SOME INEQUALITIES AND ABSOLUTE MONOTONICITY FOR MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND

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1 Commun. Korean Mah. Soc. 3 (6), No., pp hp://dx.doi.org/.434/ckms SOME INEQUALITIES AND ABSOLUTE MONOTONICITY FOR MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND Bai-Ni Guo Feng Qi Absrac. By employing a refined version of he Pólya ype inegral inequaliy oher echniques, he auhors esablish some inequaliies absolue monooniciy for modified Bessel funcions of he firs kind wih nonnegaive ineger order.. Main resuls I is well known ha modified Bessel funcions of he firs kind I ±ν (z) are soluions of he differenial equaion z d w dz +zdw dz ( z +ν ) w =. They are holomorphic funcions of z hroughou he z-plane cu along he negaive real axis, are enire funcions of ν for fixed z. When ν = ±n, I ν (z) are enire funcions of z. In, p. 375, 9.6.7], i is lised ha ( ) k+ν z I ν (z) =, z C, ν R\{,,...}, k!γ(ν +k +) where n!n Γ(z) = lim z n z C\{,,,...} n (z +k), is he classical gamma funcion, see, p. 55, 6..]. On, p. 63], he following hree double inequaliies are derived: ( ) ν z/ I ν (z) (.) < Γ(ν +) +z/ z e z + < (ν +) + ν +3 ν +3 ] (ν +3)z, z >, ν (ν +) ; Received July 8, 5. Mahemaics Subjec Classificaion. Primary 33C; Secondary 6A48, 6D5, 44A. Key words phrases. inequaliy, absolue monooniciy, complee monoonic funcion, modified Bessel funcion of he firs kind, Pólya ype inegral inequaliy. 355 c 6 Korean Mahemaical Sociey

2 356 B.-N. GUO AND F. QI (.) (.3) ] z + (ν +) < ν (ν +) + ν +3 ν +3 < Γ(ν +) ez ( z ] ( ) ν (ν +)z I ν (z) < Γ(ν +) (ν +) z e z + ] (ν +3)z, z >, (ν +) ν ; ) ν I ν (z) e z < ( + ) e z, z >, ν >. The lef-h inequaliy in (.) is very weak unless z is quie small. The equaion (3.) in 9, p. 6] reads ha ( ) ν ) j (.4) Γ(ν + ) I ν (y) > (+ y ν, /4(ν+), y >, ν > y j ν, which is more sringen han he lef h side inequaliy in (.3), where j ν, is he firs zero of he Bessel funcion of he firs kind ( ) ν z ( ) k (z/) k J ν (z) =, ν R, z C. k!γ(ν +k +) In 8, Theorem 7.], Theorem.3], i was derived ha (.5) αi (x) > (x/) 3 e (x/) β( x ) 3 I (x) exp β( x ) ] on (, ) if only if α β. More srongly, i was discovered in 8, Theorem 5.] ha () when β, he funcion F β (u) = u βu ( ) e u I βu is decreasing on (, ); () when < β <, i is unimodal (ha is, i has a unique maximum) is convex on (, ). F β (u) When α = or β = x > , he inequaliy in (.5) is beer han (.4). In 7, Theorem.], i was obained ha () when k 5, he inequaliy (.6) I k ( u ) u k/ is valid on (, ); () when β, he funcion G β (u) = dk du k ( βu ( ) d I βu du u e u ( ) ) u e u

3 INEQUALITIES AND MONOTONICITY FOR BESSEL FUNCTIONS 357 is decreasing on (, ); when < β <, i is unimodal G β (u) is convex on (, ). The inequaliy (.6) for k = includes he ones in (.5) for α = β =. For more informaion on recen resuls of Bessel funcions, please refer o 3, 4] closely relaed references herein. Recall from 3,, 3] ha a funcion f is said o be compleely monoonic on an inerval I if i has derivaives of all orders on I such ha ( ) k f (k) (x) < for x I k. Recall also from 3,, 3] ha a funcion f is said o be absoluely monoonic on an inerval I if i has derivaives of all orders f (k ) () < for I k N, where N denoes he se of all posiive inegers. I is easy o see ha a funcion f(x) is compleely monoonic in (a,b) if only if f( x) is absoluely monoonic in ( b, a). See 3, p. 45, Definiion c]. Theorem a in 3, p. 6] reads ha a necessary sufficien condiion ha f(x) should be compleely monoonic in x < is ha f(x) = e x dα(), where α() is bounded non-decreasing he inegral converges for x <. Theorem c in 3, p. 6] saes ha a necessary sufficien condiion ha f(x) should be absoluely monoonic in < x < is ha f(x) = e x dα(), where α() is non-decreasing he inegral converges for < x <. For more informaion on hese kinds of funcions, please refer o 3, Chaper XIII], 3, Chaper IV], 6, 9, ] closely relaed references herein. The main aim of his paper is, by employing a refined version of he Pólya ype inegral inequaliy oher echniques, o esablish some lower upper bounds absolue monooniciy for modified Bessel funcions of he firs kind I n () wih nonnegaive ineger order n. The main resuls may be saed as he following heorems. Theorem.. The double inequaliies (.7) I () cosh π sinh +4 exp +4 (.8) I () cosh < π 4 sinh, are valid. sinh, >

4 358 B.-N. GUO AND F. QI Theorem.. The funcion I () sinh is absoluely monoonic on (, ) complee monoonic on (, ). Hence, we have ( ) (l) I (l) sinh (), > ( ) (l) ( ) l I (l) sinh () ( )l, < for l. Consequenly, () when >, (.9) I () > sinh I () > cosh sinh. () when <, he firs inequaliy in (.9) keeps he same direcion, bu he second inequaliy in (.9) reverses.. A lemma In order o prove Theorem., we need he Pólya ype inegral inequaliy below. Lemma. (, Theorem ] 4, Proposiion ]). If f(x) is coninuous no idenically a consan on a,b], if f(x) is differeniable m f (x) M on (a,b), hen (.) where b a b a f(x)dx f(a)+f(b) M S (a,b)]m S (a,b)] (b a), (M m) S (a,b) = f(b) f(a). b a For more deailed informaion on he inequaliy (.), please refer o 6, pp. 5 3, Secion 5] closely relaed references herein. 3. Proofs of Theorems.. Now we are in a posiion o prove Theorems... Proof of he inequaliy (.7). Le f (x) = e cosx on,π] >. Then f (x) = sinxecosx, f (x) = ( sin x cosx ) e cosx, f () = e, f (π) = e, f () = f (π) =.

5 INEQUALITIES AND MONOTONICITY FOR BESSEL FUNCTIONS 359 Since he equaion sin x cosx = cosx cos x = of he variable x has only one posiive roo x = arccos f (x +4 ) = exp which is he minimum of f (x) for x,π]. In, p. 376, 9.6.6], i was lised ha (3.) I (z) = π π e ±zcosθ dθ = π π cosh(zcosθ)dθ. Combining (.) (3.), using he above properies of he funcion f (x), direcly arranging give I () e +e e e e e π +4 exp ] +4 which is equivalen o he double inequaliy (.7). The proof of he inequaliy (.7) is complee. Proof of he double inequaliy (.8). Leh (x) = cosh(cosx) forx,π] R. Then h () = h (π) = cosh. Since f (x) is symmeric wih respec o x = π, is derivaive f (x) is anisymmeric wih respec o x = π. Hence, we have max x,π] h (x) = min x,π] h (x). Applying h (x) o he inequaliy (.) yields A simple compuaion gives I () cosh π 4 max x,π] h (x). h (x) = sinxsinh(cosx) sinh, x,π]. Therefore, he double inequaliy (.8) follows immediaely. Proof of Theorem.. From he power series expansion sinh k = (k +)! i follows ha (3.) I () sinh I () = = ( ) k (k!), (k +)! k (k!) ] k.

6 36 B.-N. GUO AND F. QI Since (k +)! = (k +)!!(k)!! > (k)!!] = k (k!), he coefficiens (k+)! in he power series (3.) is nonnegaive for k (k!) all k. This implies he absolue monooniciy on (, ) complee monooniciyon(,)ofhe funcion I () sinh. Asaresul, bydefiniions of he absolue complee monooniciy, we obain I () sinh ] (l), > ( ) l I () sinh ] (l), < for l. The proof of Theorem. is complee. 4. Remarks comparisons Remark 4.. The lower bound in he double inequaliy (.7) is beer han sinh, bu he lower bound in he double inequaliy (.8) is no beer han sinh. When < <.53, he double inequaliy (.8) is beer han (.7); when >.53, he double inequaliy (.7) is beer han (.8). Remark 4.. If ν =, he inequaliies in (.) (.) become he same inequaliy (4.) z +z ez < I (z) < 6+3z 3(+3z) ez, z > he inequaliy (.3) is reduced o (4.) < I (z) < (e z + e ) z, z >. I is easy o see ha he firs inequaliy in (.9) is beer han he lower bounds in inequaliies (4.) (4.). The inequaliy (.8) is worse han he double inequaliies (4.) (4.). The upper bound in (.7) is worse han he upper bounds of he double inequaliies (4.) (4.). When x >.85 x >. respecively, he lower bound in (.7) is beer han he lower bounds in he double inequaliies (4.) (4.). In conclusion, he inequaliies for I obained in Theorems.. are more or less significan. Remark 4.3. When ν =, from (.) (.3), i follows ha (4.3) z( z) (z +) ez < I (z) < z(z +4) (5z +4) ez

7 INEQUALITIES AND MONOTONICITY FOR BESSEL FUNCTIONS 36 (4.4) z < I (z) < z (e z + e ) 4 z for z >. The inequaliy (.4) for ν = is (4.5) I (y) > y (+ y j, ) j, /8 for y >, where j, = 3.83 is he firs zero of J. I is clear ha he inequaliy (4.5) is beer han he lef-h side inequaliies in (4.3) (4.4). When >.894, he second inequaliy in (.9) is beer han he lef h side inequaliy in (4.3). When > 6.4, he second inequaliy in (.9) is beer han (4.5). When > 5.898, he second inequaliy in (.9) is beer han he corresponding one in (.5) for α = β =. In a word, he inequaliy for I in Theorem. is somewha significan. Remark 4.4. The inequaliy (.) was generalized in 5] as { m(b 3 a 3 ) f(a) f(b)+bf (b) af (a)+m(a b )/ } + 6 (a b)m f (a)+f (b)] b f(x)dx bf(b)+af(a)+ b f (b) a f (a) a { M(b3 a 3 ) f(a) f(b)+bf (b) af (a)+m(a b )/ } + 6 (a b)m f (a)+f, (b)] where f(x) is a -imes differeniable funcion saisfying m f (x) M on a, b]. For more informaion on he inequaliy (.) is generalizaions, please refer o he papers, 5,,, 5], especially he exposiory survey aricle 6], pleny references herein. Remark 4.5. The mehod used in his paper has been employed in 7] o evaluae he complee ellipic inegrals. I can also be used o derive bounds for (z/) ν π I ν (z) = e ±zcosθ sin ν θdθ, R(ν) > πγ(ν +/). Remark 4.6. The firs inequaliy in (.9) has been applied in ]. All inequaliies for I can be applied as did in ]. Remark 4.7. This paper is a revised version of he preprin 8]. References ] M. Abramowiz I. A. Segun (Eds), Hbook of Mahemaical Funcions wih Formulas, Graphs, Mahemaical Tables, Naional Bureau of Sards, Applied Mahemaics Series 55, h prining, Dover Publicaions, New York Washingon, 97.

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9 INEQUALITIES AND MONOTONICITY FOR BESSEL FUNCTIONS 363 Bai-Ni Guo School of Mahemaics Informaics Henan Polyechnic Universiy Jiaozuo Ciy, Henan Province, 454, P. R. China address: Feng Qi Deparmen of Mahemaics College of Science Tianjin Polyechnic Universiy Tianjin Ciy, 3387, P. R. China address: