v and the right member of (13) is even, this can subsumed in the substantiallymore complicated proof for 0 > v > -1/2 which follows.

From [11, p. 714] it is clear that

.38 > y_1/3,1 > .36 > 1/2.Moreover, Yol > 98-. Writing y(v) Y,,1, we have [11, p. 508, (3)]

d2 fodv2 log y(v)] 4 K(2y(v) sinh t)y’(v)(sinh t) e- 2 dt

-4 Ko(2y(v sinh t)te- 2 dt < O,

since Ko(t) > 0 and K’o(t) < 0 for 0 < < oe. Thus, the increasing functionlog y(v) is concave down on the interval -1/2 < v < oe. The respective unattainedlower bounds 1/2 and - for y(- 1/2) and y(O) show that

log y(v)> 3v log-} + log 98-, -1/2 < v < 0.

The result (13) will follow, for this range of v, if

3v log -} + log ] => -log 3 + 1/2 log 2 + 1/2 log (1 9v2),

This is equivalent to

(64/9)3+1 => 2(1 9v2),

which, in turn, follows from the sequence of inequalities

--}<v<=O.

-<=v<=O,

(64/9)3v+ >__ 73v+ et3+ 1)log7 >__ e(log 7)(3v + 1)

=> 4(3v + 1) >= 2(1 9v2), -1/2__< v__<0.

This completes the proof of (13) and that of Corollary 1.

4. Proofs of Corollaries 2, 3, 4. Corollary 2, except for the last two sentences,follows on specializing 7 to be 1 v in (4) and (5); this gives (6) and (7) respectively.

To show that z =< 2, it suffices to recall that cv2 > 1, all v [6, p. 1254, Remark (i)]since 2(v;1 v) < 1, -1/2 < v < 1/4.

To show that z when (t) J(t) we recall that Jl increases with v forv > 1 [11, p. 508]. Thus, from (7),

Jvl > J-1/2,1 1/2g > 1 > /],(V 1 v), V> --.The Szeg6 inequality follows as described on pairing arches. Corollary 3

follows from Theorem and Corollary 1 on putting 7 1.Corollary 4 follows on putting 7 2- v in the theorem and noting that

(5) gives

2(v;2 v)= 0, v > 1/2.

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5. Proof of Theorem 2. We may put A without loss of generality, thecase A 0 being trivial. The Wronskian of f(t), g(t) is defined, as usual, to be/(f, g; t) f(t)g’(t) f’(t)g(t). From [11, p. 76, (1)], with v(t) Jv(t) + BY(t),

2(e, Y;t)= (L, Y;t)= > 0, > 0.

7tt

The definition (for noninteger v)

Y(t) (csc v)(cos v)J(t) J_

implies (again for noninteger v)

(t) (1 + B cot vrc)Jv(t) (B csc vrc)J_ (t).

With the familiar asymptotic equation

J(t)[2F(1 +v)]-lV, v> -1, ast0+

in mind, we divide the proof of Theorem 2 into four parts.(i) If O <= v < 1 and B <= O, then cvl > YI.For B 0 the result is known [7(a), p. 364, (i)].IfB < 0, then (0) + m, since Y(0) oe. Furthermore, 0 < //(cg, Yv; cv)

-c’(c)Y(c), whence Y(c) > 0 and cvl > YI as asserted.(ii) If-1/2 < v < 0 and B <= -tan vrt, then again c > Yv.Here (t)> 0, 0 < < c1, and the reasoning of (i) applies to yield the

desired result.(iii) IfO <= v < 1 and B > O, then yv > Cv.Here (t) < 0, 0 < < c, with 0 < ///(, Y; y)= cv(y)y,(y). Thus,

(Yvl) > 0 and y > c.(iv) If -1/2 < v < 0 and B > -tan vn, then again y > c.As in (iii), Cgv(t < 0, 0 < < cvl, c(y) > 0, so that yx > c1.

6. Remarks. (i) In Theorem 1, the range of cannot be extended. Forthe areas become equal when Ivl 1/2 and increase [8] when Ivl < 1/2. For the samereason v cannot be extended in Corollary 2 to the value v -1/2, nor in Corollary4 to the value v 1/2.

(ii) Szeg6’s monotonicity result on areas (Corollary 3) cannot, in general,be extended to include the area of the first arch, that is, the one beginning at 0and terminating at Cvl (a relevant question when 1), since [11, p. 394, (8)]

f Yo(t) at o.

For this case (v 0, Cdo(t _= Yo(t)), we have n 1, since Yo .89... >2(0;1). Here the first arch bounds a smaller area than does the second arch.

This is true also for Y(t), - < v < 0, as can be inferred from Ill, p. 394, (7)],together with Corollary 1.

(iii) When cdv(t) _= J(t), v > -1, on the contrary, the first arch does beginthe sequence of arches with decreasing areas. This was proved by R. G. Cooke [4].The example in (ii) shows also that Cooke’s theorem cannot be extended fromJ,(t) to arbitrary cd,(t), not even to Y(t).

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(iv) A particularly simple proof of Cooke’s theorem [4] has been devised byJ. Steinig [9].

For the range Ivl > 1/2, E. Makai proved [8] results of Cooke type [4] in a moregeneral setting, in a very neat way, using the differential equation method employedhere in 2.

(v) There is a misprint in 2, p. 104] in describing Szeg6’s result on (1). There,the value of ,(v" 1) in (8) is printed with the square root replaced by the cuberoot. This slip was noticed also by J. Steinig, as we learned from correspondencewith him.

(vi) Theorem 1 and Corollaries 1, 2 and 3 give conditions under which< 2 in (4). However, for Ivl < 1/2, can be arbitrarily large, when 7 (<) is

sufficiently close to 23-. More precisely, given and v, where Ivl < 1/2, by choosing7 < properly we can have at least the first elements of the sequence (4) increasebefore the sequence begins to decrease. This is a consequence of the Sturmianlemma used in 2. All that is needed is to pick/ in (11) so that qg(x) is a decreasingfunction for 0 < x < cv, for example, by choosing so that, in (5), 2(v; 7) cv,Ivl <

(vii) R. Askey, in the final paragraph of [1], advances an interesting conjecturerelated to (2’) and (3), by allowing the exponent of the factor t-1 in the integrandto differ from the order of J,(t).

(viii) J. Steinig utilizes integrals similar to those occurring in (4) in his studyof the sign of Lommel functions [10].

(ix) The differential equation (10), with qg(x) defined by (11), can be used alsoto prove Theorem 5.4 of [6, p. 1253]. Doing so eliminates the need to separatethe cases Ivl >= 1/2 and =< Ivl < 1/2 (as was done in [6]), since 0’(x) is completelymonotonic, 0 < x < , for Ivl _-> 1/2 when/ 23-.

(x) Theorem 5.4 [6, p. 1253] makes more precise the. result of Corollary 3when Ivl >= 1/2 since that theorem shows, for such v, that the sequence ofareas iscompletely monotonic. A similar partial (in v) extension of Theorem 1 of this notecan be effectuated by the method of proof of Theorem 5.4 of [6].

(xi) Corollary 4 (Cooke) can be strengthened similarly [6, Theorem 5.1,p. 1251; Theorem 5.2, p. 1252]. The sequence (9) is completely monotonic, notmerely decreasing.

Acknowledgment. The authors wish to thank the referee whose careful read-ing led to the clarification of several points. Similar thanks go to Professor J.Steinig.

REFERENCES

Ill RICHARI ASKer, Positive Jacobi polynomial sums, T6hoku Math. J., 24 (1972), pp. 109-119.[2] I. BIHARI, Oscillation and monotonicity theorems concerning non-linear differential equations of the

second order, Acta Math. Acad. Sci. Hungar., 9 (1958), pp. 83-104.[3] RICHAID G. COOKZ, On the sign of Lommel’s function, J. London Math. Soc., 7 (1932), pp. 281-

283.[4] --, A monotonic property of Besselfunctions, Ibid., 12 (1937), pp. 180-185.[5] EvIN FZIX)IIM, On the positivity ofcertain sums ofultrasphericalpolynomials, J. Analyse Math.,

11 (1963), pp. 275-284 (edited with additional notes by G. Szeg6, pp. 279-284).E6] Lzz LORCH, M. E. MULDOON AND PZa’R SZGO, Higher monotonicity properties ofcertain Sturm-

Liouvillefunctions. III, Canad. J. Math., 22 (1970), pp. 1238-1265.

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[7] LEE LORCH AND DONALD J. NEWMAN, (a) A supplement to the Sturm separation theorem, withapplications, Amer. Math. Monthly, 72 (1965), pp. 359-366; (b) Acknowledgment ofpriority,Ibid., 72 (1965), p. 980.

[8] E. MAKAI, On a monotonic property of certain Sturm-Liouville functions, Acta Math. Acad. Sci.Hungar., 3 (1952), pp. 165-172.

[9] JOHN STEINIG, On a monotonicity property ofBesselfunctions, Math. Z., 122 (1971), pp. 363-365.[10], The sign of Lommel’sfunction, Trans. Amer. Math. Soc., 163 (1972), pp. 123-129.[1 l] G. N. WATSON, A Treatise on the Theory ofBessel Functions, 2nd ed., Cambridge University Press,

Cambridge, 1944.

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Documents

Some Monotonicity Properties of Bessel Functions