One way of trying to get a better understanding of the problems in the last lecture is to see what other inequalities they imply. If these other inequalities can be proved, then this will give a further indication that the conjectures are probably true and might suggest some methods for proving them. For example, in the sum in (8.13) let ft = — j, take x = cos в/п and use the limit relation
Since (8.13) probably holds for 0 g y g ot + /? ^ it is likely that (9.2) holds for 0 ^ X ^ a — j. The special case X = 0 is classical, it was proved by Makai  by means of a very nice extension of the Sturm comparison theorem. The classical statement of Sturm's theorem concerns the oscillation of solutions of two differential equations. Watson [2,15.83] made the very useful observation that the value of two solutions can be compared by an almost identical argument. In particular, he proved the following theorem.
Theorem 9.1. Let aMx) and fi2(x) be solutions of the equations
let 11 and 12 be continuous in the interval a ^ x ^ b, and also let pi\(x) and n'2(x) be continuous in the same interval. Then, if I, ^ /2 throughout the interval, |^2(x)| exceeds |/ii(x)| so long as x lies between a and the first zero of Hy(x) in the interval, so that the first zero of /j,(x) in the interval is on the left of the first zero of ^2(x).
To be applicable to problems like (9.2) when X = 0, one added refinement is necessary. The restriction that /, and I2 be continuous in the interval a S x ^ b needs to be replaced by continuity in a < x g b. Szego  gave this extension. Makai  was the first to show that this is a very powerful theorem. He not only proved (9.2) when X = 0, he proved an old conjecture from quantum mechanics
The resulting inequality is
where xM are the zeros of Hn(x) ordered by xM > xk+! k = 1,2, ■ ■ ■ , n - 1, and
X0.„ = CO.
where n(a) is defined by
Many other inequalities can be obtained from this theorem, but care must be taken because of the lack of continuity of the interesting differential equations at an endpoint (see Lorch ). Makai  showed how to use this type of theorem in a more sophisticated way when he proved the following inequality:
l the second positive zero of The same inequality was proved for — 1 < a < -4 in Askey-Steinig .
If y is large enough, then (9.7) will hold. In particular it holds if
To see one of the basic facts behind these integrals, and the sums considered in the last lecture, consider the two integrals
The first is nonnegative, but the second is not because the first arch of cos t, when 0 ^ t ^ jt/2, contains less area than the second arch of cos t, when jt/2 ^ t ^ 3jt/2. There are two natural ways to change (9.6) so that it becomes nonnegative. One is to count the first arch more by weighting points there more than later points. One way of doing this weighting is to consider
y is approximately 0.3084438 (see Church  and Luke, Fair, Coombs and Moran ). This result and an inequality of Szego
(see the appendix to Feldheim ), suggested the inequality (9.3).
This counts the first arch more than later arches in a different way. For the series
the second method leads to Fejer's sum (1.2), while the first method leads to a number of different sums. Rogosinski and Szego  proved
Earlier, W. H. Young  had proved
A second way of changing (9.6) to make it nonnegative is to integrate it a second time,
for — 1 < a ^ 0, and the Rogosinski-Szego result gives (9.13) for 0 < a iS 1 by a summation by parts, as they remarked. Gasper  completed this result by finding the largest value of a for which (9.13) holds. The extreme case comes when n = 3 and the polynomial has a double zero. The approximate value is a = 4.5678. At present this inequality seems to be a curiosity, and I do not know how to use it to prove any interesting results. There is another way of weighting the coefficients which also seemed to be a curiosity when it was first published. This is Vietoris' inequality
(see Vietoris ). It is now clear that this inequality is quite deep, and it has some interesting applications. This will be treated below after a few more inequalities have been given.
These inequalities are very familiar to anyone who has read this far. Vietoris  found an extension of (9.17) to
For sine series there are analogues of these inequalities. Since ££=lsinfc0 changes sign the procedures above can be applied to suggest the inequalities
where ak is defined by (9.15).
when x = cos 6/n and n = - /?. But there is another way of looking at
To see what is really behind Vietoris' inequalities (9.14) and (9.18) we return to Bessel functions and the integral (9.3). One way of thinking of this integral is as a positive multiple of the limit as n -* x of
another way of looking at (9.3) is to first write x"~ ' as the Hankel transform (9.19) and then see when the partial integrals are positive (or nonnegative). Since the function t2'+l is the weight function for which t~"Jj(t) are orthogonal, i.e.,
respectively. The expansions required to show this are in Vietoris  and this connection was pointed out in Askey-Steinig .
For some values of y, 8 this problem is related to an old quadrature problem. When 7 = a, 8 = /3, then the series (9.22), when evaluated at the zeros of /*£ ';'(x), gives the Cotes' numbers for the quadrature problem of integrating the Lagrange interpolation polynomial at the zeros of /"«"'(x) with respect to dx (see Szego, [9, Chap. 15]). Fejer , Polya  and Szego  studied this problem and a summary of the results up to 1938 is given in Szego [9, Chap. 15]. Convergence holds for each continuous function if — 1 < a, /3 ^ § and there is a continuous function for which convergence fails when a > j or /3 > f. If the Cotes' numbers are positive, then convergence is easy to prove and Szego made a start on the problem of finding out when Cotes' numbers are positive. He showed they are positive when a = /3, — 1 <aS0 and j ^ a ^ 1, and they are ultimately positive for 0 < a < j and 1 < a ^ j. The positivity for all n in these two cases was proved in Askey-Fitch . Five years ago I thought this was a fairly dull problem, to be looked at only as a test problem to see how much was known about Jacobi polynomials. After all, the convergence problem had been completely solved, and when a, /3 < § a stronger type of convergence was known (see Hollo  and Turan ). But it was annoying that the only results known when a /3 were two relatively simple cases, a = 1, /3 = 0 and a = j, /3 = - j (see Fejer  and Szego ). A start was made on the general case by using
to show that (9.22) holds for ot, /3 ^ 0, a + /3 g 1 and ot = /3 + 1, g fi ^ 0. Also (9.22) fails for a > 0 + 1 (see Askey ).
(see (2.45a)), this way of looking at (9.3) suggests the following problem for Jacobi polynomials. Expand
and ask when
Vietoris' inequalities (9.14) and (9.18) are the special cases
The general problem (9.22) for arbitrary y. 8 also has a connection with quadrature problems. The values of (9.22) at the zeros of P^-n(x) are the Cotes' numberswhen integrating with respect to (1 — x)"-y(l + xY~*dx. The case y = 6 = 0 is just the classical case of Gaussian quadrature. As usual the case /? = — j is especially interesting. When fi = —j, (5 = 0 there is convergence for each continuous function when y^ja + | (the convergence problem has been solved for arbitrary a, /?,;•. 5 in Horton ), and the Cotes' numbers are positive when y = ja + J if the following inequality holds:
which unfortunately has only been proved when a = i,f,f. However, (9.23) is true when a ^ j. Gasper has recently proved this. So I have changed my mind about the problem of proving the positivity of Cotes' numbers. It is more than just a test problem, it is the source of new and interesting inequalities. Since sources of inequalities are rare it is clear that further work should be done on this problem.
When a > t this would follow from
If a0 > A! ^ X2 ^ • ■■ A„_, ^ 0, then q(6) has n — 1 zeros which satisfy (9.24) fcjr/(n + i) < 9k < (k + l)jt/(n + k = 1, 2, • • • , n - 1.
Polya  proved the zeros were real and Szego  gave the estimates. If the additional inequalities
are satisfied, then the further inequalities
hold (see Szego ). The inequalities (9.24) show that the zeros are uniformly distributed, and the inequalities (9.24) and (9.26) show that they are uniformly separated. Vietoris' inequalities can be used to show that
To see that Vietoris' inequalities are deep consider the following problem. Let
while the conditions (9.27) only imply
Neither of the conditions (9.25) or (9.27) implies the other, but the inequalities (9.29) and (9.30) show that (9.27) is in some sense weaker than (9.25), while the conclusions (9.28) are slightly (but only very slightly) stronger than those in (9.24) and (9.26). This gives an indication of the depth of Vietoris' inequalities.
To return to the inequality (9.2), one of the most surprising developments is in a recent paper of Gasper . As was remarked above, Makai  proved (9.2) when A = 0, and the other extreme case A = a. — j was proved by Fields and Ismail  by an incredible asymptotic argument. Since the two methods used by Makai and Fields-Ismail had nothing in common, Gasper decided to try to find other proofs of these two results and he succeeded in writing both of these integrals as sums of squares of Bessel functions times a product of gamma functions, and these gamma functions were clearly positive when a > j. In the general case, 0 < A < a — \ he obtained the coefficients as a 5F4 with a free parameter, which he chose to reduce the coefficient to a Saalschiitzian 4F3, which was then transformed by Whipple's formula to a well-poised 7F6 which was clearly positive, since all the terms in the sum were positive.
To wax philosophical for a minute, I would like to suggest that Hardy [1. p. 14] was wrong when he suggested that the age of great formulas may be over. Whipple's formula was found shortly after Ramanujan died, and it is clearly a very important formula. So is the addition formula of Sapiro and Koornwinder. Great formulas do not come along very often, but they never have. The rate at which important formulas are being found is probably decreasing, especially as a proportion of the mathematics being done, but there are still important formulas to be found. And when an explicit formula can be found there is nothing to beat it. In the rush to abstraction and generalization we often forget this.
It would be interesting to consider
Gasper  has a number of other interesting positive integrals, including
for 0 < /j. < 2. The next easiest case is probably n =
Additional sums and integrals could be considered, but we shall forgo this here, and go on to a few applications, and then close with a brief comment about the general problem which has really been considered in these two lectures. One application is to univalent functions. Let
Fejer  proved that/(r) is univalent for \z\ < 1 if AJan S 0,;' = 0,1, 2, 3, 4, n = 0, !,•••. His proof used
which is (1.19). Szego  improved this by reducing the conditions to A>a„ S 0, 7 = 0,1,2,3. He used
They each proved (9.34) without stating it in this form, and so neither of them realized that they could have used Fejer's inequality
This is (1.21). For (1.1) is
so (9.34) trivially follows from (1.21).
Fuchs  has shown that no difference less than the third will give univalence for f(z) even when fractional differences are used. This is related to the fact that (1.21) is best possible in the sense that
changes sign in any interval 0 ^ 6 ^ 00, 90 > 0, for infinitely many n when c > 0.
When y = 2jt/3 this is Schweitzer's inequality (1.24). Robertson  and Fuchs  gave a refinement to p-valent functions, and each of their proofs used the inequalityThese inequalities can be used to construct positive definite functions, i.e., functions which are Fourier transforms of positive measures. If /(x) is even, continuous for x ^ 0, vanishes at infinity and convex for x > 0, then the Fourier transform of / is nonnegative and so /(x) is positive definite by the inversion formula (see Polya , ). One analogue of even functions in R" is the class of radial functions. Let
If g(t) is continuous for t ^ 0,(-l)Jg,J)(r) ^ 0,; = 0,1, •••, [n/2],t > 0,lim,-3C g(r) = 0. and (- 1)'" 2'g[n/2i(f) is convex for t > 0, then/(x,, • • ■ , x„) is positive definite (see Askey ). An analogous theorem for spheres and projective spaces is in Askey . The first theorem of this type was found by W. H. Young . There are also theorems of this type which give sufficient conditions for a function to be the Fourier transform of a unimodal distribution (see Askey ). Hopefully other applications will arise after these inequalities become better known.
and a result of J. W. Alexander  can be used to prove
For this and further results, see Bustoz .
Added in proof. The first application of Theorem 9.1 to comparing areas under arches seems to have been given by Hartman and Wintner . L. Lorch called this paper to my attention.
The real question which has been considered in these two lectures is to find real intervals over which generalized hypergeometric functions do not change sign, or on which they are positive. There are related questions on sets in the complex plane which are also interesting, and only a start has been made on the deep results in the plane. Szego  proved