SPECIAL FUNCTIONS: APPROXIMATIONS AND BOUNDS

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1 Applicble Anlysis nd Discrete Mthemtics, 1 7), Avilble electroniclly t Presented t the conference: Topics in Mthemticl Anlysis nd Grph Theory, Belgrde, September 1 4, 6. SPECIAL FUNCTIONS: APPROXIMATIONS AND BOUNDS P. Cerone The Steffensen inequlity nd bounds for the Čebyšev functionl re utilised to obtin bounds for some clssicl specil functions. The technique relies on determining bounds on integrls of products of functions. The bove techniques re used to obtin novel nd useful bounds for the Bessel function of the first kind, the Bet function, nd the Zet function. 1. INTRODUCTION AND REVIEW OF SOME RECENT RESULTS There re number of results tht provide bounds for integrls of products of functions. The min techniques tht shll be employed in the current rticle involve the Steffensen inequlity nd vriety of bounds relted to the Čebyšev functionl. There hve been some developments in both of these in the recent pst with which the current uthor hs been involved. These hve been put to fruitful use in vriety of res of pplied mthemtics including qudrture rules, in the pproximtion of integrl trnsforms, s well s in pplied probbility problems see [31], [] nd [11]. This rticle is review of these developments nd some new results re lso presented. It is intended tht in the current rticle the techniques will be utilised to obtin useful bounds for specil functions. The methodologies will be demonstrted through obtining bounds for the Bessel function of the first kind, the Bet function nd the Zet function. It is instructive to introduce some techniques for pproximting nd bounding integrls of the product of functions. We first present inequlities due to Steffensen nd then review bounds for the Čebyšev functionl. Mthemtics Subject Clssifiction. Primry 6D15, 6D; Secondry 6D1. Key Words nd Phrses. Čebyšev functionl, Grüss inequlity, Bessel, Bet nd Zet function bounds. 7

2 Specil functions: Approximtions nd bounds 73 The following theorem is due to sc Steffensen [45] see lso [11] nd [16]). Theorem 1. Let h : [, b] R be nonincresing mpping on [, b] nd g : [, b] R be n integrble mpping on [, b] with then 1.1) φ b λ 1.) λ = b < φ g t) Φ < for ll x [, b], h x) dx + Φ b b λ Gx) dx, h x) dx b h x) g x) dx +λ Φ h x) dx + φ b +λ Gx) = g x) φ Φ φ, Φ φ. h x) dx, Remrk 1. We note tht the result 1.1) my be rerrnged to give Steffensen s better known result tht b b λ h x) dx b h x) Gx) dx +λ h x) dx, λ is s given by 1.) nd Gx) 1. Eqution 1.3) hs very plesnt interprettion, s observed by Steffensen, tht if we divide by λ then 1.4) 1 λ b b λ h x) dx b Gx)hx) dx 1 b λ Gx) dx +λ h x) dx. Thus, the weighted integrl men of h x) is bounded by the integrl mens over the end intervls of length λ, the totl weight. Now, for two mesurble functions f, g : [, b] R, define the functionl, which is known in the literture s Čebyšev s functionl, by 1.5) T f, g) := M fg) M f) M g), the integrl men is given by 1.6) M f) := 1 b f x) dx. b

3 74 P. Cerone The integrls in 1.5) re ssumed to exist. The weighted Čebyšev functionl is defined by 1.7) T f, g; p) := M fg; p) M f; p) M g; p), the weighted integrl men M f; p) is given by 1.8) P M f; p) = with the weight P stisfying < P <. We note tht b T f, g; 1) T f, g) p x)f x)dx, P = b p x)dx nd M f; 1) M f). We further note tht bounds for 1.5) nd 1.7) my be looked upon s pproximting the integrl men of the product of functions in terms of the product of integrl mens which re more esily clculted explicitly. Bounds re perhps best procured from identities. It is worthwhile noting tht number of identities relting to the Čebyšev functionl lredy exist. The reder is referred to [4] Chpters IX nd X.) Korkine s identity is well known, see [4, p. 96] nd is given by 1.9) T f, g) = 1 b b ) ) f x) f y) g x) g y) dxdy. b ) It is identity 1.9) tht is often used to prove n inequlity due to Grüss for functions bounded bove nd below, [4]. The Grüss inequlity [35] is given by 1.1) T f, g) 1 4 Φ f φ f ) Φ g φ g ), φ f f x) Φ f for x [, b], with φ f, Φ f constnts nd similrly for g x). The interested reder is lso referred to Drgomir [3] nd Fink [34] for extensive tretments of the Grüss nd relted inequlities. Identity 1.9) my lso be used to prove the Čebyšev inequlity which sttes tht for f ) nd g ) synchronous, nmely f x) f y))g x) g y)),.e. x, y [, b], then 1.11) T f, g). As mentioned erlier, there re mny identities involving the Čebyšev functionl 1.5) or more generlly 1.7). Recently, Cerone [11] obtined, for f, g : [, b] R f is of bounded vrition nd g continuous on [, b], the identity 1.1) T f, g) = 1 b b ) ψ t) dft),

4 Specil functions: Approximtions nd bounds ) ψ t) = t )Gt, b) b t)g, t) with 1.14) Gc, d) = d g x) dx. c The following theorem ws proved in [11]. Theorem. Let f, g : [, b] R, f is of bounded vrition nd g is continuous on [, b]. Then sup ψ t) b f), t [,b] 1.15) b ) T f, g) L b ψ t) dt, for f L Lipschitzin, b ψ t) df t), for f monotonic nondecresing, b f) is the totl vrition of f on [, b]. The bounds for the Čebyšev functionl were utilised to procure pproximtions to moments nd moment generting functions in [11] nd [4]. The reder is referred to [31] nd the references therein for pplictions to numericl qudrture of trpezoidl nd Ostrowski functionls, which were shown to be relted to the Čebyšev functionl in [15]. For other Grüss type inequlities, see the books [9] nd [4], nd the ppers [19], [3], [6], [9], [3], further references re given. Recently, Cerone nd Drgomir [19] [3] hve pointed out generlistions of the bove results for integrls defined on two different intervls nd more generlly in mesurble spce setting see lso, [8] nd [14]). The functionl T f, g; p) defined in 1.7) stisfies number of identities including tht due to Sonin [4] 1.16) P T f, g; p) = b p x) f x) γ ) g x) M g; p) ) dx from which the following bounds my be procured. Nmely, b inf f ) γ p x) g x) M g; p) dx, γ R ) 1/ b 1.17) P T f, g; p) p x)f x) M f; p)) dx 1/ b p x)g x) M g; p)) dx),

5 76 P. Cerone 1.18) b p x)h x) M h; p)) dx = b p x) h x) dx P M h; p) nd P is s defined in 1.8). Further, it my be esily shown by direct clcultion tht, [ ] b 1.19) inf p x) f x) γ) b dx = p x) f x) M f; p) ) dx. γ R Some of the bove results re used to find bounds for the Bessel function Section ), the Bet function Section 3), the Zet function Section 4) see lso [9] for further detils).. BOUNDING THE BESSEL FUNCTION In this section we investigte techniques for determining bounds on the Bessel function of the first kind see lso [1], [13]). In Abrmowitz nd Stegun [1] eqution 9.1.1) defines the Bessel of the first kind.1) J ν z) = γ ν z) 1.) γ ν z) = 1 t ) ν 1 coszt) dt, Re ν) > 1, z ν ) π Γ ν + 1 ). For the current work the interest is in both z nd ν rel. Theorem 3. For z rel then.3) nd 1 B.4) B 1, ν + 1 ) 1 B, ν + 1 ) ; 1 λ) J ν z) γ ν z) 1 B, ν + 1 ) ; λ 1 1 B, ν + 1 ), ν > 1 1, ν + 1 ) ; λ 1 1 B, ν + 1 )

6 Specil functions: Approximtions nd bounds 77 J ν z) γ ν z) 1 B 1, ν + 1 ) 1 B, ν + 1 ) ; 1 λ), 1 < ν < 1,.5) B α, β; x) = x u α 1 1 u) β 1 du, the incomplete Bet function,.6) B α, β) = B α, β; 1) = Γ α) Γ β) Γ α + β), the Bet function, nd.7) λ 1 = sin z z. Tking ν = 1 produces equlity in.3) nd.4), nmely, J 1 z) = γ 1 z) sin z z. Proof. Consider the cse ν > 1 then h t) = 1 t ) ν 1 is nonincresing for t [, 1]. Further, tking g t) = coszt we hve tht 1 g t) 1 for t [, 1] nd, from 1.) λ = 1 1 coszt + 1) dt = sin z ). z Utilising Theorem 1 nd fter some lgebr, the bove results re procured. Remrk. We note from.1) tht we my obtin clssicl bound see [1, p. 36]) for J ν z), nmely J ν z) ) ν z πγ ν + 1 ) 1 1 t ) ν 1 dt, from.5) nd.6).8) to give 1 1 t ) ν 1 dt = 1 1 B, ν + 1 ) = 1.9) J ν z) z ν 1 Γ ν + 1). 1 Γ Γ ν + ) 1 ) Γ ν + 1)

7 78 P. Cerone The following theorem gives bound on the devition of the Bessel function from n pproximnt see lso [17]). This is ccomplished vi bounds on the Čebyšev functionl for which there re numerous results. Theorem 4. The following result holds for the Bessel function of the first kind J ν z). Nmely, z ν.1) J ν z) ) Γ ν + 1) sinz z 1/ ) ν z Γ ν) 1 π Γ ν + 1 ) Γ ν + 1 ) Γ ν + 1) [ cosz ) 1 sin z + 4 cosz ) ] 1/. z 4 Proof. Sketch) We use the norm result for the.1) nd.) consider, Čebyšev functionl. From.11) Q ν z) = J ν z) 1 γ ν z) = 1 t ) ν 1 coszt) dt. Let f t) = 1 t ) ν 1 nd g t) = coszt. 3. BOUNDING THE BETA FUNCTION The incomplete bet function is defined by 3.1) B x, y; z) = z t x 1 1 t) y 1 dt, < z 1. We shll restrict our ttention to x > 1 nd y > 1. In this region we observe tht 3.) t x 1 z x 1 nd 1 z) y 1 1 t) y 1 1 with t x 1, n incresing function nd 1 t) y 1, decresing function, for t [, z]. The following theorem follows from utilizing Steffensen s result s depicted in Theorem 1 [1], see lso [17] for detils. Theorem 5. For x > 1 nd y > 1 with z 1 we hve the incomplete Bet function defined by 3.1) stisfying the following bounds 3.3) mx {L 1 z),l z)} B x, y; z) min {U 1 z),u z)},

8 Specil functions: Approximtions nd bounds ) L 1 z) = zx 1 y nd [ 1 z + z ) ] y [ 1 z) y, U 1 z) = zx z ) y ] x y x 3.5) L z) = λ x z) x + 1 z) y 1 zx λ x z), x x U z) = 1 z) y 1 λ z) ) x + zx z λ z) ) x x x with 1 1 z)[1 z 1 y)] 3.6) λ z) = y [1 1 z) y 1]. Proof. Using Steffensen s inequlity) If we tke h t) = 1 t) y 1 nd g t) = t x 1, then for y > 1 nd x > 1, h t) is decresing function of t nd g t) z x 1. Thus, from 1.1) z 3.7) z x 1 1 t) y 1 dt z t x 1 1 t) y 1 dt z z λ 1 λ1 x 1 1 t) y 1 dt, λ 1 = λ 1 z) = z t x 1 z x 1 dt = z x. Corollry 1. For x > 1 nd y > 1 we hve the Bet function B x, y) = 1 t x 1 1 t) y 1 dt, which is symmetric in x nd y, stisfies the following bounds, { } 1 3.8) mx xy x, 1 yx y B x, y) { [ 1 min ) y ], 1 [ ) x ]}. y x x y Proof. Put z = 1 in 3.6) to give λ 1) = 1 followed by the obvious correspondences from 3.3) 3.5). y The following theorem reltes to the Bet function [17] nd is correction of the result in [1].

9 8 P. Cerone Theorem 6. For x > 1 nd y > 1 the following bounds hold for the Bet function, nmely, 3.9) 1 B x, y) min {Ax),Ay)}, xy 3.1) Ax) = x 1 x 1+ x x 1 Proof. Sketch. Using the Čebyšev functionl nd Sonin identity). We hve from 1.16) 1.17) with p ) 1, Tht is, T f, g) = M fg) M f) M g) M f ) γ g ) M g) ). 3.11) T f, g) inf f ) γ γ M g ) M g). ). If we tke f t) = t x 1, g t) = 1 t) y 1 then M f) = 1 x nd M g) = 1 y. The following plesing result is vlid [1], [17]). Theorem 7. For x > 1 nd y > 1 we hve 3.1) 1 x 1 B x, y) xy x x 1 y 1 y , y 1 the upper bound is obtined t x = y = = Proof. Using the -norm bound for the Čebyšev functionl) We hve from 1.17) 1.19) b ) T f, g) b 1/ f t) dt M f)) b Tht is, tking f t) = t x 1, g t) = 1 t) y 1. Now, consider 3.13) C x) = x 1 x x 1. g t) dt M g)) 1/. The mximum occurs when x = x = to give C x ) = Hence, becuse of the symmetry we hve the upper bound s stted in 3.1).

10 Specil functions: Approximtions nd bounds 81 Remrk 3. In recent pper Alzer [4] shows tht 1 ) 1 xy B x, y) b A = mx x 1 x Γ x) Γ x) = , nd b A re shown to be the best constnts. This uniform bound of Alzer is only smller for smll re round, 3 + ) 5 while the first upper bound in 3.1) provides better bound over much lrger region of the x y plne. We my stte the following corollry given the results bove. Corollry. For x > 1 nd y > 1 we hve 1 xy B x, y) min {C x) C y),b A}, C x) is defined by 3.13) nd b A by 3.14). Remrk 4. The upper bound in Theorem 6 by numericl investigtion, seems not to be s good s tht given in Theorem 7. Anlyticlly, the trnsformtion χ = x 1 x nd η = y 1 y in 3.9) 3.1) results in requiring to show tht for χ, η 1. Hχ, η) = 1 χ) 1 χ 1 χ η 1 + χ 1 η 1 + η 4. BOUNDS FOR THE EULER ZETA AND RELATED FUNCTIONS 4.1. BACKGROUND TO ZETA AND RELATED FUNCTIONS The Zet function [1]) 4.1) ζx) := n=1 1 n x, x > 1 ws originlly introduced in 1737 by the Swiss mthemticin Leonhrd Euler ) for rel x who proved the identity 4.) ζx) := p 1 1 ) 1 p x, x > 1, p runs through ll primes. It ws Riemnn who llowed x to be complex vrible z nd showed tht even though both sides of 4.1) nd 4.) diverge for Re z) 1, the function hs continution to the whole complex plne with simple pole t z = 1 with residue 1. The function plys very significnt role

11 8 P. Cerone in the theory of the distribution of primes see [5], [7], [7], [3], [37] nd [46]). One of the most striking properties of the zet function, discovered by Riemnn himself, is the functionl eqution πz ) 4.3) ζz) = z π z 1 sin Γ1 z)ζ1 z) tht cn be written in symmetric form to give 4.4) π z z Γ = π )ζz) 1 z ) ) 1 z Γ ζ1 z). ζs) is commonly referred to s the Riemnn Zet function nd if s is restricted to rel vrible x, it is referred to s the Euler Zet function. In ddition to the reltion 4.3) between the zet nd the gmm function, these functions re lso connected vi the integrls [3] 4.5) ζx) = 1 t x 1 Γx) e t dt, x > 1, 1 nd 4.6) ζx) = 1 t x 1 Cx) e t dt, x >, ) Cx) := Γx) 1 1 x) nd Γ x) = e t t x 1 dt. In the series expnsion 4.8) te xt e t 1 = m= B m x) tm m!, B m x) re the Bernoulli polynomils fter Jcob Bernoulli), B m ) = B m re the Bernoulli numbers. They occurred for the first time in the formul [1, p. 84] 4.9) m k=1 k n = B n+1m + 1) B n+1, n, m = 1,, 3,.... n + 1 One of Euler s most celebrted theorems discovered in 1736 Institutiones Clculi Differentilis, Oper 1), Vol. 1) is 4.1) ζn) = 1) n 1 n 1 π n B n ; n = 1,, 3,.... n)!

12 Specil functions: Approximtions nd bounds 83 The Zet function is lso explicitly known t the non-positive integers by ζ n) = 1) n B n+1, for n = 1,,... n + 1 The result my lso be obtined in stright forwrd fshion from 4.6) nd chnge of vrible on using the fct tht 4.11) B n = 1) n 1 t n 1 4n e πt 1 dt from Whittker nd Wtson [48, p. 16]. We note here tht ζn) = A n π n, A n = 1) n 1 n 1 n n + 1)! + j=1 1) j 1 j + 1)! A n j nd A 1 = 1 3!. Further, the Zet function for even integers stisfy the reltion Borwein et l. [7], Srivstv [43]) ζn) = n + 1 ) 1 n 1 ζj)ζn j), n N \ {1}. j=1 Despite severl efforts to find formul for ζn + 1), there seems to be no elegnt closed form representtion for the zet function t the odd integer vlues. Severl series representtions for the vlue ζn + 1) hve been proved by Srivstv nd co-workers in prticulr, see [43], [44]. There re lso integrl representtions for ζ n + 1), see [1, p. 87] nd [8]. Both series representtions nd the integrl representtions re however somewht difficult in terms of computtionl spects nd time considertions. We note tht there re functions tht re closely relted to ζ x). Nmely, the Dirichlet η ) nd λ ) functions given by 4.1) η x) = nd 4.13) λx) = n=1 n= 1) n 1 n x = 1 t x 1 Γ x) e t + 1 dt, x > 1 n + 1) x = 1 t x 1 Γ x) e t dt, x >. e t

13 84 P. Cerone These re relted to ζ x) by 4.14) η x) = 1 1 x) ζ x) nd λx) = 1 x) ζ x) stisfying the identity 4.15) ζ x) + η x) = λx). It should be further noted tht explicit expressions for both of η n) nd λn) exist s consequence of the reltion to ζ n) vi 4.14). 4.. RESULTS FOR THE ZETA FUNCTION Lemm 1. The following identity involving the Zet function holds. Nmely, 4.16) t x e t dt = C x + 1)ζ x + 1) xc x)ζ x), x >, + 1) C x) is s given by 4.7). Bsed on the identity in Lemm 1, the following theorem ws developed see Alzer [], Cerone et l. [18], nd lso [1] the constnts in the bounds of 4.17) were developed. Theorem 8. For rel numbers x > we hve 4.17) ln 1 ) b x) < ζ x + 1) 1 b x) ) ζ x) < b x), 4.18) b x) = 1 x 1, nd the constnts ln 1 nd 1 re shrp. The following is correction of result obtined by the uthor [13] by utilising the Čebyšev functionl bounds given by 1.17) nd 4.5). Theorem 9. For α > the Zet function stisfies the inequlity 4.19) 4.) κ = α 1 ζ α + 1) α π 6 κ 1 α Γ α 1) Γ α) )1, Γ α + 1) ) [π 1 π 7ζ 3)] 1 =

14 Specil functions: Approximtions nd bounds 85 with equlity obtined t α = 1. The following theorem ws obtined in [17] utilising bounds for the Čebyšev functionl. Theorem 1. For α > 1 nd m = α the zet function stisfies the inequlity 4.1) + 1) ζ α + 1) α mγm ζ m + 1)Γα m + 1) Γ α + 1) α m+ 1 ) Γ α + 1) E m Γ α m + 1) Γ α m + 1) )1, 4.) E m = m Γ m + 1) λm) λm + 1) ) 1 Γ m + 1)ζ m + 1), with λ ) given by 4.13). Equlity in 4.1) results when α = m. Proof. Sketch using the Čebyšev Functionl Approch). Let 4.3) τ α) = Γ α + 1)ζ α + 1) = m = α. Mke the ssocitions = e x/ 4.4) p x) = e x/, f x) = x α e x 1 dx x m e x/ e x/ xα m dx, α > 1 x m e x/ e x/, g x) = xα m then we hve from 1.17) 4.5) P = e x/ dx =, M f; p) = 1 e x/ x m e x/ e dx = 1 Γ m + 1)ζ m + 1), x/ M g; p) = 1 e x/ x α m dx = α m Γ α m + 1). The following corollry provides upper bounds for the zet function t odd integers.

15 86 P. Cerone Corollry 3. The inequlity 4.6) Γ m + 1) m 1 ) ζ m) m+1 1 ) ζ m + 1) ) holds for m = 1,,.... Γ m + 1)ζ m + 1) > Proof. From eqution 4.) of Theorem 1, we hve E m >. Utilising the reltionship between λ ) nd ζ ) given by 4.14) redily gives the inequlity 4.6). Remrk 5. In 4.6), if m is odd, then m nd m+1 re even so tht n expression in the form 4.7) α m)ζ m) β m)ζ m + 1) γ m)ζ m + 1) >, results, 4.8) α m) = m 1 ) Γ m + 1), β m) = m+1 1 ) Γ m + 1) γ m) = Γ m + 1). nd Thus for m odd we hve 4.9) ζ m + 1) < α m)ζ m) γ m)ζ m + 1). β m) Tht is, for m = k 1, we hve from 4.9) 4.3) ζ 4k 1) < α k 1)ζ 4k ) γ k 1)ζ k) β k 1) giving for k = 1,, 3, for exmple, ) ζ 3) < 1 π π = , 7 7 ) ζ 7) < 1 π6 π = , ) ζ 11) < 1 6π1 π = The bove bound for ζ 3) ws obtined previously by the uthor in [13] from 4.). If m is even then for m = k we hve from 4.9) 4.31) ζ 4k + 1) < α k)ζ 4k) γ k)ζ k + 1), k = 1,,.... β k)

16 Specil functions: Approximtions nd bounds 87 We notice tht in 4.31), or equivlently 4.7) with m = k there re two zet functions with odd rguments. There re number of possibilities for resolving this, but firstly it should be noticed tht ζ x) is monotoniclly decresing for x > 1 so tht ζ x 1 ) > ζ x ) for 1 < x 1 < x. Firstly, we my use lower bounds obtined in [1], nmely L x) = 1 b x)) ζ x) + ln 1 ) b x) or L x) = ζ x + ) bx + 1) 1 b x + 1), b x) is given by 4.18). However, from numericl investigtion in [1], it seems tht L x) > L x) for positive integer x nd so we hve from 4.31) 4.3) ζ L 4k + 1) < α k)ζ k) γ k)l k), β k) we hve used the fct tht L x) < ζ x + 1). Secondly, since the even rgument ζ k + ) < ζ k + 1), then from 4.31) we hve 4.33) ζ E 4k + 1) < α k)ζ 4k) γ k)ζ k + ). β k) Finlly, we hve tht ζ m + 1) > ζ m + 1) so tht from 4.7) we hve, with m = k on solving the resulting qudrtic eqution tht 4.34) ζ Q 4k + 1) < β k) + β k) + 4γ k)α k)ζ 4k). γ k) For k = 1 we hve from 4.3) 4.34) tht nd for k = ζ L 5) < π ζ E 5) < π π4 16 7π ) = , 1 ) = , ζ Q 5) < π 4 = ; ζ L 9) < π8 1 ζ E 9) < π π6 1 ) = 1.856, 6 ) 1 π4 = , ζ Q 9) < π8 =

17 88 P. Cerone It should be noted tht the bove results give tighter upper bounds for the odd zet function evlutions thn were possible using the methodology utilising techniques bsed round Theorem 8 s demonstrted by the numerics which re presented in Tble 1 of [1]. Numericl experimenttion using Mple seems to indicte tht the upper bounds for ζ L 4k + 1),ζ E 4k + 1) nd ζ Q 4k + 1) re in incresing order. Anlytic demonstrtion tht ζ L 4k + 1) is better remins n open problem. 5. CONCLUDING REMARKS In the pper the usefulness of some recent results in the nlysis of inequlities, hs been demonstrted through ppliction to some specil functions. Although these techniques hve been pplied in vriety of res of pplied mthemtics, their ppliction to specil functions does not seem to hve received much ttention to dte. There re mny specil functions which my be represented s the integrl of products of functions. The investigtion in the current rticle hs restricted itself to the investigtion of the Bessel function of the first kind, the Bet function nd the Zet function. It my be surmised from the bove investigtions tht the ccurcy of the bounds over prticulr regions of prmeters cnnot be scertined priori. It hs been demonstrted, however, tht some useful bounds my be obtined which hve hitherto do not seem to hve been discovered. The pproch of utilising developments in the field of inequlities to specil functions hs been shown to hve the potentil for further development. A generl investigtion of Dirichlet series hs lso been undertken in [], [1] utilising convexity rguments nd it is shown tht in prticulr ) 1 5.1) ζ s + 1) A ζ s), 1 G ζ s),ζ s + ) ) ζ s + ) A, ) is the rithmetic men nd G, ) the geometric men. Specificlly, for s = n, then 5.) ζ n + 1) H ζ n),ζ n + ) ) G ζ n),ζ n + ) ), the hrmonic men H α, β) = G α, β) 1 Aα, β) = A α, 1 ). β The reder my lso wish to refer to the ppers [3] nd [6] which provide some results using monotonicity nd convexity rguments.

18 Specil functions: Approximtions nd bounds 89 REFERENCES 1. M. Abrmowitz, I. A. Stegun Eds.): Hndbook of Mthemticl Functions with Formuls, Grphs nd Mthemticl Tbles. Ntionl Bureu of Stndrds, Applied Mthemtics Series, 55, 4th printing, Wshington, H. Alzer: Remrk on double-inequlity for the Riemnn zet function. Expositiones Mthemtice, 3 4) 5), H. Alzer: Monotonicity properties of the Hurwitz zet function. Cnd. Mth. Bull., 48 5), H. Alzer: Shrp inequlities for the Bet function. Indg. Mth. N.S.), 1 1), T. M. Apostol: Anlytic Number Theory. Springer, New York, G. Bstien, M. Roglski: Convexité, compléte monotonie et inéglites sur les fonctions zêt et gmm, sur les fonctions des opérteurs de Bskkov et sur des fonctions rithmétique. Cnd. J. Mth., 54 ), J. M. Borwein, D. M. Brdley, R. E. Crndll: Computtionl strtegies for the Riemnn zet function, J. of Comput. nd Applied Mth., 11 ), I. Budimir, P. Cerone, J. E. Pečrić: Inequlities relted to the Chebyshev functionl involving integrls over different intervls. J. Ineq. Pure nd Appl. Mth., ) Art., 1). [ONLINE 9. P. S. Bullen: A Dictionry of Inequlities. Addison Wesley Longmn Limited, P. Cerone: Bounds for Zet nd relted functions. J. Ineq. Pure & Appl. Mth., 6 5) Art. 134, 5). [ONLINE P. Cerone: On n identity for the Chebychev functionl nd some rmifictions. J. Ineq. Pure nd Appl. Mth., 3 1) Art. 4, ). [ONLINE 1. P. Cerone: On pplictions of the integrl of products of functions nd its bounds. RGMIA Res. Rep. Coll., 6 4) 3), Article 4. [ONLINE P. Cerone: On odd zet nd other specil functions bounds. Inequlity Theory nd Applictions, Nov Science Publishers, N.Y. in press). 14. P. Cerone: On some results involving the Čebyšev functionl nd its generlistions. J. Ineq. Pure & Appl. Mth., 4 3) Art. 54, 3). [ONLINE P. Cerone: On reltionships between Ostrowski, trpezoidl nd Chebychev identities nd inequlities. Soochow J. Mth., 8 3) ), P. Cerone: On some generlistons of Steffensen s inequlity nd relted results. J. Ineq. Pure nd Appl. Mth., 3) Art. 8, 1). [ONLINE P. Cerone: Specil functions pproximtions nd bounds vi integrl representtions. Advnces in Inequlities for Specil Functions, P. Cerone, S. S. Drgomir Ed.), to pper.

19 9 P. Cerone 18. P. Cerone, M. A. Chudhry, G. Korvin, A. Qdir: New inequlities involving the zet function. J. Inequl. Pure Appl. Mth., 5 ) 4), Art. 43. [ONLINE: P. Cerone, S. Drgomir: A refinement of the Grüss inequlity nd pplictions. Tmkng J. Mth., in press.. P. Cerone, S. Drgomir: Some convexity properties of Dirichlet series with positive terms. RGMIA Res. Rep. Coll., 8 4) 5), Article 14. [ONLINE: 1. P. Cerone, S. Drgomir: Inequlities of Dirichlet series with positive terms, submitted. RGMIA Res. Rep. Coll., 9 1) 6), Article 8. [ONLINE: P. Cerone, S. Drgomir: New upper nd lower bounds for the Čebyšev functionl. J. Ineq. Pure & Appl. Mth., 3 5) Art. 77, ). [ONLINE 3. P. Cerone, S. Drgomir: Generlistions of the Grüss, Chebychev nd Lupş inequlities for integrls over different intervls. Int. J. Appl. Mth., 6 ) 1), P. Cerone, S. S. Drgomir: On some inequlities rising from Montgomery s identity.j. Comput. Anl. Applics., 5 4) 3), P. Cerone, S. S. Drgomir: On some inequlities for the expecttion nd vrince. Koren J. Comput. Appl. Mth., 8 ) 1), X. L. Cheng, J. Sun: A note on the perturbed trpezoid inequlity. J. Ineq. Pure nd Appl. Mth., 3 ) Art. 9, ). [ONLINE 1.html]. 7. J. B. Conrey: The Riemnn hypothesis. Notices of the AMS3), D. Cvijović, J. Klinowski: Integrl representtions of the Riemnn zet function for odd-integer rguments. J. of Comput. nd Applied Mth., 14 ) ), S. S. Drgomir: A generlistion of Grüss inequlity in inner product spces nd pplictions. J. Mth. Anl. Appl., ), S. S. Drgomir: Some integrl inequlities of Grüss type. Indin J. of Pure nd Appl. Mth., 31 4) ), S. S. Drgomir, Th. M. Rssis Ed.): Ostrowski Type Inequlities nd Applictions in Numericl Integrtion. Kluwer Acdemic Publishers,. 3. H. M. Edwrds: Riemnn s Zet Function. Acdemic Press, New York, S. R. Finch: Mthemticl Constnts. Cmbridge Univ. Press, Cmbridge, A. M. Fink: A tretise on Grüss inequlity. Anlytic nd Geometric Inequlites nd Applictions. Mth. Appl., ), Kluwer Acdemic Publishers, Dordrecht, G. Grüss: Über ds Mximum des bsoluten Betrges von 1 b fx) dx b ) b gx) dx. Mth. Z., ), b fx)gx) dx b

20 Specil functions: Approximtions nd bounds A. Gut: Some remrks on the Riemnn zet distribution. Rev. Roumine Mth. Pures et Appl., 51 6) to pper). Preprint, U.U.D.M. Report 5:6, ISSN , Deprtment of Mthemtics, Uppsl University. 37. J. Hvil: Gmm: Exploring Euler s constnt. Princeton University Press, New Jersey, A. Ivić: The Riemnn Zet-Function, Theory nd Applictions. Dover Publictions, Inc., Mineol, New York, 1985, 517 pp. 39. L. Lndu: Monotonicity nd bounds on Bessel functions. Electronic J. of Differentil Equtions, ), D. S. Mitrinović, J. E. Pečrić, A. M. Fink: Clssicl nd New Inequlities in Anlysis. Kluwer Acdemic Publishers, Dordrecht, J. Pečrić, F. Prosch, Y. Tong: Convex Functions, Prtil Orderings nd Sttisticl Applictions. Acdemic Press, Sn Diego, N. J. Sonin: O nekotoryh nervenstvh otnosjšcihsjk opredelennym integrlm. Zp. Imp. Akd. Nuk po Fiziko-mtem, Otd.t., ), H. M. Srivstv: Certin clsses of series ssocited with the Zet nd relted functions. Appl. Mth. & Comput., 141 3), H. M. Srivstv, J. Choi: Series ssocited with the zet nd relted functions. Kluwer Acd. Publ., Dordrecht/Boston/London 1), pp J. F. Steffensen: On certin inequlities between men vlues nd their ppliction to cturil problems. Skndinvisk Akturietidskrift, 1918), E. C. Titchmrsh: The Theory of the Riemnn Zet Function. Oxford Univ. Press, London, G. N. Wtson: A tretise on the theory of Bessel functions. 1966) nd Edn., Cmbridge University Press. 48. E. T. Whittker, G. N. Wtson: A Course of Modern Anlysis. Cmbridge University Press, Cmbridge, School of Computer Science nd Mthemtics, Received October 7, 6) Victori University, PO Box 1448, MCMC 81, Victori, Austrli E mil: pietro.cerone@vu.edu.u URL:

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