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An extension of Vietoris’s inequalities. (English) Zbl 1138.42002
For which positive nonincreasing sequences (a k ) 0kn is the cosine polynomial k=0 n a k coskx positive on [0,π[? L. Vietoris [Österr. Akad. Wiss., Math.-naturw. Kl., S.-Ber., Abt. II 167, 125–135 (1958; Zbl 0088.27402); Österr. Akad. Wiss., Math.-naturw. Kl. Anz. 192–193 (1959; Zbl 0090.04301)] gave a general sufficient condition: a 2k (1-1/2k)a 2k-1 . It has been noted that this condition may be improved: in particular, G. Brown and Q. H. Yin, [Acta Sci. Math. 67, 221–247 (2001; Zbl 0980.42003)] proposed to determine the optimal constant α 0 in place of the 1/2 above. The author proves that α 0 is the root of the equation 0 3π/2 t -α costdt=0 in the interval ]0·308443,0·308444[: the condition a 2k (1-α 0 /k)a 2k-1 is sufficient for k=0 n a k coskx to be positive, whereas this sum admits no uniform lower bound if a 2k =a 2k+1 =k-α k and α<α 0 . Vietoris’ theorem occurs in the proof of other inequalities, so that the following consequences are derived. (1) An improvement of a result of R. A. Askey and J. Steinig [Trans. Am. Math. Soc. 187, 295–307 (1974; Zbl 0244.42002)]. Let b k be a positive nonincreasing sequence such that b k (1-α 0 /k)b k-1 . The n real and simple zeros s 1 <<s n of k=0 n b k cos(n-k)x in ]0,π[ satisfy (k-1/2)π/(n+1/4)<s k <kπ/(n+1/4). (2) Sufficient conditions for sums of Gegenbauer polynomials and Jacobi polynomials to be positive.

MSC:
42A05Trigonometric polynomials, inequalities, extremal problems
42A32Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
26D05Inequalities for trigonometric functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
References:
[1]Andrews, G.E., Askey, R., Roy, R.: Special functions. Cambridge University Press, Cambridge (1999)
[2]Askey, R., Steinig, J.: Some positive trigonometric sums. Trans. Amer. Math. Soc. 187(1), 295–307 (1974) · doi:10.1090/S0002-9947-1974-0338481-3
[3]Askey, R.: Orthogonal polynomials and special functions. Regional Conf. Lect. Appl. Math. 21, Philadelphia, SIAM (1975)
[4]Askey, R., Gasper, G.: Positive Jacobi polynomial sums II. Amer. J. Math. 98, 709–737 (1976) · Zbl 0355.33005 · doi:10.2307/2373813
[5]Belov, A.S.: Examples of trigonometric series with nonnegative partial sums. (Russian) Math. Sb. 186(4), 21–46 (1995); (English translation) 186(4), 485–510 (1995)
[6]Brown, G., Hewitt, E.: A class of positive trigonometric sums. Math. Ann. 268, 91–122 (1984) · Zbl 0537.42002 · doi:10.1007/BF01463875
[7]Brown, G., Wang, K., Wilson, D.C.: Positivity of some basic cosine sums. Math. Proc. Camb. Phil. Soc. 114, 383–391 (1993)
[8]Brown, G., Koumandos, S., Wang, K.: Positivity of more Jacobi polynomial sums. Math. Proc. Camb. Phil. Soc. 119(4), 681–694 (1996)
[9]Brown, G., Koumandos, S., Wang, K.: Positivity of basic sums of ultraspherical polynomials. Analysis, 18(4), 313–331 (1998)
[10]Brown, G., Koumandos, S., Wang, K.: Positivity of Cotes numbers at more Jacobi abscissas, Monatsh. für Math. 122, 9–19 (1996) · Zbl 0860.33005 · doi:10.1007/BF01298452
[11]Brown, G., Yin, Q.: Positivity of a class of cosine sums. Acta Sci. Math. (Szeged) 67, 221–247 (2001)
[12]Brown, G., Dai, F., Wang, K.: Extensions of Vietoris’s inequalities I, this journal
[13]Chen, W., Wang, K.: Some positive Cotes numbers for Jacobi abscissas. Math. Proc. Camb. Phil. Soc. 123, 193–198 (1998)
[14]Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions, Vol. 2, McGraw-Hill, New York (1953)
[15]Fejér, L.: Ultrasphärikus polynomok összegérol. Mat. Fiz. Lapok 38, 161–164 (1931); Also in: Gesammelte Arbeiten II, 421–423. Birkhäuser Verlag, Basel und Stuttgart (1970)
[16]Hardy, G.H.: A Chapter from Ramanujan’s Note-Book. Proc. Camb. Phil. Soc. 21, 492–503 (1923)
[17]Koumandos, S.: On a positive sine sum. Colloq. Math. 71(2) 243–251 (1996)
[18]Koumandos, S., Ruscheweyh, S.: Positive Gegenbauer polynomial sums and applications to starlike functions. Constr.Approx. 23(2), 197–210 (2006)
[19]Koumandos, S., Ruscheweyh, S.: On a conjecture for trigonometric sums and starlike functions, submitted.
[20]Pólya, G.: Über die Nullstellen gewisser ganzer Funktionen. Math. Z. 2, 352–383 (1918). Also in : Collected Papers, Vol II, 166–197. Cambridge, MA: MIT Press (1974)
[21]Ruscheweyh, S.: Coefficient conditions for starlike functions. Glasgow Math. J. 29, 141–142 (1987) · Zbl 0617.30008 · doi:10.1017/S0017089500006753
[22]Ruscheweyh, S., Salinas, L.: Stable functions and Vietoris’ Theorem. J. Math. Anal. Appl. 291, 596–604 (2004) · Zbl 1052.30011 · doi:10.1016/j.jmaa.2003.11.035
[23]Szego, G.: Inequalities for the zeros of Legendre polynomials and related functions. Trans. Amer. Math. Soc. 39, 1–17 (1936). Also in: Collected Papers of G. Szego, Edited by Richard Askey, 2, 591–610. Birkhäusr-Boston, Basel, Stuttgart (1982).
[24]Vietoris, L.: Über das Vorzeichen gewisser trigonometrischer summen. S.-B. Öster. Akad. Wiss. 167, 125–135 (1958); Teil II: Anzeiger Öster. Akad. Wiss. 167, 192–193 (1959)
[25]Watson, G.N.: A note on generalized hypergeometric series. Proc. London Math. Soc. 23, 13–15 (1925)
[26]Zygmund, A.: Trigonometric Series, 2nd edn. Cambridge University Press (1959)