ROMA TRE UNIVERSITÀ DEGLI STUDI. Università degli Studi Roma Tre. Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea in Matematica

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1 ROMA TRE UNIVERSITÀ DEGLI STUDI Università degli Studi Roma Tre Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea in Matematica Tesi di Laurea Specialistica in Matematica Sintesi Turán type inequalities and logarithmically convex functions Candidato Filippo Cavallari Relatore Prof. Andrea Laforgia Anno Accademico 01/013

2 A Elena, a Mamma e a Papà.

3 Some historical and bibliographical references In 1941, while studying the zeros of Legendre polynomials P n (x) = dn (x 1) n dx n n! n Turán has discovered the famous inequality [P n+1 (x)] > P n (x)p n+ (x), which holds for all x ( 1, 1) and n = 0, 1,,.... Even if Turán's paper [43] has been published just in 1950, Szegö [41] in 1948 presented four dierent elegant proofs of the above inequality and extended the result to ultraspherical (or Gegenbauer), Laguerre and Hermite polynomials. Turán's inequality established for Legendre polynomials has generated considerable interest, and shortly after 1948, analogous results were obtained by several authors for other classical polynomials and special functions. Andrews, Askey and Roy in their wonderful book [4] about special functions begin the preface with the following: Paul Turán once remarked that special functions would be more appropriately labeled as useful functions. Because of their remarkable properties, special functions have been used for centuries. For example, since they have numerous applications in astronomy, trigonometric functions have been studied for over a thousand years. Even the series expansion for sine and cosine (and probably the arc tangent) were known to Madhava in the fourteenth century. These series were rediscovered by Newton and Leibniz in the seventeenth century. Since the, the subject of special functions ha been continuously developed, with contributions by a host of mathematicians, including Euler, Legendre, Laplace, Gauss, Kummer, Eisenstein, Riemann and Ramanujan. Today there is an extensive literature dealing with Turán type inequalities. For example analogous inequalities to Turán's inequality has been found 3

4 for: 1. Laguerre and Hermite polynomials [0, 41];. ultraspherical (or Gegenbauer) polynomials [11, 1, 13, 18, 39, 41] 3. Jacobi polynomials [1, ]; 4. Bessel functions of the rst kind [5, 39, 40] 5. modied Bessel functions of the rst kind [4, 5, 7]; 6. modied Bessel functions of the second kind [4, 5, 31]; 7. polygamma function and Riemann zeta function [17, 4, 30]; 8. generalized complete elliptic integrals [5]; 9. Gauss hypergemetric function [6]; 10. Kummer function (conuent hypergeometric function) [8, 7]. This classical inequality still attracts the attention of mathematicians and it is worth mentioning that the above Turán inequality was improved by Constantinescu [16], and further by Alzer et al. []. It is important to note that even if the Turán type inequalities are interesting in their own right, there are many applications of those inequalities. For example a necessary condition for the Riemann hypothesis can be written as a higher order Turán type inequality (see [17, 0, 31] and the references therein). Other results have been applied successfully in problems which arise in information theory [36], economic theory [14] and biophysics [38]. Aim and contents Of the thesis The purpose of this thesis is to give a detailed review of some recent results concerning Turán type inequalities, emphasizing the techniques of proof and focusing on those arguments which allow generalizations. In chapter 1 we review Baricz's work [5, 6] about generalized complete elliptic integrals and 4

5 Gauss hypergeometric function. The former are dened, for x (0, 1), x = 1 x and a (0, 1), by K a (x) = π F 1 (a, 1 a, 1, x ) K a(x) = K a (x ) K a (0) = π K a (1) = For those kind of functions we will prove, under appropriate hypotheses on the variable and the parameters, the functional inequality Ka (x)k b (x) K a(x) + K b (x) K a+b (x) K a (x) + K (x), b which, in particular, implies a Turán type inequality. and We then considered a more general version of these elliptic integrals: B(a, c a) K a,c (x) = F 1 (a, c a, c, x ) K a,c(x) = K a,c (x ) B(a, c a) K a,c (0) = K a,c (1) = B(a, c a) E a,c (x) = F 1 (a 1, c a, c, x ) E a,c(x) = E a,c (x ) B(a, c a) E a,c (0) = E a,c (1) = B(a, c a)b(c, 1) B(c + 1 a, c) where B(x, y) is the beta function. In that case the most general result we 5

6 proved are the Turán type inequalities [ ] K a+b,c(x) Ka,c (x)k b,c (x), [ ] E a+b,c(x) Ea,c (x)e b,c (x), [ [ K a, c+d E a, c+d (x)] Ka,c (x)k a,d (x), (x)] Ea,c (x)e a,d (x). Later we emphasize that the proof of the above inequalities is totally based on the structure of the power series representation of the involved functions (see [6]). Indeed at the base of all the elliptic integral we considered there is the Gauss hypergeometric function, that is F 1 (a, b, c, x) = (a) n (b) n (c) n In chapter one this function is considered in the case x n n!. F a (x) := F 1 (a, c a, c, x). Thanks to the generality of a technical lemma, we proved that Fa (x)f b (x) F a(x) + F b (x) F a+b (x) F a (x) + F (x), b and, in particular, Turán type inequality for Gauss hypergeometric function. At the end of this chapter we also considered some special cases where reverse Turán type inequalities holds. In chapter we moved our attention to modied Bessel functions of the rst kind and Kummer function (also known as conuent hypergeometric function). The reference work here was [7]. The starting point were two analytical inequalities: Lazarevi inequality, which states that if x 0, then ( sinh x ) 3, cosh x < x 6

7 and Wilker inequality, which states that if x > 0, then ( ) sinh x + tanh x x x >. Since hyperbolic sine a cosine functions are particular cases of a variant of the modied Bessel of the rst kind, namely the function I p (x) = p Γ(p + 1)x p I p (x) = (1/4) n (p + 1) n n! xn, where I p (x) is just the modied Bessel function of the rst kind dened by I p (x) = (x/) n+p n!γ(p + n + 1), we were able to extend Lazarevi inequality and Wilker inequality to those kind of functions. During this investigation, we adopted the same point of view of chapter 1: we discussed monotonicity and convexity properties of the power series expansion of the function involved as function of the parameters. Thank to this approach we proved some functional inequalities for the modied Bessel function of the rst kind and, in particular, Turán type inequality. Continuing with this approach we extended (in the same way we have done going from the generalized complete elliptic integrals to Gauss hypergeometric function) all this results to the Kummer function, which is dened by Φ(a, c, x) = 1 F 1 (a, c, x) = We were able to prove the functional inequalities (a) n (c) n xn n!. [Φ(a + 1, c + 1, x)] Φ(a, c, x)φ(a +, c +, x), [Φ(a, c, x)] (a+1)/(c+1) [Φ(a + 1, c + 1, x)] a/c, [Φ(a + 1, c + 1, x)] (a c)/[c(a+1)] + Φ (a, c + 1, x) Φ(a, c, x)φ(a, c +, x), Φ(a + 1, c + 1, x) Φ(a, c, x), 7

8 which includes Turán type inequalities for Φ(a, c ± ν, x), Φ(a ± ν, c ± ν, x). The case Φ(a ± ν, c, x) was treated by Barnard, Gordy and Richards in [8]. Their approach is quite dierent from the one of Baricz. Instead of focusing on monotonicity and convexity properties of the coecients of the power series expansion, they used a clever combination of contiguous relations and telescoping sums to prove the Turán inequality Φ(a, c, x) Φ(a ν, c, x)φ(a + ν, c, x). They also showed the way to extend this result to the generalized hypergeometric function dened by pf q (a 1,..., a p, b 1,..., b q, x) = (a 1 ) n... (a p ) n (b 1 ) n... (b q ) n xn n!. Finally they presented an interesting result concerning with the arithmetic mean and geometric mean. Namely they proved A ( Φ(a + ν, a + b, x), Φ(a ν, a + b, x) ) Φ(a, a + b, x) G ( Φ(a + ν, a + b, x), Φ(a ν, a + b, x) ), beginning to give an answer to Baricz's conjecture about the link between functional inequalities and means (see [6]). In this regard we mention the interesting article [3] by Anderson, Vamanamurthy and Vuorinen. The last two chapters are devoted to some recent papers which give a wider approach to this kind of problems and present theorems stated in a very general setting. In chapter 3 we discussed the work of Karp and Sitnik [9]. They idea of their proof seems to watch in the same direction of the one that Barnard, Gordy and Richards presented in their work and we reported in chapter. But Karp and Sitnik go more deeply inside the question: they were able to do not limit themselves to a special case of a 8

9 particular special function and constructed a very interesting tool of research for Turán type inequalities. In fact they follow as a particular case of a more general phenomenon: they gave sucient conditions for the function x f(a + δ, x)f(b, x) f(b + δ, x)f(a, x) to have positive power series coecients in the following three cases: f(a, x) = f(a, x) = f(a, x) = (a) n f n x n, n! f n Γ(a + n)x n, f n (a) n x n. This means that all those function are Wright log-convex, which is a condition equivalent to log-convexity for continuous functions. They then applied this results to the Kummer function, the Gauss hypergeometric function and the generalized hypergeometric function to nd, in addition to Turán type inequalities, some functional inequalities that give bounds for some kind of ratio of these functions. The last chapter present the recent work of Kalmykov and Karp [6]. In this work the authors continue the investigation began in the paper on which we focused in chapter 3. In particular they recognize the following question: under what conditions on non-negative sequence {f k } and the numbers a i, b j the function µ f(µ, x) = k=0 n i=1 f Γ(a i + µ + ε i k) k m j=1 Γ(b j + µ + ε n+j k) xk is (discrete) log-concave or log-convex? 9

10 In their work they consider the function f(µ, x) = f n n!γ(µ + n) xn. As in [9], the gave sucient condition for the function ϕ a,b,µ (x) = f(a + µ, x)f(b + µ, x) f(a + b + µ, x)f(µ, x) to have positive power series coecients. As consequence of this they were able treat the so called generalized Turanian dened by ε (µ, x) = f(µ, x) f(µ + ε, x)f(µ ε, x). In particular they nd A ε (µ)f0 ε (µ, x) B ε (µ)f(µ, x) where A ε (µ) = B ε (µ) = Γ(µ ε)γ(µ + ε) Γ(µ), Γ(µ ε)γ(µ + ε)γ(µ) Γ(µ ε)γ(µ + ε) Γ(µ). Γ(µ ε)γ(µ + ε) They nally applied these results to the modied Bessel function of the rst kind in order to rene some known results. 10

11 Bibliography [1] M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Pubblications, [] H. Alzer, S. Gerhold, M. Kauers, A. Lupas, On Turán's inequality for Legendre polynomials, Expo. Math 17, 89-31, 004. [3] G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl. 335(), , 007. [4] G. E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, [5] Á. Baricz, Turán type inequalities for generalized complete elliptic integrals, Math. Z. 56, , 007. [6] Á. Baricz, Turán type inequalities for hypergeometric functions, Proc. Amer. Math. Soc. 136(9), 33-39, 008. [7] Á. Baricz, Functional inequalities involving Bessel and modied Bessel functions of the rst kind, Expo. Math. 6(3), 79-93, 008. [8] R. W. Barnard, M. Gordy, K. C. Richards, A note on Turán type and mean inequalities for the Kummer function, J. Math. Anal. Appl. 349(1), 59-63, 009. [9] J. M. Borwein, P. B. Borwein, Pi and the AGM, Wiley, [10] F. Bowman, Introduction to elliptic functions and applications, Dover,

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