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File: BJP_Kratzel.tex, printed: 2011-01-14, 1.09 TURAŃ TYPE INEQUALITIES FOR KRA¨TZEL FUNCTIONS A´RPA´DBARICZ†, DRAGANAJANKOV,AND TIBOR K. POGAŃY Abstract. Complete monotonicity, Laguerre and Turán type inequalities are established for theso-called Krätzel function Zν, defined by ρ ∞ Zρν(u)=Z tν−1e−tρ−utdt, 0 1 where u > 0 and ρ,ν ∈ R. Moreover, we prove the complete monotonicity of a determinant 1 function of which entries involvetheKrätzel function. 0 2 n 1. Introduction a J In1941 whilestudyingthezeros ofLegendrepolynomials, theHungarian mathematician Paul 3 Turań discovered the following inequality 1 [P (x)]2 > P (x)P (x), ] n n−1 n+1 A where x < 1, n 1,2,... and P stands for the classical Legendre polynomial. This C n | | ∈ { } inequality was publishedbyP.Turańonlyin1950in[31]. However, sincethepublicationin1948 . h by G. Szego˝ [30] of the above famous Turań inequality for Legendre polynomials, many authors t a have deduced analogous results for classical (orthogonal) polynomials and special functions. In m the last 62 years it has been shown by several researchers that the most important (orthogonal) [ polynomials (e.g. Laguerre, Hermite, Appell, Bernoulli, Jacobi, Jensen, Pollaczek, Lommel, 1 Askey-Wilson, ultraspherical polynomials) and special functions (e.g. Bessel, modified Bessel, v gamma, polygamma, Riemann zeta functions) satisfy a Turań inequality. In 1981 one of the 3 PhD students of P. Turań, L. Alpár [2] in Turań’s biography mentioned that the above Turań 2 5 inequality had a wide-ranging effect, this inequality was dealt with in more than 60 papers. The 2 Turań type inequalities now have a more extensive literature and recently some of the results . 1 have been applied successfully in problems that arise in information theory, economic theory 0 and biophysics. Motivated by these applications, the Turań type inequalities have recently 1 come under the spotlight once again and it has been shown that, for example, the Gauss and 1 : Kummer hypergeometric functions, as well as the generalized hypergeometric functions, satisfy v naturally some Turań type inequalities. For the most recent papers on this subject we refer to i X [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [17], [21], [19], [29]. For more details see also the r references therein. a Motivated by the above immense research on Turań type inequalities, in this paper our aim is to deduce complete monotonicity, lower bounds and Turań type inequalities for the so-called Kra¨tzel function, defined below. The Kra¨tzel function is defined for u > 0, ρ R and ν C, ∈ ∈ being such that Re(ν) < 0 for ρ 0, by the integral ≤ ∞ (1.1) Zρν(u) = tν−1e−tρ−utdt. Z0 2000 Mathematics Subject Classification. Primary 33C15, Secondary 26D07. Keywordsandphrases. Krätzelfunctions,ModifiedBesselfunctions,Turántypeinequalities,Completemono- tonicity, Vandermondedeterminant, Turán determinant, Hölder-Rogers and Chebyshev integral inequality. †TheresearchofA´.BariczwassupportedbytheJánosBolyaiResearchScholarshipoftheHungarianAcademy ofSciencesandbytheRomanianNationalAuthorityforScientificResearchCNCSIS-UEFISCSU,projectnumber PN-II-RU-PD388/2010. 1 2 A´.BARICZ,D.JANKOV,T.K.POGAŃY For ρ 1 the function (1.1) was introduced by E. Kra¨tzel [24] as a kernel of the integral ≥ transform ∞ (Kρf)(u) = Zν(ut)f(t)dt, ν ρ Z0 which was applied to the solution of some ordinary differential equations. The study of the Kra¨tzel function (1.1) and the above integral transform were continued by several authors. For example, in [23] the authors deduced explicit forms of Zν in terms of the generalized Wright ρ function, while in [22] the authors obtained the asymptotic behavior of this function at zero and infinity and gave applications to evaluation of integrals involving Zν. Such investigations ρ now are of a great interest in connection with applications, see [22] and [23] and the references therein for more details. We note that the Kra¨tzel function occurs in the study of astrophysical thermonuclear functions, which are derived on the basis of Boltzmann-Gibbs statistical mechan- ics, see [28]. It is also important to note that the Kra¨tzel function Zν is related to the modified 1 Bessel function of the second kind K . More precisely, in view of the formula [4, p. 237] ν (1.2) Kν(u) = 2uν+ν1 ∞t−ν−1e−t−u42tdt Z0 we have that for all u > 0 and ν C ∈ u2 u ν u ν Zν = 2 K (u) =2 K (u) 1 4 2 −ν 2 ν (cid:18) (cid:19) (cid:16) (cid:17) (cid:16) (cid:17) and consequently (1.3) Zν(u) = 2uν/2K (2√u). 1 ν Note that this function is useful in chemical physics. More precisely, the function u 2ν−1Zν(uq2/4) = (q√u)νK (q√u) 7→ 1 ν is related to theHartree-Frock energy andis usedas a basis function for thehelium isoelectronic series. See [15] and [16] for more details. Moreover, the function u 2/π2ν−1Zν(u2/4) = 2/πuνK (u) 7→ 1 ν p p is called in the literature as reduced Bessel function and plays an important role in theoretical chemistry. See [33] and the references therein for more details. This paper is organized as follows: in the next section we present some monotonicity, log- convexity properties, complete monotonicity and lower bounds for the Kra¨tzel function, while in the third section we prove the complete monotonicity of a Turań determinant which entries involve the Kra¨tzel function Zν. The main result of the third section is actually a generalization ρ of a Turań-type inequality, of which counterpart is a conjecture at the end of this paper. We close these preliminaries with the following definitions, which will be used in the sequel. A function f : (0, ) R is said to be completely monotonic if f has derivatives of all orders ∞ → and satisfies ( 1)mf(m)(u) 0 − ≥ for all u > 0 and m 0,1,... . A function g : (0, ) (0, ) is said to be logarithmically ∈ { } ∞ → ∞ convex, or simply log-convex, if its natural logarithm lng is convex, that is, for all u ,u > 0 1 2 and α [0,1] we have ∈ g(αu +(1 α)u ) [g(u )]α[g(u )]1−α. 1 2 1 2 − ≤ Note that every completely monotonic function is log-convex, see [34, p. 167]. TURAŃ TYPE INEQUALITIES FOR KRA¨TZEL FUNCTIONS 3 2. Monotonicity and convexity properties of the Krätzel functions Our first main result reads as follows: Theorem 1. If ν,ρ R and u > 0, then the following assertions are true: ∈ a. The Krätzel function Zν satisfies the recurrence relation ρ (2.1) νZν(u) = ρZν+ρ(u) uZν−1(u). ρ ρ − ρ b. The function u Zν(u) is completely monotonic on (0, ). 7→ ρ ∞ c. The function ν Zν(u) is log-convex on R. 7→ ρ d. The function u Zν(u) is log-convex on (0, ). 7→ ρ ∞ e. For n 1,2,... the following Laguerre type inequality holds ∈ { } (2.2) Zν(u) (n) 2 Zν(u) (n−1) Zν(u) (n+1). ρ ≤ ρ ρ h i f. Suppose that ρ > 0.(cid:2)Then t(cid:3)he follow(cid:2)ing ine(cid:3)quality(cid:2) holds(cid:3) (2.3) Zρ−ν(u) ≥ 2u21−νΓ(ν)K1(2√u) provided that ν 1. Moreover, if 0 < ν 1, then the above inequality is reversed. In ≥ ≤ particular, the inequality (2.4) uν−1K (u) 2ν−1Γ(ν)K (u) ν 1 ≥ is valid for all ν 1. If 0 < ν 1, then the above inequality is reversed. In the above ≥ ≤ inequality equality holds if u tends to zero or ν = 1. We note here that similar result to that of (2.4) was proved by M.E.H. Ismail [18]. More precisely, M.E.H. Ismail proved among others that for all u > 0 and ν > 1/2 the inequality (2.5) uνK (u)eu > 2ν−1Γ(ν) ν is valid and it is sharp as u 0. It should be mentioned here that for ν 1 the inequality (2.4) → ≥ is better than (2.5) since for all ν 1 and u > 0 we have ≥ uνK (u) 2ν−1Γ(ν)uK (u) > 2ν−1Γ(ν)e−u. ν 1 ≥ Observethatinfact(2.5)andthelater inequalityfollow fromthefact(see[25])thatthefunction u uνeuK (u) ν 7→ isstrictly increasingon(0, )forallν > 1/2andconsequently forallu > 0wehaveueuK (u) > 1 ∞ 1. Here we used tacitly that when ν > 0 is fixed and u tends to zero, the asymptotic relation uνK (u) 2ν−1Γ(ν) ν ∼ holds. Proof of Theorem 1. a. By using (1.3) and the recurrence relation [32, p. 79] 2ν K (u) K (u) = K (u) ν−1 ν+1 ν − − u we obtain uZν−1(u) Zν+1(u) = νZν(u). 1 − 1 − 1 We note that the above recurrence relation can be verified also by using integration by parts as follows ∞ νZ1ν(u) = e−t−ut νtν−1 dt Z0∞(cid:16) u(cid:17)(cid:0) (cid:1) = 1 e−t−ut tνdt − t2 Z0 (cid:16) (cid:17)(cid:16) (cid:17) = Zν+1(u) uZν−1(u). 1 − 1 4 A´.BARICZ,D.JANKOV,T.K.POGAŃY Moreover, by using the same idea we immediately have ∞ νZρν(u) = e−tρ−ut νtν−1 dt Z0∞(cid:16) (cid:17)u(cid:0) (cid:1) = ρtρ−1 e−tρ−ut tνdt − t2 Z0 (cid:16) (cid:17)(cid:16) (cid:17) = ρZν+ρ(u) uZν−1(u). ρ − ρ b. The change of variable 1/t = s in (1.1) yields ∞ (2.6) Zν(u) = s−ν−1e−s−ρ e−usds, ρ Z0 (cid:16) (cid:17) i.e. the Kra¨tzel function Zν is the Laplace transform of the function s s−ν−1e−s−ρ. This ρ 7→ in view of the Bernstein-Widder theorem (see [34]) implies that the function u Zν(u) is 7→ ρ completely monotonic, i.e. for all n 0,1,... , ν,ρ R and u > 0 we have ∈{ } ∈ ( 1)n Zν(u) (n) > 0. − ρ We note that this can be verified also direc(cid:2)tly by u(cid:3)sing that for all n 0,1,... , ν,ρ R and ∈ { } ∈ u > 0 (2.7) Zν(u) (n) = ( 1)nZν−n(u), ρ − ρ which follows via mathematical in(cid:2)duction(cid:3) easily from (1.1) or (2.6). c. Recall the Hölder-Rogers inequality [27, p. 54], that is, b b 1/p b 1/q (2.8) f(t)g(t)dt f(t)pdt g(t)qdt , | | ≤ | | | | Za (cid:20)Za (cid:21) (cid:20)Za (cid:21) wherep > 1,1/p+1/q = 1,f andg arerealfunctionsdefinedon[a,b]and f p, g q areintegrable | | | | functions on [a,b]. Using (1.1) and (2.8) we obtain that ∞ Zραν1+(1−α)ν2(u) = tαν1+(1−α)ν2−1e−tρ−utdt Z0 ∞ = tα(ν1−1)+(1−α)(ν2−1)e−tρ−utdt Z0 = ∞ tν1−1e−tρ−ut α tν2−1e−tρ−ut 1−αdt Z0 (cid:16) (cid:17) (cid:16) (cid:17) ∞ α ∞ 1−α tν1−1e−tρ−utdt tν2−1e−tρ−utdt ≤ (cid:20)Z0 (cid:21) (cid:20)Z0 (cid:21) = Zν1(u) α Zν2(u) 1−α ρ ρ holds for all α ∈ [0,1], ν1,ν2,ρ ∈ R(cid:2) and u(cid:3)>(cid:2)0, i.e. t(cid:3)he function ν 7→ Zρν(u) is log-convex on R. d. This follows from the fact that the integrand in (1.1) or (2.6) is a log-linear function of u and by using the Hölder-Rogers inequality (2.8) we have that ∞ Zρν(αu1 +(1−α)u2)= tν−1e−tρ−1t(αu1+(1−α)u2)dt Z0 = ∞ tν−1e−tρ−ut1 α tν−1e−tρ−ut2 1−αdt Z0 (cid:16) (cid:17) (cid:16) (cid:17) ∞ α ∞ 1−α tν−1e−tρ−ut1dt tν−1e−tρ−ut2dt ≤ (cid:20)Z0 (cid:21) (cid:20)Z0 (cid:21) = Zν(u ) α Zν(u ) 1−α ρ 1 ρ 2 holds for all α ∈ [0,1], ν,ρ ∈ R and(cid:2) u1,u2(cid:3)>(cid:2)0, i.e. t(cid:3)he function u 7→ Zρν(u) is log-convex on (0, ). ∞ TURAŃ TYPE INEQUALITIES FOR KRA¨TZEL FUNCTIONS 5 Alternatively, we may use part b of this theorem. More precisely, it is known that every completely monotonic function is log-convex (see [34, p. 167]), and then in view of part b the Kra¨tzel function Zν is log-convex on (0, ). Moreover, as a third proof we may use part c of ρ ∞ this theorem. Namely, since the function ν Zν(u) is log-convex, it follows that the following Turań type inequality holds for all ν ,ν ,ρ 7→R aρnd u > 0 1 2 ∈ ν1+ν2 2 (2.9) Z 2 (u) Zν1(u)Zν2(u). ρ ≤ ρ ρ (cid:20) (cid:21) Now, let choose ν = ν 2 and ν =ν, then we obtain the Turań type inequality 1 2 − fν(u) = Zν−1(u) 2 Zν−2(u)Zν(u) 0, ρ ρ − ρ ρ ≤ which is valid for all ν,ρ R and u(cid:2)> 0. This(cid:3) in turn together with (2.7) implies that ∈ Zν(u) ′ ′ Zν−1(u) ′ fν(u) ρ ρ ρ = = 0, "(cid:2)Zρν(u)(cid:3) # −" Zρν(u) # − Zρν(u) 2 ≥ i.e. the function u Zν(u) ′/Zν(u) is increasing on (0, (cid:2) ) for(cid:3)all ν,ρ R. 7→ ρ ρ ∞ ∈ e. This follows also from part c of this theorem. More precisely, in view of (2.7) the Laguerre (cid:2) (cid:3) type inequality (2.2) is equivalent to the Turań type inequality Zν−n(u) 2 Zν−n−1(u)Zν−n+1(u), ρ ≤ ρ ρ which clearly follows from (2.9(cid:2)) by cho(cid:3)osing ν1 = ν n 1 and ν2 = ν n+1. f. Let us recall the Chebyshev integral inequality−[27,−p. 40]: If f,g : [−a,b] R are integrable functions, both increasing or both decreasing and p : [a,b] R is a positive i→ntegrable function, → then b b b b (2.10) p(t)f(t)dt p(t)g(t)dt p(t)dt p(t)f(t)g(t)dt. ≤ Za Za Za Za Note that if one of the functions f or g is decreasing and the other is increasing, then (2.10) is reversed. We shall use this inequality. For this by using (2.6) let us write Z−ν(u) as ρ ∞ Z−ν(u) = e−uttν−1e−t−ρdt ρ Z0 and let p(t)= e−ut, f(t)= tν−1 and g(t) = e−t−ρ. Clearly f is increasing (decreasing) on (0, ) if and only if ν 1 (ν 1). Since g′(t)/g(t) = ρt−ρ−1, it follows that g is increasing if and o∞nly ≥ ≤ if ρ > 0. Moreover, ∞ ∞ 1 p(t)dt = e−utdt = u Z0 Z0 and ∞ ∞ ∞ p(t)f(t)dt= e−uttν−1dt = u−ν e−ssν−1ds= u−νΓ(ν). Z0 Z0 Z0 Similarly, integration by parts and (2.6) imply ∞p(t)g(t)dt = ∞e−ute−t−ρdt= ρ ∞t−ρ−1e−t−ρe−utdt= ρZρ(u). u u ρ Z0 Z0 Z0 Now, by choosing ν = 0 in (2.1), and using (1.3) we obtain that ρZρ(u) = uZ−1(u) = 2√uK (2√u) ρ 1 1 and appealing to the Chebyshev integral inequality (2.10) the proof of the inequality (2.3) is done. Finally, observe that by using the relation [32, p. 79] K (u) = K (u) we obtain easily ν −ν (2.11) Z−ν(u) = u−νZν(u), 1 1 and if we let ρ= 1 in (2.3), then by using (2.11) we immediately obtain (2.4), and with this the proof is complete. (cid:3) 6 A´.BARICZ,D.JANKOV,T.K.POGAŃY 3. Turán type inequalities for Krätzel functions Let us consider now the Turań type inequality (3.1) Zν(u) 2 Zν−ρ(u)Zν+ρ(u) < 0, ρ − ρ ρ which holds for all ν,ρ R and(cid:2)u > 0.(cid:3)This inequality is actually a particular case of part c of ∈ Theorem 1. More precisely, by choosing ν = ν ρ and ν = ν +ρ in (2.9) the proof of (3.1) is 1 2 − done. However, we give here an alternative proof. Just observe that 1 ∞ ∞ Zρν(u) 2−Zρν−ρ(u)Zρν+ρ(u) = 2 (ts)ν−1e−tρ−sρ−u(1t+s1)[2−(t/s)ρ−(s/t)ρ]dtds Z0 Z0 (cid:2) (cid:3) and by using the elementary inequality (t/s)ρ + (s/t)ρ 2, the integrand becomes negative, ≥ which proves (3.1). Moreover the above integral representation yields the following complete monotonicity result: the function u7→ ZZρν−ν(ρu(u)) ZZνρ+ν(ρu(u)) = 12 ∞ ∞(ts)ν−1e−tρ−sρ−u(1t+1s)[(t/s)ρ+(s/t)ρ−2]dtds (cid:12) ρ ρ (cid:12) Z0 Z0 (cid:12) (cid:12) is not on(cid:12)(cid:12)ly positive, but even c(cid:12)(cid:12)ompletely monotonic on (0, ) for all ν,ρ R. The next result is ∞ ∈ an analogue of [19, Theorem 2.1] for the Turań determinant of Kra¨tzel functions and provides a generalization of the above result and of part b of Theorem 1. Note that in view of (1.3) the following result in particular for ρ= 1 gives better Turań type inequalities for the modified Bessel function of the second kind K than [19, Theorem 2.5]. For more details, compare the ν first Turań type inequality in [19, Remark 2.6] with the right-hand side of (3.3) below. Theorem 2. If ν,ρ R and n 1,2,... , then the function ∈ ∈ { } Zν−ρ(u) Zν(u) ... Zν+(n−1)ρ(u) ρ ρ ρ Zν(u) Zν+ρ(u) ... Zν+nρ(u) u Aν (u) = (cid:12)(cid:12) ρ. ρ . ρ . (cid:12)(cid:12) 7→ ρ,n (cid:12) .. .. .. (cid:12) (cid:12) (cid:12) (cid:12) ν+(n−1)ρ ν+nρ ν+(2n−1)ρ (cid:12) (cid:12) Zρ (u) Zρ (u) ... Zρ (u) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) is completely monotonic on (0,(cid:12) ). (cid:12) (cid:12) (cid:12) ∞ Proof. By using (1.1) we have Zν−ρ(u) Zν(u) ... Zν+(n−1)ρ(u) ρ ρ ρ Zν(u) Zν+ρ(u) ... Zν+nρ(u) Aνρ,n(u) =(cid:12)(cid:12)(cid:12) ρ... ρ ... ρ ... (cid:12)(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) ν+(n−1)ρ ν+nρ ν+(2n−1)ρ (cid:12) (cid:12) Zρ (u) Zρ (u) ... Zρ (u) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) tν−ρ−1 tν−1 ... tν+(n−1)ρ(cid:12)(cid:12)−1 (cid:12) 0 0 0 (cid:12) =Z[0,∞)n+1(cid:12)(cid:12)(cid:12)(cid:12) t1ν...−1 tν1+...ρ−1 ... t1ν+n...ρ−1 (cid:12)(cid:12)(cid:12)(cid:12)j=n0e−tρj−tujdt0dt1...dtn (cid:12) ν+(n−1)ρ−1 ν+nρ−1 ν+(2n−1)ρ−1 (cid:12)Y (cid:12) tn tn ... tn (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) tν−ρ−1 tν−1 ... tν+(n−1)ρ−1 (cid:12)(cid:12) (cid:12) σ(0) σ(0) σ(0) (cid:12) = (cid:12)(cid:12)(cid:12) tσν.−(11) tνσ+(1.ρ)−1 ... tσν+(1n.)ρ−1 (cid:12)(cid:12)(cid:12) n e−tρj−tujdt0dt1...dtn, . . . Z[0,∞)n+1(cid:12)(cid:12) . . . (cid:12)(cid:12)jY=0 (cid:12) ν+(n−1)ρ−1 ν+nρ−1 ν+(2n−1)ρ−1 (cid:12) t t ... t (cid:12) σ(n) σ(n) σ(n) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) TURAŃ TYPE INEQUALITIES FOR KRA¨TZEL FUNCTIONS 7 where σ is a permutation on 0,1,...,n . Now, let sgn(σ) be denote the sign of σ and S be n { } the symmetric group on n symbols. Then we obtain t0 tρ ... tnρ 0 0 0 Aνρ,n(u) = (cid:12)(cid:12) t..ρ1 t21..ρ ... t(1n+..1)ρ (cid:12)(cid:12)sgn(σ) n tνj−ρ−1e−tρj−tujdt0dt1...dtn Z[0,∞)n+1(cid:12)(cid:12)(cid:12) tn.ρ t(n+.1)ρ ... t2.nρ (cid:12)(cid:12)(cid:12) jY=0 (cid:12) n n n (cid:12) (cid:12) (cid:12) ρ nρ (cid:12) 1 t ... t (cid:12) = (cid:12)(cid:12)(cid:12) 1.. t..ρ10 ... tn10..ρ (cid:12)(cid:12)sgn(σ) t00(cid:12)tρ1...tnnρ n tνj−ρ−1e−tρj−tuj dt0dt1...dtn Z[0,∞)n+1(cid:12)(cid:12)(cid:12) 1. t.ρn ... tnn.ρ (cid:12)(cid:12)(cid:12) (cid:0) (cid:1)jY=0(cid:18) (cid:19) (cid:12) (cid:12) (cid:12) ρ (cid:12) nρ (cid:12) 1 t0 .(cid:12).. t0 = (n+1 1)! Z(cid:12)[0,∞)n+1(cid:12)(cid:12)(cid:12)(cid:12) 1... t...ρρ1 .(cid:12).. tnn1...ρρ (cid:12)(cid:12)(cid:12)(cid:12)jY=n0(cid:18)tνj−ρ−1e−tρj−tuj(cid:19) (cid:12) 1 tn ... tn (cid:12) (cid:12) (cid:12) sgn(σ)tσ(0(cid:12)(cid:12))(tρ)σ(1)...(tρ)σ(n)(cid:12)(cid:12)dt dt ...dt × 0 (cid:12) 1 n (cid:12) 0 1 n σ∈XSn+1 n = 1 tρ tρ tν−ρ−1e−tρj−tuj (n+1)! j − i j Z[0,∞)n+11≤iY<j≤n(cid:16) (cid:17)jY=0(cid:18) (cid:19) sgn(σ)tσ(0)(tρ)σ(1)...(tρ)σ(n)dt dt ...dt × 0 1 n 0 1 n σ∈XSn+1 n = (n+1 1)! tρj −tρi 2 tνj−ρ−1e−tρj−tuj dt0dt1...dtn, Z[0,∞)n+11≤iY<j≤n(cid:16) (cid:17) jY=0(cid:18) (cid:19) where we used that by Leibniz’s formula the Vandermonde determinant can be written as ρ nρ 1 t ... t 0 0 ρ nρ 1 t ... t (cid:12)(cid:12)(cid:12) ... ...1 1... (cid:12)(cid:12)(cid:12)= sgn(σ)tσ0(0)(tρ1)σ(1)...(tρn)σ(n) = tρj −tρi . (cid:12) ρ nρ (cid:12) σ∈XSn+1 1≤iY<j≤n(cid:16) (cid:17) (cid:12) 1 tn ... tn (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Summar(cid:12)izing (cid:12) (cid:12) (cid:12) n Aνρ,n(u) = (n+11)! e−jP=0(cid:18)tρj+tuj(cid:19) tρj −tρi 2 n tνj−ρ−1dt0dt1...dtn Z[0,∞)n+1 1≤iY<j≤n(cid:16) (cid:17) jY=0 and then for all ν,ρ R, u > 0 and m 0,1,... we obtain ∈ ∈ { } m n 0< (−1)m Aνρ,n(u) (m) = (n+11)!  n t1 e−jP=0(cid:18)tρj+tuj(cid:19) Z[0,∞)n+1 j=0 j (cid:2) (cid:3) X   n 2 ρ ρ ν−ρ−1 t t t dt dt ...dt , × j − i j 0 1 n 1≤iY<j≤n(cid:16) (cid:17) jY=0 which completes the proof. (cid:3) Now, let us consider the function Φν :(0, ) R, defined by ρ ∞ → ν−ρ ν+ρ Z (u)Z (u) Φν(u) = 1 ρ ρ . ρ − Zν(u) 2 ρ (cid:2) (cid:3) 8 A´.BARICZ,D.JANKOV,T.K.POGAŃY Recall the asymptotic expansion (see [22, 24]) Zν(u) ανu22(νρ−+1ρ)e−βρuρ+ρ1, ρ ∼ ρ where ανρ = ρ2+π1ρ1−22(νρ++11) and βρ = (1+1/ρ)ρρ+11, r which holds for large values of u and fixed ρ > 0, ν R. By using the above asymptotic relation ∈ we obtain that lim Φν(u) = 0, which shows that in (3.1) the constant 0 is the best possible. u→∞ ρ Moreover, based on numerical experiments we believe, but are unable to prove the following conjecture. Conjecture. Ifν > ρ > 0,thenthefunctionΦν isstrictlyincreasing on(0, ),andconsequently ρ ∞ the following Turań type inequality holds (3.2) ρ/(ρ ν) Zν(u) 2 < Zν(u) 2 Zν−ρ(u)Zν+ρ(u). − ρ ρ − ρ ρ Note that for ν,ρ > 0 fixed if(cid:2)u tends(cid:3) to z(cid:2)ero, th(cid:3)en the asymptotic relation (see [22, 24]) ρZν(u) Γ(ν/ρ) ρ ∼ is valid. Using this relation we obtain Γ(ν/ρ 1)Γ(ν/ρ+1) lim Φν(u) = 1 − = ρ/(ρ ν) u→0 ρ − Γ2(ν/ρ) − for all ν > ρ > 0, which shows that in (3.2) the constant ρ/(ρ ν) is the best possible. − It is worthwhile to note here that in fact the inequality (3.2) is motivated by the following result. If the above conjecture were be true then (3.2) together with (3.1) would yield a generalization of (3.3), since for ρ= 1 the inequalities (3.2) and (3.1) reduce to (3.3). Theorem 3. Let K be the modified Bessel function of the second kind. Then the following ν Turań type inequalities hold for all ν > 1 and u > 0 (3.3) 1/(1 ν)[K (u)]2 < [K (u)]2 K (u)K (u) < 0. ν ν ν−1 ν+1 − − Moreover, the right-hand side of (3.3) holds true for all ν R. These inequalities are sharp in ∈ the sense that the constants 1/(1 ν) and 0 are the best possible. − For the sake of completeness it should be mentioned that the right-hand side of (3.3) was first proved by M.E.H. Ismail and M.E. Muldoon [20], and later by A. Laforgia and P. Natalini [26] and recently was deduced also by A´. Baricz [11, 12] and J. Segura [29], by using different approaches. The left-hand side of (3.3) was deduced very recently by using completely different methods by A´. Baricz [12] and J. Segura [29]. 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Department of Economics, Babe¸s-Bolyai University, Cluj-Napoca 400591, Romania E-mail address: [email protected] Dragana Jankov : Department of Mathematics, University of Osijek, 31000 Osijek, Croatia E-mail address: [email protected] Faculty of Maritime Studies, University of Rijeka, Rijeka 51000, Croatia E-mail address: [email protected]