ON THE RELATIVE EXTREMA
EXPRESSION
FOR BESSEL
OF THE TURAN
FUNCTIONS
BY S. K. LAKSHMANA RAO
(Depattment of Applied Mathematics, Indian Imtitute of Science, Bangalom-12)
Reccived October 27, 1960
(Communicated by Professor B. S. Madhava Rao, r.A.SC.)
SOME years ago Otto Sz• 1 showed that the Bessel function Jn (x) satisfies
the inequality
~ n (X) ~- [Jn (X)] 2 -- Jn+t (X) Jn-1 (X) ~ ¡ 1
[Jr~ (x)] 2,
n > 0, x real
(1)
wherefrom he deduced the interesting inequality
&n(x)>0
for n > 0 ,
x real.
(2)
The Tur• inequality (2) for the function Jn (x) is an immediate eonsequence
of the identity"
oo
/x,~ (x) = D4 Z
(n + 2k -}- 1)(Jn+~k+l (x)) 2
(3)
which represents Ah (x) a s a series of positive quantities whenever n > -- 1.
V. R. Thiruvenkatachar and T. S. Nanjundiah 8 obtained a new positive representation for Ah (x) by proving the identity
Ah (x) -- n +1 1 (J.
(x))~ + ~
2
(J~+~ (x))
oo
+ 2n 27 [(n + k -- 1) (n + k + 1)1-1(Jn+k (x)) z
(4)
k=2
and the above result (1) of Otto Sz•
follows from this immediately.
While the above representations for Ah (x) are quite interesting we may
also proceed to deduce the inequality (2) by considering the behaviour of
the function Ah (x) itself. This procedure leads to several questions concerning the sequences of relative maxima and minima of z~n (x) and the
* Present address: Regional Engineering College, Warangal (A.P.), India.
239
S. K. LAKSHMANA RAO
240
airo of the present note is the study of these sequences. We prove in particular (Theorem 4 below) that the r-th relative maximum of •n (x) is targer
than the r-th relative maximum o f ~n+l (x), for any fixed value of r. This
answers the question first raised by John Todd for the classical orthogonal
polynomials and settled for the Hermite function Hn (x) by Otto Sz• 4 and
for the Tur• expression for the Hermite function by the author. 5
TI-IEOKEM 1: When n > 0, the relative maxima of Ah (x) occur at
the zeros o f Jr~-~ (x) and the relative minima occur at the zeros of Jn+l (x).
On differentiating the expression
(5)
A h (x) = (Jn (x)) ~ -- Jn+a (x) J,t-~ (x)
twice, we get
d
2
(6)
d-x /Xn (X) = X Jrt+l (X) Jn-1 (x)
and
a"z
2
d
2
d
d x 2 / x , n (x) = x Jn+l (x) ~ Jn-1 (x) -4- x Jrt-1 (x) dx Jn+l (x)
2
-- ~x Jn+l (x) Jn-1 (x).
(7)
Let 0 < Xl, s < X2,s < .. 9 denote the zeros of Js (x) in the ascending order.
From (7) we see that
= - - 4n
= 4n
Jn ( x )
{('
x Jn ( x )
)'}
X=Zr, n-1 < 0
x= Xr, n+l > 0 .
Hence the theorem.
The statement in the above theorem is reversed if n < 0.
follows we assume that n > 0.
In what
If we denote the successive relative maxima of A h ( X ) by M i , n,
M2,n, 99 9 and the successive relative minima by m i , n, m~,n . . . . respectively,
we have
Mr, n = /hn (Xr, n-0 = (Jn (Xr, n - 0 ) ~,
(8)
Relative Extrema o f Tur•
Expression for Bessel Functions
241
and
mr,n = A h (Xr,n+l) = (Jn (Xr,n+l)) ~.
(9)
We may also observe incidentally that
mr, n-1 = (Jn-1 (Xr,n)) 2 = (Jn+l (xr, n)) 2 = Mr, r91
Since the quantities Mr, n and mr,n are positive for all values of n, it
follows that
A ~ (x) > 0
which is Tur•
(10)
inequality for the Bessel function Jn (x).
THEOI',EM 2: The sequences of relative maxima and relative minima
of Ah (x) are decreasing beyond a certain value of r. More specifically
Mr, n > Mr+l,n
if
Xr, n-1 > ~ = v'2n (n -- 2)
and
mr, n > mr+x,n if
IfO<n<2,
Xr, n+l > ~ = a/2n (n q- 2).
we define ~ = 0 .
Consider the function
f (x) -= A h (x) -1- 2n
x~ (J,-1
(x))..
Then, by differentiation we get
2
4n
4n
d
f ' (X) = X Jn+l (x) Jn-1 (x) -- ~ (Jn-1 (x)) ~ -4- x ~ Jn-1 (x) dx Jn-1 (x)
_
2 (2n (n -- 2) -- x ~) ( J n - , (x)) ~
so that f ( x ) is increasing in 0 < x < ~ and decreasing in x > ~. Also at
the relative maxima of Z,n (x), f ( x ) = A h (x) and hence the assertion that
Mr,n > Mr+a,n if Xr+a,n-~ > Xr,n-a > ~. To prove the corresponding
result for the relative minimum, consider the function
g (x) = zx~ (x) -
2n (j,+, (x)F.
Differentiating this we have
g' (x) = X2~ (2n (n -4- 2) -- x ~) (Jn+~ (x)) ~
242
S . K . LAKSItMANA17~O
so that g (x) is decreasŸ in x > 7/. Since g (x) coincides with 91 (x) at all
the relative mŸ
of Ah(x), it follows that mr, n > mr+l,n if xr+l,n+l >
Xr,n+l ~ 7.
This completes the p r o o f of the theorem.
THEOgEM 3: The r-th relative m a x i m u m of A h (x) is greater than
the r-th relative mŸ
i.e., Mr, n > mr, n.
We have A n (Xr, n-1) = Mr, n and A h (xr, n+i) = mr, n and (d/dx) A n (x)
= (2/X)Jn+i (x) Jn-i (x). F r o m the interlacing properties of the zeros of
Bessel functions we know that 6, 7
Xr-1, n+l ~
X~',n-1 ~ X~. ?~ ~ Xr,n+i ~ X r + l , 7/.--1"
Hence in the interval (Xr, r,-i, Xr, n+l), Sgn (d/dx) &n (x) = (--1) zr-x = -- 1,
so that Ah (x) is decreasing in the interval and &r~ (xr, n-~) > A h (xr,~+O
or Mr, n > mr, n which proves the theorem.
By employing the functions f ( x ) and g (x) already introduced, we can
show further that
[
8~~
Mr, n > ~1 + x4r, n+l/ mr, n
if Xr, n-1 > r
and
Mr, n > (1
8n a ,~-1 mr, n
x4~,¡
if Xr, n-i > ~7.
THEOREM 4: For a fixed value of r, the r-th relative maxima of A h (x)
forro a sequence of decreasing functions of n, i.e., M r , n > Mr, n+l. We
have already noticed that /~n (Xr, n - 1 ) = Mr, n. We may see easily that
Mr, n+~ = An+i (xr, n) = &r, (xr, n). Also from the relation (d/dx) A h (x)
= (2/x) Jn+i (x) Jn-1 (x), ir follows that in the interval (xr, n-x, Xr, n),
Sgn (d/dx)/~ n (x) ---- -- 1, so that Ar~ (x) is decreasing in the interval and
Ah (Xr, n-1) > Ar~ (Xr, n). Whence the theorem.
Analogously for the relative minima, we have
THEOREM 5 " For a fixed value of r, the r-th relative minima of A h (x)
f o r m a sequence of decreasing functions of n, i.e., mr, n > mr, n+a.
We have noticed earlier that mr, n-~ = Mr, n+l. Combining this with
the result of the previous theorem, we see that mr, n-i ]> mr, n which proves
the assertion.
Relative Extrema of Tur• Expression for Bessel Functions
243
R~FZ~NC~
1. Otto Sz•
..
"Inequalities concerning ultraspherical polynomials and
Besse! functions," Proceeding~ of the American Mathematical Society~ 1950, 1, 256-67.
2.
..
A Treatise on the Theory of Bessel Functions, 2nd EditJon,
Cambridge, 1948, p. 152.
Watson, G. N.
3. Thiruvenkatachar, V. R.
and Nanjundiah, T. S.
"Inequalities concerning ttessel functions and orthogonal
polynomials," Proc. Ind. Acad. Sci., 1951, 33 A, 373-84.
4.
Otto Sz•
5.
Lakshmana Rao, S. K.
6.
Watson, G. N.
7.
Gray, A.: Mathews, G. 13.
and MaeR.obert. T. M,
A$
..
..
..
" O n the relativo extrema of the Hermite orthogonal functions,"
Jour. lnd. Math. Soc. (N.S.), 1951, 15, 129-34.
" O n the relativo extrema of the Tur• expression for the
general Herrnite function," Indian Instituto of Science
Golden Jubilee Reseatch Volume, 1959, pp. 200-04.
" R e Ÿ (2) above, p. 480.
A Treatise on Bessel Functions, 1952, p. 242.