on turan's inequality for ultra- spherical ......d2anx)(x)/dx2 we prove the concavity of...

ON TURAN'S INEQUALITY FOR ULTRA-SPHERICAL POLYNOMIALS

K. VENKATACHALIENGAR and s. K. LAKSHMANA RAO

0. Introduction. Some years ago P. Turan noticed the interesting

inequality

(0.1) An(X) = [Pn(x)]2 - Pn+l(x)Pn-l(x) £ 0, » £ 1, | X \ g 1,

where P„(x) is the Legendre polynomial of order n and the above

inequality has been extended in recent years to the case of some other

orthogonal polynomials as well as the ordinary and modified Bessel

functions [l; 2]. B. S. Madhava Rao and V. R. Thiruvenkatachar

[3] deepened the inequality (0.1) by showing that

d2 2 T d I2(0.2) — A„(x) = - - Pnix) Z 0.

dx2 n(n + 1) Ldx J

The inequality (0.1) has also been generalized to the case of ultra-

spherical polynomials P£M(x) in the form

(0.3) A,.x(«) = [FnAx)]2 - Fn+i.x(x)Fn-i,x(x) ^ or < 0,

according as \x} ^ or >1, where £n,x(x) =PnMix)/Pnx\l). V. R.

Thiruvenkatachar and T. S. Nanjundiah [4] and A. E. Danese [5]

have obtained series of positive functions for A„(x) and its analogue

in the case of the ultraspherical, Laguerre and Hermite polynomials.

In the present paper we deepen the above results on ultraspherical

polynomials by determining the signs of the first and second deriva-

tives of AnX)ix)=[PnX)ix)]2-Pn+1ix)Pnx!.lix) for various values of X

and x. We notice first of all that dAnX)ix)/dx can be represented

as a numerical multiple of the Wronskian Pn+iix)dPnxllix)/dx

—Pn-i(x)idPn+lix)/dx) and express it also via the Christoffel-

Darboux formula as a series of functions of constant sign. The sign

of dA!nX)ix)/dx for various values of X and x follows easily from this.

We determine the sign of (X — l)xidA„)ix)/dx) for a general value of

n and X and show that id2/dx2)AnX)ix) ^0 according as X>1 or

1/2 ^X<1, thus extending the result of B. S. Madhava Rao and

V. R. Thiruvenkatachar to the case of ultraspherical polynomials.

For integer values of n we obtain a series of positive functions for

id2/dx2)Anx\x). Next we show that (1-x2)AlM(x) decreases steadily

in (0, 1) when KX^3/2. From the signs of dA^ix)/dx,

Received by the editors August 9, 1956 and, in revised form, April 1, 1957.

1075License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

1076 K. VENKATACHALIENGAR AND S. K. LAKSHMANA RAO [December

d2AnX)(x)/dx2 we prove the concavity of A„,x(x) for 0 <X^ 1/2 and also

arrive at inequalities for An,\(x) in the range \x\ :£1, which are not

only simpler but also cover a wider range of X than the corresponding

inequalities given by O. Szasz [2]. Finally we determine an integral

representation for An(x).

1. Expressing dA^\x)/dx as a Wronskian. We append below the

well known relations satisfied by the ultraspherical polynomials [6]

for frequent use in the following.

d2y dy m(1.1) (1 - x2) — - (2X 4- l)x— + 11(11 + 2X)y = 0, y = P„ (x)

dx2 dx

(1.2) nPn\x) - 2(m + X - l)xPnX-i(x) + (n + 2X - 2)Pn-i(x) = 0,

w = 2

(1.3) -P»' (*) = /(r(X + r)/r(X))P(„"!;) (*), r ^ ndxr

ex) d (\) o (X)(1.4) nPn (x) = x—Pn (x)-Pn-i(x),

dx dx

(X) d (x) d a)(1.5) (n + 2X)Pn (X) = — Pn+l(x) - X — Pn (x),

dx dx

(xi d <x) d (x>(1.6) 2(n + X)P„ (x) = — Pn+i(x)-P„-i(x),

dx dx

(1.7) (1 - x2) — Pn\x) = - nxPT(x) + (n+2\- l)P»_i(*),

(1.8) (1 - x2) — Pn\x) = (» + 2X)*P,<iX>(.T) - (« + l)P„(+i(*).

Differentiation of

(1.9) A? \x) = [PlX,(x)]2 - P%i(x)P?2i(x)

yields

(1.10) — An\x) = af\x) - Bn\x)dx

where

„M(.) = P(:\x) ± P?(X) - PZ(X) y Pn+\(X)dx dx

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

i957l ULTRASPHERICAL POLYNOMIALS 1077

and

»). , rx) . , d (X) (X) d <x>8n (X) = Pn+l(x) — Pn-l(x) - Pn (x) — Pn (x).

dx dx

We have then

Pn+i(x) Pn-i(x)

(1.11) an (x)+8n (x) = d (x) d (X) .— Pn+l(x) — Pn-l(x)dx dx

Also

nan (x) — (n + 2X)/3„ (x)

= 2(n + \)PnX\x) — Pn\x) - nPnli(x) — P™i(x)dx dx

(X) d (x)- (n + 2\)Pn+i(x) — P„_i(*)

dx

= 2(« + X)P^X)(^) - Pn\x) - (n +2\- l)P™i(x) - Pn+\(X)dx dx

. (X) , „ d (x> , s- (n + l)Pn+i(x) — Pn-i(x)

dx

- (2X - 1) \Pn+\(x) — Pn-l(x) ~ Pnll(X) — P„+\(x)\ .{ dx dx )

The first three terms together are found to vanish on using (1.4),

(1.5), (1.6) and we get

nan\x) - (n + 2\)8n\x) = - (2X - 1)[«?'(*) + Bn\x)\,

so that there follows the relation

(1.12) (n + 2X - l)aT(x) = (n+ l)^X>(x).

Solving (1.11) and (1.12) for anX)(x), 8„)(x) and using (1.10), we have

Pn+l(x) Pn~l(x)d (X) 1 — X

(1.13) — An (x) = ——- d (x) d a, .dx n + X — Pn+i(x) — P„_i(at)

ax ax


1078 K. VENKA'I ACHALIENGAR AND S. K. LAKSHMANA RAO [December

From (1.3) and (1.13) we can also express the first derivative of

(X) r dr (M ~12 dr (>,) dr (xi

Ankx) m —- Pn \X) - — PUl(x) — Pnll(x)l_dxr J dxr dxr

as a constant times the Wronskian of P„_+r+i(x) and P^X-iix).

2. Series of functions of constant sign for dAnx\x)/dx. Using (1.2)

in succession we can write

(2.1) Pn+\(X) = anXPnll(x) + KP^x) + CnP^x),

where

4(« + X)(w + X- 1)an = -'

nVn + 1)(2.2)

(« + X)(»+2X-2)(» + 2\-3)

nin + 1)(» + X - 2)

Hence the determinant

t/ ^ P„+\(x) P^ix)Ux, y) = ex) (x)

Pn+iiy) Pn-iiy)

can be written in the form

anX Pn-lix) + bnPn-lix) + CnPn-iix) Pn-\(x)

anyPnliiy) + bnPn-i(y) + cnPn-ziy) P™i(y)

.2 2 (X) (X)= an(x — y )Pn-i(x)Pn-i(y) — cnon-i(x, y)

and so we get2 2 (X) (X)

(2.3) Bnix, y) + cjn-i(x, y) = o„(x - y )Pn_1(x)Pn_1(y).

On observing that

PB+i(*) Pn-i(x)Kix^ti

v-*x y - x — Pn+iix) — P»_i(x)dx dx

we obtain from (2.3), the relation

(n + X) d (X) n-2 + \ d a) m-A„ (x) + cn- A„_2(x) = - 2aBx[PB_i(x)J2,

1 — X dx 1 — X dx


19571 ULTRASPHERICAL POLYNOMIALS 1079

which can be rewritten in the form

m^ ''.Wn (»+2X-2)(« + 2X-3) d (X)(2.4) — A„ (x)-——- — An_2(x)

dx n(n + 1) dx

8(n + X - 1) r Cx) l2= - \ ^n (1 - \)x[Pn2i(x)]\

n(n + 1)

Solving this difference equation for dA(n)(x)/dx we obtain

d (X), N 8(X- l)r(n+2X- 1)— A„ (x) = -xdx (n + 1)!

[(-+0/21 (n-2k+ 1)!(» - 2k + X 4- 1) w ,

' ^ -w-,lxii± n- l^»-»+i(*)J Ii_i r(» — 2k + 2X + 1)

which is the desired series expansion. Differentiation of (2.5) gives a

series for (d2/dx2)Anx\x) which has however no particularly elegant

form. In the case of Legendre polynomials (X = l/2), (2.5) becomes

& —2x K»+l>/2]

— An(x) = ——— Y (2n - U + 3)[Pn-2k+i(x)]2.dx n(n + 1) t»i

Differentiating this we see that

d2

— 2 I(n+l)/2]

=- E (2» - Ik + 3)P„_24+i(x)[P„-tt+i(*) + 2xP^2*+,(x)]n(n + 1) t-i

which is the same as identity (0.2). From (2.5) it follows that

(2.6) sgn | — a1X>(x)1 = sgn (X - 1)* (X > 0)

which amounts to the following property of Anx'(x).

The Turdn expression AjX)(x) is an increasing (a decreasing) function

for positive (negative) values of x when X > 1. This is to be reversed when

0<X<1.From this we see that

when X > 1, A„ (x) S AB (0) > 0 for all values of x;

when 1/2 ^ X < 1, a1X>(x) S A?°(l) SO for | x| g 1;

when 0 < X g 1/2, C\x) ^ A^\l) ^ 0 for | *| gj 1.



3. Sign of (X— l)xdA^ (x) / dx for a general index n. For a general

index n (which will be restricted in a way later on) we take P„\x) as

a solution of the differential equation (1.1) which remains regular at

x= +1 (or at x= — 1) either of which is a regular singular point of

the differential equation. We stipulate that the value of the function

as well as its slope at x= +1 are positive. Defining Aj,X)(x) as in (1.9)

we can again derive the relation (1.13) as before. Multiplying this

last relation by (1— x2)XH/2 and differentiating the result, we get

^-\a - xTm t a:\x)]dx L dx J

1 _ X Pn+l(x) Pn-l(x)

= ^r~X~ 1[(1_,Y^P^m] ±\(l-xT1,2yP^(x)VdxL dx J dx\_ dx J

Simplifying the second row of the determinant by means of the

differential equation we are led to

d Y 2 X+l/2 d (X) "I— (1 - x) — An (x)

(3.1) dxl dx J

= 4(l-X)(l-xY-1/2P„X+>1(x)PBX_)1(x).

Hence the extrema of (1 —x2)x+ll2dAnX)(x)/dx occur only at the zeros

of Pn+i(x), Pnxii(x). The solution P„X)(x) as defined above cannot have

zeros on the real segment (1, oo). It increases from a positive value

at x= 1. If it attains a maximum value at a point Xi (> 1), we have

the conditions (dy/dx)xi = 0, (d2y/dx2)zl<Q and from the differential

equation we notice a contradiction if ra(w + 2X) is positive. Hence

P„X)(x) is increasing in x> 1 and consequently does not vanish for any

x on (1, oc). Let us denote by

(a): C*l, 6*2, «3, • • •

03): 0,, 0,, ft, ■ • •

the zeros in ( —1, +1) of P„+i(x) and Pfii(x) respectively. From

(1.13) we have

sgn — An (x) \ = sgn —— PB_i(x) — PB+i(x)LdX JI=a LW + X dX Jx=a

From (1.7) we may write

(I -a2) T^- Pn+\(x) 1 = (n + 2X) PnX\a)LdX J x=a



and hence

T d (X,, 1 T(X - 1)(» + 2\)PHX!i(a)Pn)(a)lsgn — A„ (x) = sgn -- .

Lax Jx=a L (w + X)(l - a2) J

Using the relation 2(n+\)aPnX)(a) = (n + 2\-l)Plnx±1(a) which follows

from (1.2), we get

d (X) r nsgn — A„ (x) = sgn [(X - l)(n 4- 2\)(n + 2X - l)/a].

In a similar way we observe that

sgn [ — Af'(x)l = sgn [(X - l)nft/(» + !)]•Lax JI=,(j

Combining the above two facts we see that whenever n S1 and

X>0

[d >\) "Ix—A„ (x) = sgn (X - 1).

dx J z=a,S

Thus at all possible extrema of (1-x2)x+1/2(a*A^)(x)/a,x),

Q^ — l)x(dAnX)(x)/dx) is positive and hence

(3.2) sgn (X - l)x — An \x) = + 1, for n S 1, X > 0.

For positive integer values of n this is the same as (2.6).

4. Sign of (d2/ox2)A;,X)(x). Differentiating (1.13) which also holds

for general n we get

D(X) I \ D(X) / \Pn+l(x) Pn-l(x)

d a), s 1 — X-A„ (x) =- d2 (x) d2 (x)dx2 « + X —- Pn+l(x) --P„_l(x)

dx2 dx2

Replacing the second row elements by means of the equations

d1 ex) d2 (xi d ai— Pn+i(x) = X — Pn (x) + (n + 2X + 1) — Pn (*),ax2 dx2 dx

d2 (x> d2 tx) d c\\— p„_;(x) = x — p„ '(*) - («-1) - pV(x)dx2 dx2 dx

(these follow on differentiating (1.5) and (1.4)) and replacing the

first row elements by means of (1.7) and (1.8) we getLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

1082 K. VENKATACHAL1ENGAR AND S. K. LAKSHMANA RAO (December

d2 (X), 1 - X- A„ (x) = -dx2 n + X

(n+2X)x.PnX)(x)-(l-x2) — pT(x) «xPnX)(x) + (l-x2) — P*\x)dx dx

n + 1 n + 2X - 1

d2 rx) d (xj d2 (x) d (X)x — P\ \x) + (ii+2\+l) — Pn \x) x--Pn\x)-(n-l) — P:\x)

dx2 dx dx2 dx

a form in which the right hand side involves only algebraic combina-

tions of P„x>(x) and its first and second derivatives. After simplifying

this determinant we arrive at the equation

(»+l)(«+2X-l) d2 (x, . Vd (x, "j2-An (x) = 2 — P„ (x)

2(X - 1) dx2 \_dx J

(4.1) + (2X - l)x |*[(£ PnX)(x)J - Pn(X,(x) ~ Pf (*)]

-p?\X)JLp?\X)\.dx )

Ii

(4.2) D?(x) = A^x) = \- PlX)(x)]2 - - pV+i(x) - P» (,),Lax J ax dx

we can easily see that DnX)(x) =4X2Anx_+i')(x) and

yD?\x)dx

= 2x{x[(£p1X)(x))2 - PIX)(X) ^PlX)(x)] - A)-ip.WW} •

We may therefore write (4.1) in the form

(»+l)(»4-2X-l) d2 w-An (X)

2(X - 1) dx2(4.3)

T d rx) "I2 d (x+D= 2 - p; \x) + 2x(2x - i)x - a;j; (*).

Lax J dx

Either of (4.1) and (4.3) is an extension of (0.2) to the case of ultra-

spherical polynomials. When n is a positive integer we may replace

the last term on the right side of (4.3) by means of the series (2.5)



and so arrive at the following series for d2ABx>(x)/dx2

(»+l)(»+2X- 1) d2 (x,/ N-■-A„ (x)

2(X - 1) dx2

r d (x) "I r(« -I- 2X)= 2 — PB \x) \2 + (2X - 1) —- 16X2x2

\_dx J nl

"»™1 in - 2k) l{n - 2k + X + 1) u+1)• 2^ -:-:— LPB-2fc(x)J-.

£[ Tin - 2k + 2X + 2)

From (3.2) it follows that \xdAn\\l)ix)/dx>0 for w^2, X>-l/2.

Using this in (4.3) we deduce that

(a « d* a(XV ^ /+1 (X>1)'(4.5) sgn-AB (x) = <

B dx2 l-l (1/2 |X < 1).

Hence ABX)(x) is a convex function of x for X> 1 and a concave func-

tion of x for l/2gX<l.

5. Decreasing nature of (1 — x2)A„X)(x). From (3.1) we know that

A„X)(x) satisfies the differential equation

r d'2 d i cm(5.1) |^(1 - x2) — - (2X + l)x - + 4(1 -X)jAB (*)

= 4(l-X)[P:X)(x)]2.

If (1 —x2)AnX)(x) is denoted by y, we get

— = (1 - x ) —- An (x) - 2xA„ (x),dx dx

(5.2)

-= (1 - x ) -A„ (x) — \x— AB ix) — 2AB (x).. dx2 dx2 dx

We find from (5.1) and (5.2) that y satisfies the differential equation

(5.3) [(1-^£i-(2X-3)^x-(4X-6)]^

= (4X - 6)x2ABM(x) + 4(1 - X)(l - x)[Pn\x)]\

Now y(0)=A;X)(0) is positive when X>0 and y(l)=0. When X>1,

dy/dx is negative for all x > 1 and hence y decreases for x > 1 when

X> 1. We shall now see that y steadily decreases even in (0, 1) when

KXg3/2. On using (2.6) we have idy/dx)x=o = 0. From (5.3) we

have (d2y/ix2),=o = (4X-6)AiX)(0)-r-4(l-X)[PiX)(0)]2 and so when



l<X|3/2, id2y/dx2)x=o is negative. Hence when KX|3/2, v reaches

a maximum at x = 0 with the maximum value =ABX)(0) and decreases

in the neighbourhood of x = 0. Also (dy/dx)x=i= — 2A„X)(1) is negative

when X>l/2 and so y(x) is decreasing in the neighbourhood of

x= 1 also when X> 1. If y(x) is not steadily decreasing in (0, 1) it must

reach a minimum value at some point x = a<l, then increase to a

maximum at some point x=j8 between a and 1 and then decrease

again. We show that this possibility cannot occur when 1 <X|3/2.

From the conditions for minimum, we have at x = a, (dy/dx) =0 and

(d2y/dx2)>0 and from the differential equation (5.3) we get

2 / d2y\ i m(1 - a ) ( ~ ) = (4X - 6)yia) + (4X - 6)a AB \a)

\ dx2/x=a

+ 4(l-X)(l-a2)[PBX>(a)]\

The left hand side is positive while the right hand side is negative.

We conclude that y cannot have a minimum anywhere in (0, 1).

Therefore v(x) must steadily decrease in (0, 1) when 1 <X|3/2. It is to

be expected that for large values of X, y(x) decreases first to a mini-

mum and then attains a maximum in (0, 1) before decreasing again

at and near x = 1.

Eliminating (1— x2)d2AnX)(x)/tix2 in d2y/dx2 by means of (5.1) we

obtain

— = (2X - 3)x— ABX)(x) + (4X - 6)ABX)(x) + 4(1 - X)[P„X)(x)]\dx2 dx

Using (3.2) we notice that sgn d2y/dx2= —1 when 1 <X|3/2.

6. Inequalities for AB,x(x). Let

A„.x(x) = [p1X)(x)/PbX)(1)]2 - [P(n+\ix)/Pn+\il)] .[P™iix)/P™(l)].

From the following well-known relations (see (4), (5))

2 (X1 ''

,A n A / \ (1 ~ x)Dn (X) 4X \ <M-1>, .(6.1) AB,x(x) =-= -— (1 - x )An_i ix)

where ^B,x = n(w + 2X) [PBX)(1)]2, and the conclusions of the previous

section, we have

(a) A„,x(x) is steadily decreasing in (0, 1) for 0 < X | 1/2;

d2(b) sgn-A„,x(x) = - 1 when 0 < X | 1/2,

dx2


1957l ULTRASPHERICAL POLYNOMIALS 1085

showing the concavity property of A„,x(x). From (a) we can im-

mediately deduce the well known inequality viz., the Turan expres-

sion A„,x(x)>0 in |x| <1 when 0<Xgl/2. From (6.1) and the con-

cluding results of §(2) we can prove Turin's inequality in full for

X>0.We have seen in (3.2) that

(3.2) sgn|(X- l)x—A„X)(x) = + 1 for n S 1, X > 0L dx J

and in (4.5) that

d2 (x) (+1 XI>I1(4.5) sgn-An (x) = < n S 2.

dx2 l-l 1/2 S X < 1

From the identity D„X) (x) = 4X2A„xJt"11) (x) it therefore follows that

d rx)(6.2) sgn Xx — Dn (x) = +1 (n S 2, X > - 1/2),

These show the monotonic increasing-decreasing behaviour and the

convexity-concavity property of D„X)(x) ior various values of X, and

enable us to write the following chain of inequalities: When

X > 0, Z?„X)(0) ̂ £>lX>(x) ^ A!X)(0)(1 - x/a) + DnX\a) x/a

^ Dnk\a),

when

-1/2 < X < 0, D„X,(0) ̂ DnX\x) S DnX)(0)(l - x/a) + D™(o) x/a

S DnX\a) 0 ^ x ^ a.

Taking a = 1 and multiplying the above inequalities by (1 —x2) we get

the following inequalities for A„,x(x) in the range |x| ;S1, for w^2.

A„,x(0)(l - x2) =S A„,x(x) ̂ rAn,x(0)(l - x) + ——— (1 - x2)L 2X + 1_

= (1 " x2)/(2\ +1) (X > 0),(6.4)

^fri = l^TTl + A».x(0)(l - *)} (1 - x2) g An.x(x)2X + 1 (2X + 1 )

g ABlX(0)(l - x2) (-1/2<X<0).


1086 K. VENKATACHALIENGAR AND S. K LAKSHMANA RAO [December

These are simpler than the inequalities for A„,x(x) in O. Szasz's paper

[2] and are at the same time an improvement over the corresponding

ones in [4].

7. Integral representation for A„(x). From (3.1) we have

(7.1) ±\il - x2)^ L AlX)(x)l + 4(1 - X)(l - *tX'\Vix)dx L ox J

= 4(1 - X)(l - x2)X-1/2[P„lX)(x)]2.

If we denote [(l-x2)*-1'2AnK)(x)]/x by/„X)(x), then

(1 - x2)X+m~ A«\x) = £ [x{l _ J)f?\x)] + (2X + l)x2f*\x).dx dx

Hence from (7.1) we get

— |"»(1 - *2)Ax)l + (2X + l)x2 ~fn\x) + 6x/„X>(x)dx2 L J dx

, , 2vX-I/2r (X) , . -,2

= 4(1-X)(1-X) [Pn \x)] ,

which may also be written in the form

d T 2 2 -x+3/2 d (X) "1 rr,(X)/ \i2— x (1 - x ) —/„ (x) = 4(1 - X)x[Pn (x)J .dx L dx

For Legendre functions (X = l/2) this becomes

(7.2) -f-^(l - *2) — [*»(*)/*]) = 2x[Pn(x)]2.dx \ dx /

Hence x2(l —x2)(d[AH(x)/x]/dx) is an increasing (a decreasing) func-

tion for positive (negative) values of x and vanishes at x= +1. From

(7.2) we have in succession

~ (An(x)/X) = 2 Ct[Pnit)]2dt,dx x2(l — x2) J 1

and

(7.3) A„(x) = 2x r U Cl[Pnit)]*dt,J 1 «-(l — u-) J 1

the latter giving a positive integral representation for A„(x). Using the

relation


i957) ULTRASPHER1CAL POLYNOMIALS 1087

T (1-X2)[^-Pn(x)]2d , ^ Lax J . ,

- (l-x2)[P„(x)]2 +-——- = - 2x[Pn(x)]2,dxL n(n + 1)

we may also write A„(x) in the form

f r d f}(l-l2)\- Pn(t)\

C r n \-dt J(7.4) A„(x) = x [Pn(t)]2 +-;-\r2dt.

J x { n(n + 1)

References

1. G. Szego, On an inequality of P. Turin concerning Legendre polynomials. Bull.

Amer. Math. Soc. vol. 54 (1948) pp. 401-405.2. O. Szasz, Inequalities concerning orthogonal polynomials and Bessel functions,

Proc. Amer. Math. Soc. vol. 1 (1950) pp. 256-267.3. B. S. Madhava Rao and V. R. Thiruvenkatachar, On an inequality concerning

orthogonal polynomials, Proceedings of the Indian Academy of Sciences, Sect. A vol.

29 (1949) pp. 391-393.4. V. R. Thiruvenkatachar and T. S. Nanjundiah, Inequalities concerning Bessel

functions and orthogonal polynomials, ibid. Sect. A vol. 33 (1951) pp. 373-384.

5. A. E. Danese, Explicit evaluations of Turin expressions, Annali di Matematica

Pura ed Applicata, Serie IV vol. 38 (1955) pp. 339-348.6. G. Szegb, Orthogonal polynomials, (1939) pp. 80-84.

Mysore University and

Indian Institute of Science


on turan's inequality for ultra- spherical ......d2anx)(x)/dx2 we prove the concavity of...

Documents