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.