n type inequalities for some probability density functions

Studia Scientiarum Mathematicarum Hungarica 47 (2), 175189 (2010) DOI: 10.1556/SScMath.2009.1123 First published online 7 August, 2009

TURÁN TYPE INEQUALITIES FOR SOME PROBABILITY DENSITY FUNCTIONS ÁRPÁD BARICZ∗ Babe³Bolyai University, Faculty of Economics, RO-400591 Cluj-Napoca, Romania e-mail: [email protected]

Communicated by P. Vértesi (Received December 6, 2007; accepted September 3, 2008)

Dedicated to my daughter Boróka

Abstract In this paper we present some Turán type inequalities for the probability density function (pdf) of the non-central chi-squared distribution, non-central chi distribution and Student distribution, respectively. Moreover, we improve a result of Laforgia and Natalini concerning a Turán type inequality for the modied Bessel functions of the second kind.

1. Turán's inequality for Legendre polynomials The famous result of P. Turán [43], established in 1950, in the special function theory, is the following inequality: (1.1)

£

¤2 Pn+1 (x) > Pn (x)Pn+2 (x) for all x ∈ (−1, 1) and n = 0, 1, 2, . . . ,

where Pn is the Legendre polynomial, that is, Pn (x) :=

dn dxn

µ

(x2 − 1) n!2n

n¶

=

[n/2] 1 X (2n − 2k)! (−1)k x2n−k 2n k!(n − k)!(n − 2k)! k=0

2000 Mathematics Subject Classication. Primary 62H10, 33C10. Key words and phrases. Probability density function; non-central chi-squared distribution; non-central chi distribution; student distribution; modied Bessel function; Turán type inequality. ∗ Research partially supported by the Institute of Mathematics, University of Debrecen, Hungary.

c 2009 Akadémiai Kiadó, Budapest 00816906/$ 20.00 °

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Á. BARICZ

and [x] denotes the greatest integer not greater than x. There is an extensive literature dealing with Turán's inequality (1.1), and has been extended in many directions for various orthogonal polynomials and special functions, for example: (1) Laguerre and Hermite polynomials [41] (2) ultraspherical polynomials [10, 12, 13, 17, 39, 41, 44] (3) Jacobi polynomials [22, 23] (4) Appell polynomials [15, 38] (5) Pollaczek and Lommel polynomials [11] (6) AskeyWilson polynomials [1] (7) Bessel functions of the rst kind [29, 39, 40] (8) modied Bessel functions of the rst kind [8, 25, 29, 42] (9) modied Bessel functions of the second kind [25, 26, 32] (10) conuent hypergeometric functions [8] (11) generalized complete elliptic integrals [5, 7] (12) Galué's generalized modied Bessel functions of the rst kind [6] (13) polygamma function and Riemann zeta function [14, 25, 31] (14) zeros of general Bessel functions [33] (15) zeros of ultraspherical, Laguerre and Hermite polynomials [20, 21, 30], and this list is far from being complete. This classical inequality still attracts the attention of mathematicians and it is worth mentioning that recently the inequality (1.1) was improved by Constantinescu [16], and further by Alzer et al. [2].

2. Turán type inequalities for some pdf-s In this paper our aim is to present some Turán type inequalities for the pdf of the non-central chi-squared, non-central chi and Student distributions, in order to improve and complete some results from [4]. Because the pdf-s of the non-central chi-squared and chi distribution are close connected to the modied Bessel function of the rst kind, not surprisingly  as we will see below  these functions also satises some interesting Turán type inequalities. To achieve our goal rst let us recall some basic things. Let X1 , X2 , . . . , Xn be random variables that are normally distributed with unit variance and nonzero mean µi , where i = 1, 2, . . . , n. It is known that X12 + X22 + · · · + Xn2 has the non-central chi-squared distribution with n = 1, 2, 3, . . . degrees of freedom and non-centrality parameter λ = µ21 + µ22 + · · · + µ2n . The probability density function χ2n,λ : (0, ∞) → (0, ∞) of the non-

TURÁN TYPE INEQUALITIES

177

central chi-squared distribution [27] is dened as χ2n,λ (x) := 2−n/2 e−(x+λ)/2

X xn/2+k−1 (λ/4)k k=0

=

Γ(n/2 + k)k!

¡√ ¢ e−(x+λ)/2 ³ x ´n/4−1/2 In/2−1 λx , 2 λ

where Iν (x) :=

X k=0

(x/2)2k+ν k!Γ(k + ν + 1)

is the modied Bessel function of the rst kind [45, p. 77]. Recall that when µ1 = µ2 = . . . = µn = 0, i.e. λ = 0, the above distribution reduces to the classical chi-squared distribution. The pdf χ2n,0 : (0, ∞) → (0, ∞) of this distribution is given by χ2n (x) := χ2n,0 (x) =

xn/2−1 e−x/2 . 2n/2 Γ(n/2)

On the other hand it is known that if X1 , X2 , . . . , Xn are random vari£ ¤ 1/2 ables such as above, then X12 + X22 + · · · + Xn2 has the non-central chi distribution with n = 1, 2, 3, . . . degrees of freedom and non-centrality pa£ ¤ 1/2 rameter τ = µ21 + µ22 + · · · + µ2n . The probability density function χn,τ : (0, ∞) → (0, ∞) of the non-central chi distribution [27] is dened as 2 +τ 2 )/2

χn,τ (x) := 2−n/2+1 e−(x

X xn+2k−1 (τ /2)2k k=0

= τ e−(x

2 +τ 2 )/2

³ x ´n/2 τ

Γ(n/2 + k)k!

In/2−1 (τ x).

Observe that when µ1 = µ2 = . . . = µn = 0, i.e. τ = 0, the above distribution reduces to the classical chi distribution with pdf χn,0 : (0, ∞) → (0, ∞) given by 2

xn−1 e−x /2 . χn (x) := χn,0 (x) = n/2−1 2 Γ(n/2)

It is easy to verify, in view of the well-known recurrence relation [45, p. 79] xIν−1 (x) − xIν+1 (x) = 2νIν (x),

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that the pdf of the non-central chi-squared distribution satises the following recurrence formula xχ2n,λ (x) − λχ2n+4,λ (x) = nχ2n+2,λ (x).

Recently, András and Baricz [4, Theorem 2.3] proved, among other things, that in fact the following Turán type inequality holds for all λ, x = 0 and n=1 £ ¤2 χ2n,λ (x)χ2n+4,λ (x) < χ2n+2,λ (x) .

(2.1)

This is an immediate consequence of the Turán type inequality Iν−1 (x)Iν+1 (x) < Iν2 (x)

which is called in literature as Amos' inequality [3, p. 243]. Moreover, using the Turán type inequality (ν + 1)Iν−1 (x)Iν+1 (x) = νIν2 (x),

which was proved by Thiruvenkatachar and Nanjundiah [42], Joshi and Bissu [29], Baricz [8], we deduced that the next reversed Turán-type inequality holds true "

(2.2)

χ2n+2,λ (x)

#2

" 5

χ2n+2 (x)

χ2n,λ (x) χ2n (x)

#"

χ2n+4,λ (x) χ2n+4 (x)

# .

Before we state our rst main result of this section we prove the following result for the modied Bessel function of the rst kind, which provides a generalization of the above Amos' inequality. We note that the idea of the proof of the next lemma is taken from [26, Lemma 2.3], where a similar result is proved for the Bessel function of the rst kind. Lemma 2.3.

xed x > 0.

The function ν 7→ Iν (x) is log-concave on (−1, ∞) for each

Proof. First we prove that for each xed b ∈ (0, 2] and each x > 0, the function ν 7→ Iν+b (x)/Iν (x) is decreasing, where ν = −(b + 1)/2, ν > −1. To show this, as in [26, Lemma 2.3], consider Neumann's formula [45, p. 441]

2 Iµ (x)Iν (x) = π

Zπ/2 £ ¤ Iµ+ν (2x cos θ) cos (µ − ν)θ dθ, 0

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TURÁN TYPE INEQUALITIES

which holds for all µ + ν > −1. Using this we nd that for 2ν + ε + b > −1 Iν (x)Iν+b+ε (x) − Iν+b (x)Iν+ε (x) 4 =− π

Zπ/2 I2ν+b+ε (2x cos θ) sin (bθ) sin (εθ) dθ, 0

which is negative for all b ∈ (0, 2] and ε ∈ (0, 2]. Consequently we obtain that the inequality Iν+b+ε (x)/Iν+ε (x) < Iν+b (x)/Iν (x) holds. Now, since £ ¤ £ ν 7→ ¤ Iν+b (x)/Iν (x) is decreasing, it follows that ν 7→ log Iν+b (x) − log Iν (x) £ ¤ is decreasing too. This implies that ν 7→ d log Iν (x) / dν is decreasing on (−1, ∞), and with this the proof is complete. ¤ Our rst main result improves the above Turán type inequalities (2.1) and (2.2) and completes Theorem 2.3 from [4].

and λ, τ = 0 consider the functions χ2a,λ , χa,τ : (0, ∞) → (0, ∞), dened by the relations Theorem 2.4. For a > 0

χ2a,λ (x) := e−(x+λ)/2

X (x/2)a/2 (λ/4)k k=0

2 +τ 2 )/2

χa,τ (x) := e−(x

Γ(a/2 + k)k!

X k=0

xk−1 ,

xa (τ /2)2k x2k−1 . 2a/2−1 Γ(a/2 + k)k!

Further, let us denote simply χ2a,0 (x) = χ2a (x) and χa,0 (x) = χa (x). Then the functions ¡√ ¢ a. x 7→ χ2a,λ (x) and x 7→ χa,τ x are log-concave, provided a = 2.

b. λ 7→ χ2a,λ (x) and τ 7→ χa,√τ (x) are log-concave on [0, ∞). c. a 7→ χ2a,λ (x) and a 7→ χa,τ (x) are log-concave on (0, ∞). d. a 7→ χ2a,λ (x)/χ2a (x) and a 7→ χa,τ (x)/χa (x) are log-convex on (0, ∞). ¤

¤ £

£

£

¤

e. a 7→ χ2a+2,λ (x)χ2a (x) / χ2a+2 (x)χ2a,λ (x) and a 7→ χa+2,τ (x)χa (x) / £

¤ χa+2 (x)χa,τ (x) are increasing on (0, ∞). In particular, using parts b, c, d, e, the following Turán type inequalities hold true for each x, a, a1 , a2 > 0 and λ, τ, λ1 , λ2 , τ1 , τ2 = 0 2

[χ2a, λ1 +λ2 (x)] 2

£ ¤£ ¤ = χ2a,λ1 (x) χ2a,λ2 (x) ,

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Á. BARICZ

h χ

i2

q τ1 +τ2 (x) a, 2

2

[χ2a1 +a2 ,λ (x)] 2

2

√ [χ a1 +a 2 , τ (x)] 2

 

χ2a1 +a2 (x) ,λ

2

χ2a1 +a2 (x)

= [χa,√τ1 (x)][χa,√τ2 (x)], £ ¤£ ¤ = χ2a1 ,λ (x) χ2a2 ,λ (x) , = [χa1 ,√τ (x)][χa2 ,√τ (x)],

2

"

 5

χ a1 +a2 ,τ (x)

#2

2

χ a1 +a2 (x) "

2

Proof.

χ2a2 ,λ (x)

#

·

χa1 ,τ (x) 5 χa1 (x)

χ2a2 ,λ (x)

#

χ2a2 (x) ¸·

,

¸ χa2 ,τ (x) , χa2 (x)

#· ¸ χ2a1 +2 (x) χ2a2 (x) = , χ2a2 +2,λ (x) χ2a1 ,λ (x) χ2a2 +2 (x) χ2a1 (x) ¸· ¸ · ¸· ¸ · χa1 +2,τ (x) χa2 ,τ (x) χa1 +2 (x) χa2 (x) = , χa2 +2,τ (x) χa1 ,τ (x) χa2 +2 (x) χa1 (x) χ2a1 +2,λ (x)

#"

#"

χ2a1 (x)

2

"

χ2a1 ,λ (x)

"

a1 = a2 , a1 = a2 .

a. Let us consider the function γν : (0, ∞) → (1, ∞), dened by γν (x) := 2ν Γ(ν + 1)x−ν/2 Iν

¡√ ¢ x ,

which is called sometimes as the normalized modied Bessel function of the rst kind of order ν . Taking into account the denition of γν we have that # " xa/2−1 e−x/2 −λ/2 2 (2.5) χa,λ (x) = e γa/2−1 (λx) = χ2a (x)e−λ/2 γa/2−1 (λx). a/2 2 Γ(a/2) It is easy to verify that the pdf χ2a is log-concave when a = 2. On the other hand [9, Theorem 2.2] the function γν is log-concave if ν > −1. Thus the function x 7→ γa/2−1 (λx) is log-concave too for all a > 0. Hence in view of (2.5) the function χ2a,λ is log-concave as a product of two log-concave functions. Now observe that from denitions we easily have ¡√ ¢ ¡√ ¢ √ (2.6) χa,τ x = χa,√λ x = 2 xχ2a,λ (x). √ Since the function¡ x 7→¢ x is log-concave on (0, ∞), we conclude that the √ function x 7→ χa,τ x is log-concave as a product of two log-concave functions.

TURÁN TYPE INEQUALITIES

181

b. Using the same argument as in the previous part we have that the function λ 7→ γa/2−1 (λx) is log-concave too for all a > 0, and consequently λ 7→ χ2a,λ (x) becomes also log-concave for all a > 0. Using the √ relation χa,√τ (x) = 2 xχ2a,τ (x) the required log-concavity of the function τ 7→ χa,√τ (x) follows. c. First consider the case when λ = τ = 0. Then clearly we have " Ã !# " Ã !# 2 d2 xa/2−1 e−x/2 d2 xa−1 e−x /2 1 0 ³a´ log = log = − ψ 5 0, da2 da2 4 2 2a/2 Γ(a/2) 2a/2−1 Γ(a/2) where ψ(x) = Γ0 (x)/Γ(x) is the so called digamma function, i.e. the logarithmic derivative of the gamma function. Here we used the known fact that the gamma function is log-convex on (0, ∞), i.e. the digamma function ψ is increasing on (0, ∞). For this we refer the interested reader for example to [19] and to the references therein. Thus we have proved that the functions a 7→ χ2a (x) and a 7→ χa (x) are log-concave on (0, ∞). Now consider the case when λ, τ > 0. Recall that ¡√ ¢ e−(x+λ)/2 ³ x ´a/4−1/2 Ia/2−1 λx , 2 λ ³ x ´a/2 2 2 χa,τ (x) = τ e−(x +τ )/2 Ia/2−1 (τ x). τ

χ2a,λ (x) =

Since for each xed x, λ, τ > 0 the functions a 7→ (x/λ)a/4−1/2 and a 7→ (x/τ )a/2 are log-concave, using Lemma 2.3 we conclude that the functions a 7→ χ2a,λ (x) and a 7→ χa,τ (x) are log-concave too on (0, ∞). d. From (2.5) we get that χ2a,λ (x)/χ2a (x) = e−λ/2 γa/2−1 (λx).

Recently, the author [8, Theorem 1] proved that the function ν 7→ γν (x2 ) is log-convex on (−1, ∞). This implies that the function a 7→ γa/2−1 (λx) is log-convex too on (0, ∞) and consequently the function a 7→ χ2a,λ (x)/χ2a (x) is log-convex, as we required. Finally, application of (2.6) yields the logconvexity of a 7→ χa,τ (x)/χa (x) on (0, ∞). e. It is known [8, Theorem 1] that ν 7→ γν+1 (x)/γν (x) is increasing on (−1, ∞) for all xed x > 0. Hence we have that the function a 7→ γa/2 (λx)/γa/2−1 (λx) is increasing on (0, ∞) for each xed x > 0, λ = 0. Consequently in view of (2.5) and (2.6) the required result follows, and thus the proof is complete. ¤

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It is worth mentioning here that from the above inequalities can be deduced many similar interesting inequalities to those given in (2.1) and (2.2). For example, choosing in the fourth inequality of Theorem 2.4 τ = 0 and a1 = n, a2 = n + 2, respectively, we get the following Turán type inequality for the pdf of the chi distribution: £

(2.7)

¤2 χn+1 (x) = χn (x)χn+2 (x),

which holds for each n = 1 and x > 0. Now taking n = 1 in (2.7) for all x > 0 we obtain that £ ¤2 χ2 (x) = χ1 (x)χ3 (x). In fact this last particular Turán type inequality establishes a relation between the pdf of the Rayleigh, half-normal and Maxwell distributions. Namely, χ2 is the pdf of a particular Rayleigh distribution with parameter 1, χ1 is the pdf of the well-known half-normal distribution, and χ3 is the pdf of a particular Maxwell distribution which arises in many problems of physics and chemistry. As we has seen in the previous theorem in particular for λ = τ = 0 the pdf-s of the chi and the chi-squared distribution are log-concave with respect to their parameter. In fact many other univariate distributions has the same property. Namely, for example the exponential, power, Weibull, gamma and Pareto distributions has the property that their pdf is log-concave with respect to their parameter. Indeed, it is easy to verify that the functions a 7→ ae−ax ,

a

a 7→ axa−1 e−x ,

a 7→

xa−1 e−x , Γ(a)

a 7→ ax−a−1

are log-concave on (0, ∞) for each xed x belonging to the interval which support the distribution in the question. In what follows we are mainly interested on the pdf of the Student distribution [27] of Gosset. Suppose that X1 , X2 , . . . , Xn are independent random variables that are normally√ distributed with expected valueµ and variance 1. Then the random variable n(X − µ)/S  where X is the arithmetic mean P 2 of X1 , X2 , . . . , Xn and (n − 1)S 2 = ni=1 (Xi − X)  has a Student distribution with n − 1 degrees of freedom and the pdf Sn : R → (0, ∞) of this distribution is dened as n+1 1 Γ( 2 ) Sn (x) := √ πn Γ( n2 )

¶− n+1 µ 2 x2 1+ n

The next result establishes two Turán type inequalities for this function.

183

TURÁN TYPE INEQUALITIES

Theorem 2.8.

ned by

For a > 0 consider the function Sa : R → (0, ∞), de-

a+1 1 Γ( 2 ) Sa (x) := √ πa Γ( a2 )

µ ¶− a+1 2 x2 1+ . a

Then the function a 7→ Sa (x) is strictly log-concave on (0, 1] for each xed x ∈ R, and is strictly log-concave too on (1, ∞) for each xed p |x| 5 2a/(a − 1). In particular, for each n = 1, n = 2 respectively, the next Turán type inequalities hold true n+1 (x)] [S n(n+2)

£

Sn+1 (x)

2

¤2

> S 1 (x)S n

1 n+2

(x),

x ∈ R, r

> Sn (x)Sn+2 (x),

|x| 5

2(n + 2) . n+1

Proof. Simple computations show that

· µ ¶ ³a´ 1¸ ¤ a+1 d£ log Sa (x) = ψ −ψ 2 − da 2 2 a · µ ¶¸ a+1 x2 x2 + · 2 − log 1 + . a x +a a

Observe that to prove the strict log-concavity of a 7→ Sa (x) it is enough to show that the function a 7→ ∂ log Sa (x)/∂a is strictly decreasing. To prove this for convenience we consider the functions f, g : (0, ∞) → R, dened by µ ¶ 1 1 f (a) := ψ a + − ψ(a) − 2 2a and

µ ¶ a+1 x2 x2 g(a) := · 2 − log 1 + . a x +a a

Then clearly we have 2∂ log Sa (x)/∂a = f (a/2) + g(a). Due to Qiu and Vuorinen [37, Theorem 2.1] it is known that the function f is strictly decreasing and convex from (0, ∞) onto (0, ∞). On the other hand (a − 1)(x2 + a) − a(a + 1) dg(a) = x2 , da a2 (x2 + a)2

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Á. BARICZ

p and this negative if a ∈ (0, 1] and x ∈ R or if a > 1 and |x| 5 2a/(a − 1). Thus the function a 7→ d log Sa (x)/ da is decreasing, and consequently the strict log-concavity of a 7→ Sa (x) follows. Namely, if a1 , a2 ∈ (0, 1] and α ∈ [0, 1], then for each x ∈ R we have

(2.9)

£ ¤ α£ ¤ 1−α Sαa1 +(1−α)a2 (x) = Sa1 (x) Sa2 (x) .

Thus taking in (2.9) α = 1/2, a1 = 1/n and a2 = 1/(n + 2), we obtain the rst inequality in Theorem 2.8. Moreover, (2.9) holds also for eacha1 , a2 > 1 and r ¾ ½r 2a1 2a2 . |x| 5 min , a1 − 1 a2 − 1 Finally, choosing in (2.9) α = 1/2, a1 = n and a2 = n + 2 we get the second Turán type inequality, and thus the proof is complete. ¤ As far as we know the previous results are new. However, there is an extensive literature for similar inequalities for discrete random variables, please see for example [18, 28] and the references therein. Let X be a random variable and for each n = 1, 2, . . . consider the probabilities pn = P (X = xn ), where x1 < x2 < . . . . By denition the discrete random variable X is logconcave if the (Turán type inequality) p2n+1 = pn pn+2 holds for all n = 1. This inequality is sometimes referred to the quadratic Newton inequality as Niculescu pointed out in [36]. For example, the Poisson distribution and the geometric distribution has the above property [28]. Moreover, as Devroye [18] observed, the discrete Bessel distribution has the same property. Namely, bn+1 /bn is decreasing in n and consequently b2n+1 = bn bn+2 holds true, where for each ν > −1, a > 0 and n = 0 by denition bn := P (X = n) =

(a/2)2n+ν . Iν (a)n!Γ(n + ν + 1)

3. Turán type inequalities for the modied Bessel functions of the second kind Consider the modied Bessel function of the second kindKν (called sometimes as the MacDonald or Hankel function), dened [45, p. 78] as follows: Kν (x) :=

π I−ν (x) − Iν (x) , 2 sin νπ

where the right of this equation is replaced by its limiting value if ν is an integer or zero. Recently Laforgia and Natalini [32, Theorem 2.4] proved that

185

TURÁN TYPE INEQUALITIES

for each ν1 , ν2 > −1/2 and x > 0 the following reversed Turán type inequality holds true 2

Kν1 (x)Kν2 (x) = [K ν1 +ν2 (x)] .

(3.1)

2

Observe that in particular, for ν1 = ν and ν2 = ν + 2, we get for all ν > −1/2 and x > 0 the inequality [32, Eq. 2.18] £ ¤2 Kν (x)Kν+2 (x) = Kν+1 (x) .

(3.2)

It is worth mentioning here that (3.2) holds true for ν = −1/2, and consequently for ν = −3/2 too. To prove these rst recall the following formulas [45, p. 79] £ ¤ x Kν−1 (x) − Kν+1 (x) = −2νKν (x), K−ν (x) = Kν (x),

which hold for each ν and x unrestricted real numbers. From this clearly K−1/2 (x) = K1/2 (x) and K−3/2 (x) = K3/2 (x) = (1 + 1/x)K1/2 (x). These imply that the inequality £ ¤2 £ ¤2 K−3/2 (x)K1/2 (x) − K−1/2 (x) = K−1/2 (x)K3/2 (x) − K1/2 (x) =

¤2 1£ K1/2 (x) > 0 x

holds for all x > 0, as we required. Now, this suggest the following result, which improves Theorem 2.4 from [32]. We note here that a similar argument to those presented below in the proof of the next result can be used, as Giordano et al. [24, Remark 3.2] pointed out, in order to prove that the function Kν is log-convex on (0, ∞). Theorem 3.3. The function ν 7→ Kν (x) is strictly log-convex on R for each xed x > 0. In particular, the reversed Turán type inequalities (3.1) and (3.2) hold for all x > 0, ν1 , ν2 and ν arbitrary real numbers. Moreover  due to the strict log-convexity  in (3.1) equality holds if and only if ν1 = ν2 and the inequality (3.2) is strict. Proof. First recall the well-known HölderRogers inequality [34, p. 54],

that is,

Zb

(3.4) a

¯ ¯ ¯ f (t)g(t)¯ dt 5

" Zb a

¯ ¯ ¯ f (t)¯ p dt

#1/p " Zb a

¯ ¯ ¯ g(t)¯ q dt

#1/q ,

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Á. BARICZ

where p > 1, 1/p + 1/q = 1, f and g are real functions dened on [a, b] and |f |p , |g|q are integrable functions on [a, b]. On the other hand it is known the following integral representation [45, p. 181] of the modied Bessel function of the second kind Z∞

(3.5)

e−x cosh t cosh (νt) dt,

Kν (x) = 0

which holds for each x > 0 and ν ∈ R. Using (3.4), (3.5) and the fact that the function ν 7→ cosh (νt) is strictly log-convex on R for all xed t = 0, we conclude that Z∞ e−x cosh t cosh (αν1 t + (1 − α)ν2 t) dt

Kαν1 +(1−α)ν2 (x) = 0

Z∞
0. Consequently by denition ν 7→ Kν (x) is strictly log-convex on R, which completes the proof. ¤ We note that after the rst draft of this manuscript had been completed we found that Ismail and Muldoon [26, Lemma 2.2] proved that for each xed x > 0 and each xed b > 0 the function ν 7→ Kν+b (x)/Kν (x) is increasing on R. Clearly this implies [35, p. 318] that ν 7→ Kν (x) is log-convex on R. Acknowledgement. I am grateful to Professor Martin E. Muldoon for pointing out an error in the rst draft of this manuscript. Thanks are due to him for his useful comments and for a copy of papers [26, 35].

TURÁN TYPE INEQUALITIES

187

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