LOGARITHMIC ORDER AND TYPE OF INDETERMINATE MOMENT PROBLEMS

Transcription

1 LOGARITHMIC ORDER AND TYPE OF INDETERMINATE MOMENT PROBLEMS CHRISTIAN BERG AND HENRIK L. PEDERSEN WITH AN APPENDIX BY WALTER HAYMAN We invesigae a efined gowh scale, logaihmic gowh, fo indeeminae momen poblems of ode zeo. We show ha he fou enie funcions appeaing in he Nevanlinna paameizaion have he same logaihmic ode and ype. In he appendix i is shown ha he logaihmic indicao is consan. 2 Mahemaics Subjec Classificaion: pimay 44A6, seconday 3D15 Keywods: indeeminae momen poblem, logaihmic ode 1. Inoducion and esuls This pape deals wih he indeeminae momen poblem on he eal line. We ae given a posiive measue µ on R having momens of all odes and we assume ha µ is no deemined by is momens. Fo deails abou he indeeminae momen poblem see he monogaphs by Akhieze 1, by Shoha and Tamakin 26 o he suvey pape by Beg 3. Ou noaion follows ha of Akhieze 1. In his indeeminae siuaion he soluions ν o he momen poblem fom an infinie convex se V, which is compac in he vague opology. Nevanlinna has obained a paameizaion of V in ems of he so-called Pick funcions. We ecall ha a holomophic funcion ϕ defined in he uppe half plane is called a Pick funcion if Iϕ(z) fo Iz >. The class of Pick funcions is denoed by P. The Nevanlinna paameizaion is he one-o-one coespondence ν ϕ ϕ beween V and P { } given by dν ϕ () z A(z)ϕ(z) C(z) =, z C \ R. (1) B(z)ϕ(z) D(z) eseach suppoed by he calsbeg foundaion 1

2 2 Hee A, B, C and D ae ceain enie funcions defined in ems of he ohonomal polynomials {P k } and he polynomials of he second kind {Q k } in he following way: A(z) = z Q k ()Q k (z), k= B(z) = 1 + z C(z) = 1 + z D(z) = z Q k ()P k (z), k= P k ()Q k (z) k= and P k ()P k (z). (2) k= These funcions ae closely elaed due o he elaion A(z)D(z) B(z)C(z) 1. (3) Two ohe funcions play a ole, namely ( ) 1/2 ( ) 1/2 p(z) = P k (z) 2 and q(z) = Q k (z) 2. k= We ecall ha 1/p(x) 2 is he maximal poin mass of any soluion o he momen poblem a he poin x R. The funcion q has a simila popey when one consides he so-called shifed momen poblem, cf. Pedesen 23. In Beg and Pedesen 4 he enie funcions A, B, C and D wee shown o have he same ode, ype and indicao funcion. I was also shown ha he logaihmically subhamonic funcions p and q had ha ode, ype and indicao. A esul of M. Riesz saes ha each of he enie funcions is of minimal exponenial ype and heefoe he common ode is a numbe beween and 1. The poin of his pape is o invesigae momen poblems of ode. The quesion aises if he gowh of he fou enie funcions and p and q is also he same when one consides a efined gowh scale fo funcions of ode. We shall use a logaihmic scale, which has been used by ohe auhos in connecion wih q-special funcions. Seveal examples of indeeminae momen poblems of ode have been invesigaed. The indeeminae momen poblems wihin he socalled q-askey scheme have been idenified by Chisiansen 11. As examples of momen poblems of ode zeo we menion in paicula he momen k=

3 3 poblems associaed wih he q-meixne, q-chalie, Al-Salam Caliz II, q-laguee and Sieljes Wige polynomials. Also he discee q-hemie II, q 1 -Meixne Pollaczek, symmeic Al-Salam Chihaa II and coninuous q 1 -Hemie polynomials lead o momen poblems of ode zeo. See Secion 4. Fo an enie funcion f he quaniy M(f, ) denoes he maximum modulus of f on he closed disk ceneed a he oigin and of adius. We ecall ha an enie funcion f is of ode if fo any ε > hee is > such ha log M(f, ) ε, fo. The inequaliy log M(f, ) ε is hus ue fo sufficienly lage, and his we wie as log M(f, ) as ε, adoping a noaion fom Levin 21. Fo an enie funcion f of ode zeo we define he logaihmic ode ρ = ρ f as ρ = inf{α > log M(f, ) as (log ) α }. Fo non-consan f we mus have ρ 1, by he usual poof of Liouville s heoem. I is easy o obain ha ρ = lim sup log log M(f, ). log log When ρ < we define he logaihmic ype τ = τ f as and i is eadily found ha τ = inf{β > log M(f, ) as β(log ) ρ }, τ = lim sup log M(f, ) (log ) ρ. I is easily seen ha if f(z) has logaihmic ode ρ and logaihmic ype τ hen so has he funcion f(az + b) (fo a ). Fuhemoe, he funcion f(z) n is again of logaihmic ode ρ bu of logaihmic ype nτ, while f(z n ) has logaihmic ode ρ and logaihmic ype τn ρ. I is also clea ha if a anscendenal enie funcion has logaihmic ode equal o 1, hen he logaihmic ype mus be infinie. Fo a polynomial of degee k 1 he logaihmic ode is 1 and he ype is k.

4 4 The indicao funcion fo an enie funcion of finie logaihmic ode ρ and finie logaihmic ype is defined in he naual way as h(θ) = lim sup log f(e iθ ) (log ) ρ, θ 2π. Howeve i uns ou ha he indicao of any enie funcion of finie logaihmic ode and ype is acually consan equal o he ype. This fac can be deduced (a leas when ρ 2) fom esuls in a pape by Bay 2 (see p. 469 in Bays pape). M. Sodin has kindly infomed us ha he esul can also be deduced fom a esul of Gishin 13. In he Appendix we pesen a self-conained poof of his esul by Wale Hayman. Wih hese definiions we have he following esul poving he conjecue p. 651 in Ismail 14. Because of he applicaions o he q-askey scheme, Ismail called q-ode, q-ype and q-phagmén-lindelöf-indicao wha we have called logaihmic ode, ype and indicao, see Ismail 14 p Theoem 1.1. The funcions A, B, C, D, p and q appeaing in an indeeminae momen poblem of ode have he same logaihmic ode ρ 1. If ρ < hen hey have he same logaihmic ype. Any combinaion A(z) C(z) and B(z) D(z), whee R { }, has also he same logaihmic ode and ype. The common logaihmic ode and ype of he funcions of Theoem 1.1 ae called he logaihmic ode and ype of he indeeminae Hambuge momen poblem. The fou enie funcions occuing in he indeeminae Hambuge momen poblem can be egaded as he enies of a ceain 2 2 maix of enie funcions. This leads o he concep of a Nevanlinna maix, which was inoduced by Kein 19, see also Akhieze 1. In Beg and Pedesen 5 he common gowh of he enies was invesigaed, and i was shown ha all fou enies have he same odinay ode and ype. Definiion 1.1. An enie funcion N : C SL 2 (C) of he fom ( ) A(z) B(z) N(z) = C(z) D(z) is called a (eal) Nevanlinna maix if he enies ae eal anscendenal enie funcions and { } A(z) + B(z) I >, fo R { }, Iz >. C(z) + D(z)

5 5 If we conside he enie funcions A, B, C and D fom an indeeminae Hambuge momen poblem, hen he maix ( ) A(z) C(z) B(z) D(z) defines a eal Nevanlinna maix, aking ino accoun he elaions (1) and (3). Pa of Theoem 1.1 can be genealized o eal Nevanlinna maices. We have Theoem 1.2. Fo any eal Nevanlinna maix of ode zeo ( ) A B, C D he enies A, B, C and D have he same logaihmic ode and logaihmic ype. This common ode and ype is also he logaihmic ode and ype of any of he funcions A + B, C + D, whee R. Theoem 1.1 and 1.2 ae poved in Secion 2. I was shown in Begweile, Ishizaki and Yanagihaa 7 and in Ramis 25 ha enie anscendenal soluions of ceain q-diffeence equaions ae of logaihmic ode 2 and finie logaihmic ype. Refined esuls abou he zeos of such soluions ae given in Begweile and Hayman 6. Remak 1.1. Thee is a noion of poximae o efined ode fo enie funcions, oiginally inoduced by Valion 28, see also Levin s book 22. In his geneal seup i is sill ue ha he fou enie funcions in a Nevanlinna maix have he same gowh, due o he quie accuae esimaes beween he funcions ha we shall use in he poof of Theoem 1.2. Acknowledgemen. The auhos wan o hank Mouad Ismail fo he encouagemen o undeake he pesen invesigaion. 2. Poof of he main esuls In his secion we pove Theoem 1.1 and 1.2. following lemma. The key o his is he Lemma 2.1. Le f and g be wo anscendenal enie funcions such ha f/g is a Pick funcion. Then f and g have he same logaihmic ode and ype.

6 6 Poof. We use he fac ha any Pick funcion p admis an inegal epesenaion of he fom z + 1 p(z) = az + b + z dτ(), whee a, b R and τ is a finie posiive measue on he eal line. The funcion f/g is a meomophic Pick funcion, so fom he inegal epesenaion we easily obain f(z) g(z) = az + b b z z b n, (z a n )a n whee a, b, b R, {a n } is he se of nonzeo poles, b n >, n 1 and n=1 b n/a 2 n <. Fom his seies epesenaion we see ha f(z) g(z) K z 2 + 1, y n=1 fo some consan K and all z C \ R. Fo y 1 his esimae gives us (wih = z ) f(z) K( 2 + 1)M(g, ). Fo y < 1, we ge, since log f is subhamonic, log f(z) 1 2π 2π log f(z + e i ) d log K(( + 1) 2 + 1) 1 2π 2π log y + sin d + log M(g, + 1). If we combine his wih he esimae fo y 1, hen we ge log M(f, ) K 1 + K 2 log( + 1) + log M(g, + 1), (4) fo suiable posiive consans K 1 and K 2. Fom his elaion i follows ha he logaihmic ode ρ f of f is less han o equal o he logaihmic ode ρ g of g: his is clea if ρ g =, so we may suppose ha ρ g <. Le ε > be given. Then and hence, M(g, ) as e (log )ρg +ε, M(f, ) as e K1+K2 log(+1)+(log(+1))ρg +ε as e (log )ρg +2ε, since ρ g 1. In his way we see ha ρ f ρ g. Clealy, he funcion g/f is also a Pick funcion, and so we ge ρ g ρ f. Theefoe he wo logaihmic odes mus be idenical. This common ode

7 7 we denoe by ρ. Assume now 1 < ρ <. Fom he elaion (4) i also follows ha log M(f, ) (log ) ρ K 1 + K 2 log( + 1) log M(g, + 1) (log ) ρ + (log ) ρ. This implies ha he logaihmic ype of f is less han o equal o he logaihmic ype of g. Again, by consideing g/f, we find ha he wo logaihmic ypes ae equal. When ρ = 1 boh f, g have logaihmic ype. Poof of Theoem 1.2. Fo any R { }, he meomophic funcion A(z) + B(z) C(z) + D(z) is a Pick funcion. Hence, by Lemma 2.1, he logaihmic ode and he logaihmic ype of he wo funcions A + B and C + D (fo fixed ) ae idenical. In paicula ( = ) he logaihmic ode and ype of A is he same as he logaihmic ode and ype of C, and similaly fo B and D ( = ). In a eal Nevanlinna maix, he funcion D/C is also a Pick funcion, see e.g. Beg and Pedesen 5, so he logaihmic ode and ype of D and C ae also idenical. Fo fixed R, he funcion C(z) + D(z) C(z) = + D(z) C(z) is hus also a meomophic Pick funcion and heefoe he logaihmic gowh of C + D is he same as he logaihmic gowh of C. I is easy o see ha also he maix ( ) C(z) D(z) A(z) B(z) is a eal Nevanlinna maix. Theefoe also B/A and hence (A + B)/A is a Pick funcion. Consequenly, he logaihmic gowh of A + B is equal o he logaihmic gowh of A. Poof of Theoem 1.1. The asseions abou he funcions A, B, C and D follow fom Theoem 1.2. We un o he funcions p and q. We claim ha we also have ρ p = ρ q = ρ, whee ρ is he common logaihmic ode of he fou enie funcions. Indeed, i is enough o pove ha ρ p = ρ D, as menioned in Beg and Pedesen 4. Fom he definiion of D, (2), and he

8 8 Cauchy-Schwaz inequaliy we see M(D, ) p()m(p, ) so ha ρ D ρ p. Fuhemoe, fomula (22) in Beg and Pedesen 4, saing M(p, ) 2 M(B, + 1)M(D, + 1), yields ρ p ρ D. We also obain τ D τ p and 2τ p τ B + τ D so ha τ p = τ, he common logaihmic ype. 3. Sieljes momen poblems A Sieljes momen poblem may be deeminae on he half-line, bu indeeminae on he whole eal line. One defines he quaniy α as P n () α = lim n Q n (). I is a fac ha α and ha he poblem is indeeminae on he half-line (o in he sense of Sieljes) if and only if α <, cf. Chihaa 1 o Beg 3. The se V + = {σ V supp(σ) [, )} of soluions o a Sieljes momen poblem, which is indeeminae in he sense of Sieljes, can be paameeized via he one-o-one coespondence ν σ σ beween V + and S { } given by dν σ () + w = P (w) + σ(w)r(w) Q(w) + σ(w)s(w), w C \ [, ), whee he funcions P, Q, R and S can be defined as limis of convegens of he Sieljes coninued facion. The paamee space S is he se of Sieljes ansfoms, i.e. funcions of he fom σ(w) = a + dτ(x), w C \ (, ], x + w whee a and τ is a posiive measue on [, ) such ha he inegal makes sense. This is he Kein paameizaion of he soluions o an indeeminae Sieljes momen poblem, see Kein and Nudelman 2 o Beg 3. The funcions P, Q, R and S ae elaed o A, B, C and D as follows. P (z) = A( z) 1 α C( z), R(z) = C( z), Q(z) = (B( z) 1α ) D( z), S(z) = D( z). Concening hese funcions we have Poposiion 3.1. The enie funcions P, Q, R and S all have he same logaihmic ode and ype as he indeeminae Hambuge momen poblem.

9 9 Poof. This follows diecly fom he definiions of P, Q, R and S in ems of linea combinaions of A, B, C and D and Theoem 1.1. Fo a Sieljes momen sequence { n } n one consides he coesponding symmeic Hambuge momen sequence given by {,, 1,, 2,,...}. Thee is a close connecion beween hese wo momen poblems and elaions beween he enie funcions in he wo Nevanlinna paameizaions can be found in e.g. Chihaa 1 o Pedesen 24. Le us jus menion ha he D-funcions D S and D H fo he Sieljes and Hambuge poblems ae conneced by zd H (z) = D S (z 2 ). Fom his elaion we easily obain Poposiion 3.2. Le ρ S and τ S denoe he logaihmic ode and ype of an indeeminae Sieljes momen poblem and le ρ H and τ H denoe he logaihmic ode and ype of he coesponding symmeic Hambuge momen poblem. Then we have 4. Examples ρ H = ρ S, τ H = τ S 2 ρ S. In his secion we deemine he logaihmic ode and ype of some indeeminae momen poblems fom he q-askey scheme, which is discussed in Koekoek and Swaouw 18. To do so we apply esuls fom Secion 5 below. The momen poblem associaed wih he q-meixne polynomials, which we denoe as {M n (z + 1; b, c; q)} n is indeeminae in he sense of Sieljes. The fou enie funcions in he Kein paameizaion have been compued in Theoem 1.3 in Chisiansen 11. In paicula he funcion Q is shown o be given by Q(z) = 1 φ 1 ( 1 z bq ), q; q/c whee b < 1/q and c >. Fo he definiion of basic hypegeomeic seies see Gaspe and Rahman 12 o Koekoek and Swaouw 18. We denoe he zeos of Q by {x n }, whee > x 1 > x 2 >.... By a esul of Begweile and Hayman 6 hee is a consan A > such ha x n Aq 2n as n, (5) see Poposiion 1.5 in Chisiansen 11 fo deails. Poposiion 4.1. The indeeminae Sieljes momen poblem associaed wih he q-meixne polynomials have logaihmic ode equal o 2 and logaihmic ype equal o 1/(4 log q).

10 1 Poof. We see fom (5) and Poposiion 5.6 ha Q has logaihmic ode equal o 2 and logaihmic ype equal o 1/(4 log q). Then he esul follows fom Poposiion 3.1. By specializaion o aking limis of he paamees in he q-meixne case we obain: Coollay 4.1. The indeeminae Sieljes momen poblems associaed wih he q-chalie, Al-Salam Caliz II, q-laguee and Sieljes Wige polynomials ae all of logaihmic ode 2 and logaihmic ype 1/(4 log q). The Discee q-hemie II momen poblem is symmeic and he coesponding Sieljes momen poblem is he q 2 -Laguee momen poblem wih α = 1 2. Applying Poposiion 3.2 we ge he following. Coollay 4.2. The indeeminae Hambuge momen poblem associaed wih he Discee q-hemie II polynomials is of logaihmic ode 2 and logaihmic ype 1/(2 log q). The q 1 -Hemie momen poblem was eaed in deail by Ismail and Masson 15. The zeos of he funcion D ae given explicily as x n = 1 2 Theefoe he couning funcion saisfies Fom Poposiion 5.6 we obain: ( q n q n), n Z. n() 2 log log q,. Poposiion 4.2. The indeeminae Hambuge momen poblem associaed wih he Coninuous q 1 -Hemie polynomials has logaihmic ode 2 and logaihmic ype equal o 1/ log q. 5. The logaihmic gowh scale In his secion we collec some facs abou enie funcions of finie logaihmic gowh. Mos of hese facs can be found in he lieaue, bu fo he eades convenience we have included he poofs. Fo an enie funcion f wih Taylo seies f(z) = c n z n n=

11 11 he (odinay) ode is if and only if lim n log n ( log 1 n cn ) =. One can also expess he logaihmic ode and ype in ems of he Taylo coefficiens. Poposiion 5.1. Fo an enie funcion f(z) = n= c nz n of ode is logaihmic ode ρ saisfies Poof. We pu and we fis show ha ρ = 1 + lim sup n µ = lim inf n log n ( log log 1 n cn log log c n log n ). µ = ρ ρ 1. (6) Suppose ha ρ is finie and le λ > ρ. Then hee exiss > 1 such ha log M(f, ) (log ) λ, fo. By applying he Cauchy esimaes we find, fo any n and, The funcion log c n (log ) λ n log. ϕ() = (log ) λ n log (defined fo 1) aains is minimum fo ( (n ) ) 1/(λ 1) = exp λ which is bigge han fo n λ(log ) λ 1. Fo such n he minimum value ove [, [ is ( n ) 1/(λ 1) n(1/λ 1), λ which is a negaive quaniy. I follows ha fo all sufficienly lage n ( n ) 1/(λ 1) log c n n(1/λ 1), λ

12 12 so ha o log log c n log log c n log n λ ( ) 1 1/λ log n + log, λ 1 λ 1/(λ 1) fo all sufficienly lage n. Hence µ = lim inf n ( ) λ log 1 1/λ λ 1 + λ 1/(λ 1), log n log log c n log n λ λ 1. Since his holds fo any λ > ρ we mus have µ ρ/(ρ 1). Noice ha µ = if ρ = 1 and also ha ρ = if µ = 1. Convesely, if µ > 1 we choose ν (1, µ) and nex n such ha log log c n log n > ν, fo all n n. This implies c n < e nν z = 1, fo n n. Then we have, fo f(z) n 1 n= c n n + Cons n + c n n n=n ν n. n=n e n Fo given so lage ha n ν 1 1 < log we choose n 1, depending on, such ha n 1 n + 1 and (n 1 1) ν 1 1 < log n ν (This is possible since ν > 1.) Fo n n 1 we hus have log n ν 1 1 so ha log n n ν n, o n e nν e n. This yields e n ν n=n 1 n n=n 1 e n < 1. Fo n {n,..., n 1 1} we have n ν 1 1 < log so ha n < (log + 1) 1/(ν 1).

13 13 Theefoe n 1 1 e n ν n=n n = < n 1 1 n=n e n log e n n 1 1 e (log +1) 1/(ν 1) log e nν n=n = e (log +1)1/(ν 1) log < e (log +1)1/(ν 1) log ν n 1 1 e n ν n=n e n. n=1 This gives, fo sufficienly lage, log M(f, ) log (Cons ) n e (log +1)1/(ν 1) log Cons (log ) 1+1/(ν 1) = Cons (log ) ν/(ν 1). Since ν was an abiay numbe beween 1 and µ we conclude ha f has logaihmic ode µ/(µ 1). We have heefoe veified he elaion (6) and fom i we ge ρ = µ 1 = 1 + lim sup n = 1 + lim sup n = 1 + lim sup n 1 log log c n log n 1 log n log log c n log n log n ( ). log log 1 n cn Remak 5.1. The logaihmic ode is ρ (2) in he noaion of Shah and Ishaq 27 and he ρ(2, 2) ode of Juneja, Kapoo and Bajpai 16. Poposiion 5.2. Fo an enie funcion f(z) = n= c nz n of logaihmic ode ρ (1, ) is logaihmic ype τ saisfies τ = (ρ 1)ρ 1 ρ ρ lim sup n n ( ) ρ 1. 1 log n cn

14 14 Poof. Suppose ha M(f, ) e K(log )ρ fo. Fom he Cauchy esimaes we see ha log c n K(log ) ρ n log,. The funcion K(log ) ρ n log aains is minimum fo 1 when Kρ(log ) ρ 1 = n, i.e. when log = (n/(kρ)) 1 ρ 1. Fo all sufficienly lage n we mus heefoe have log c n (log ) ( K(log ) ρ 1 n ) ( ) 1 ( ) n ρ 1 1 = n Kρ ρ 1, so ha o K 1 ρ ( log c n ρ 1 nρ 1 1 ρ 1, Kρ ρ) ( 1 1 ) ρ 1 n ρ (ρ 1)ρ 1 ρ ρ 1 = log c n ρ ρ n ( 1 log n cn ) ρ 1. Since his holds fo all sufficienly lage n and K is an abiay numbe geae han he logaihmic ype τ we mus have τ (ρ 1)ρ 1 ρ ρ lim sup n Fo he convese we ague as follows and pu σ = lim sup n n ( 1 log n cn n ( ) ρ 1. 1 log n cn ) ρ 1 = lim sup n n ρ log c n ρ 1. Le ε > be given. We choose n such ha fo all n n we have n ρ ρ 1 σ + ε, log c n which means ha log c n (σ + ε) 1 ρ 1 n ρ ρ 1. Hence, fo z = 1, f(z) n 1 n= c n n + Cons n + c n n n=n e (σ+ε) ρ 1 1 n ρ ρ 1 +n log. n=n

15 15 When is so lage ha (σ + ε) 1 ρ 1 n ρ ρ 1 + n log > n, we choose he smalles inege n 1 = n 1 () > n such ha fo n n 1 (σ + ε) 1 ρ 1 n ρ ρ 1 + n log n. This implies fis of all ha n 1 1 < (σ + ε)(log + 1) ρ 1, bu o ea he n s beween n and n 1 1 we look a he concave funcion of s We find fo ϕ(s) = s log (σ + ε) 1 ρ 1 s ρ ρ 1. ϕ (s) = log ρ ρ 1 (σ + ε) 1 ρ 1 s 1 ρ 1 = ( ) ρ 1 ρ 1 s = s () = (σ + ε)(log ) ρ 1. ρ Fuhemoe, ϕ(s) aains is maximum a s = s () and We hus ge ϕ(s ()) = (ρ 1)ρ 1 ρ ρ (σ + ε)(log ) ρ. n 1 1 e (σ+ε) ρ 1 1 n ρ n=n Theefoe, fo z = sufficienly lage, and hence ρ 1 +n log (σ + ε)(1 + log ) ρ 1 e ϕ(s()). M(f, ) Cons n (σ + ε)(log + 1) ρ 1 e ϕ(s()) Cons n (σ + ε)(log + 1) ρ 1 ( ) (ρ 1) ρ 1 exp (σ + ε)(log ) ρ, ρ ρ ( ) (ρ 1) ρ 1 M(f, ) as exp ρ ρ (σ + 2ε)(log ) ρ. Theefoe he logaihmic ype τ saisfies τ (ρ 1)ρ 1 ρ ρ (σ + 2ε),

16 16 and leing ε, we see ha τ (ρ 1)ρ 1 ρ ρ σ. Remak 5.2. The logaihmic ype is he T (2, 2) ype of Juneja, Kapoo and Bajpai 17. Example 5.1. We le q C and suppose ha < q < 1. The basic hypegeomeic seies ( a1,..., a φ s b 1,..., b s ), q; z defines an enie funcion of z when s and he paamees ae such ha none of he denominaos become zeo. We assume also ha φ s is no a polynomial. The logaihmic ode of φ s is equal o 2, as can be seen fom Poposiion 5.1. The logaihmic ype is equal o which follows fom Poposiion 5.2. In paicula, (z; q) = (1 zq k ) = k= 1 2(1 + s ) log 1/ q, n= ( 1) n q (n 2) (q; q) n z n = φ is of logaihmic ode 2 and logaihmic ype 1/(2 log 1/ q ). ( ) q; z Example 5.2. We le q C and suppose ha < q < 1. Then (1) f(z) = n= q n α 1 α z n is (fo α > 1) of logaihmic ode α and logaihmic ype equal o τ = (α 1)α 1 α α 1 (log 1/ q ) α 1.

17 17 (2) (3) f(z) = q en z n n= is of logaihmic ode 1 and infinie logaihmic ype. f(z) = q n(log n)2 z n n= is of ode zeo, bu is logaihmic ode is. A anscendenal enie funcion f of odinay ode less han 1 mus have infiniely many zeos, which we label {a n } and numbe accoding o inceasing ode of magniude and epeaing each zeo accoding o is mulipliciy. We suppose ha f() = 1 and fom Hadamad s facoizaion heoem, we ge ha ( f(z) = 1 z ). a n n=1 The gowh of an enie funcion of odinay ode less han 1 is hus in pinciple deemined by he zeo disibuion. We shall use he following quaniies o descibe his disibuion. We define he usual zeo couning funcion n() as n() = #{n 1 a n }. We define he following quaniies in ems of he zeo couning funcion N() = n() and n() Q() = 2 d. These quaniies ae elaed o M() = M(f, ) in he following way d, N() log M() N() + Q() (7) fo >. (See e.g. he elaion (3.5.4) in Boas 9 ). If f is of (odinay) ode we ge fom Hadamad s fis heoem ha he convegence exponen of he zeos is also equal o. I means ha we have 1 a n ε < n=1

18 18 fo all ε >. In his siuaion we define he logaihmic convegence exponen ρ l as { } 1 ρ l = inf α > ( log a n + 1) α <. n=1 The following poposiion expesses he logaihmic convegence exponen in ems of he logaihmic ode of he zeo couning funcion. Poposiion 5.3. We have Poof. > > 1, ρ l = lim sup log n() log log. Fis of all, we see by inegaion by pas ha fo a > and dn() (log ) a = To ease noaion we le n() (log ) a n( ) (log ) a + a L = lim sup log n() log log. n() (log ) a If α > L we choose a (L, α) and noice ha ha lim n()/(log ) α =. Fuhemoe, since n() d (log ) α+1 = n() d (log ) a (log ) α a+1, n() d (log ) a+1. (8) is bounded and hence he limi of he inegal on he igh-hand side of (8) (wih a eplaced by α) as is finie. Theefoe 1 (log a n ) α = n n a n dn() (log ) α < and consequenly ρ l α and hus we obain ha ρ l L. Convesely, if a > ρ l we have If we look again a (8) i means ha dn() (log ) a <. n() (log ) a emains bounded as, hence L a. We conclude ha ρ l L.

19 19 I is also possible o elae he logaihmic ode and logaihmic convegence exponen. The poposiion below is menioned in he assumpions of Theoem in Boas 9 in he special case whee ρ = 2. Poposiion 5.4. Fo an enie funcion of ode we have ρ = ρ l + 1. To pove his poposiion we give wo lemmas. Lemma 5.1. Suppose ha n() cons (log ) α fo > 1 and some α >. Then, fo > 1, N() cons (log ) α+1 and (whee [ ] denoes he inege pa) [α] Q() cons α... (α l + 1) (log ) α l + δ(), l= whee δ() cons (log ) α [α] 1. Poof. By definiion we have N() cons + 1 n() Concening Q() we have Q() = d cons (log ) α n() 2 Hee, by epeaed inegaions by pas, (log ) α (log )α 2 d = {[ whee = (log ) α + α (log ) α d cons 2 d. ] 1 1 d = cons (log )α+1. α(log ) α 1 } + 2 d (log ) α 1. [α] = α... (α l + 1) (log ) α l + δ(), l= (log ) α [α] 1 δ() = α... (α [α]) 2 d. 2 d

20 2 Since he exponen α [α] 1 is negaive we find δ() cons (log ) α [α] 1. Lemma 5.2. If, fo some α >, log M() cons (log ) α+1 hen n() cons (log ) α. Poof. Since N() log M() we have n() log = n() 2 Hence n() cons (log ) α. d N(2 ) cons ( log 2) α+1 cons (log ) α+1. Poof of Poposiion 5.4. Suppose ha α > ρ l. Fom Poposiion 5.3 we have n() (log ) α fo all sufficienly lage. Fom (7) and Lemma 5.1 we hus have log M() cons (log ) α+1, and heefoe we see ha ρ α + 1. Fom his we conclude ha ρ ρ l + 1. On he ohe hand, if β > ρ hen log M() (log ) β fo all sufficienly lage. By Lemma 5.2 we heefoe have n() cons (log ) β 1, so ha ρ l β 1. We have shown ha ρ l ρ 1. I is also possible o elae he logaihmic ype o he gowh of he zeo couning funcion. Fo an enie funcion of finie logaihmic ode ρ > 1 we pu κ = κ(f) = lim sup n() (log ) ρ 1. Poposiion 5.5. Fo an enie funcion of finie logaihmic ode ρ > 1 we have he following elaion beween he quaniy κ and he logaihmic ype τ: τ κ/ρ eτ. Poof. Fo any given ε > we choose > 1 such ha fo. Then we ge N() = n() (κ + ε)(log ) ρ 1, n() d + Cons + (κ + ε) n() d Cons + κ + ε ρ (log )ρ. (log ) ρ 1 d

21 21 Since we have Q() Cons (log ) ρ 1 we see ha log M(f, ) N() + Q() Cons + κ + ε ρ (log )ρ + Cons (log ) ρ 1. Theefoe τ κ/ρ. Fo ε > we have log M(f, ) (τ + ε)(log ) ρ fo and hence, fo any s > 1, n()(s 1) log This gives so ha s n() d N( s ) log M( s ) (τ + ε)s ρ (log ) ρ. n() (τ + ε) sρ s 1 (log )ρ 1, sρ κ τ s 1. I is easily found ha he funcion ϕ(s) = s ρ /(s 1), s > 1 aains is minimum fo s = ρ/(ρ 1) and ha he minimum is ( ) ρ ρ ρ ϕ = ρ 1 (ρ 1) ρ 1. Hence ρ ρ κ τ (ρ 1) ρ 1. Since (ρ/(ρ 1)) ρ 1 e we finally see ha κ τρe. We shall now see ha τ = κ/ρ povided ha he zeo disibuion has some egulaiy. Poposiion 5.6. Le f be an enie funcion of finie logaihmic ode ρ > 1. Then he following ae equivalen fo. (i) n() λ(log ) ρ 1. (ii) log M() λ ρ (log )ρ. Poof. Since he funcion Q() in (7) is O((log ) ρ 1 ) unde each of he condiions (i),(ii), we have fo λ > log M() λ ρ (log )ρ N() λ ρ (log )ρ.

22 22 I is heefoe enough o show ha We have N() = n() n() λ(log ) ρ 1 N() λ ρ (log )ρ. (log ) ρ 1 ( ) n() d+ (log ) ρ 1 λ (log ) ρ 1 d+λ d. If n() λ(log ) ρ 1 hen we choose such ha n() (log ) ρ 1 λ ε, fo, and his gives N() λ (log ) ρ 1 d Cons + ε ρ (log )ρ. Theefoe Fo s > 1 we have lim (s 1)n() log N() (log ) ρ = λ ρ. s Theefoe, if N() λ ρ (log )ρ we find and conclude ha n() d = N( s ) N(). (s 1)n() log (log ) ρ N(s ) (log ) ρ N() (log ) ρ lim sup = s ρ N(s ) (log s ) ρ N() (log ) ( ) ρ λ = (s ρ 1) ρ + o(1), n() (log ) ρ 1 λ s ρ 1 ρ s 1. If we le s end o 1 we find κ λ. We nex use he elaion o find (s 1)n( s ) log lim inf s n() n() (log ) ρ 1 λ ρ d = N( s ) N() s ρ 1 (s 1)s ρ 1,

23 23 and heefoe n() κ = lim = λ. (log ) ρ 1 6. Appendix: The Phagmén-Lindelöf indicao of some funcions of ode zeo This appendix was wien by Wale Hayman duing he Inenaional Confeence on Diffeence Equaions, Special Funcions and Applicaions held in Munich, Gemany in he peiod July 25 - July 3, 25. We appeciae ha Haymans esul could be included in his appendix. Inoducion and saemen of esuls Suppose ha f(z) is a anscendenal enie funcion. We wie m() = inf z = f(z), M() = sup f(z), (9) z = fo he minimum and maximum modulus of f especively. Nex we define a funcion Ψ() o be of slow gowh (s.g.) if Ψ() is posiive nondeceasing in [, ) and I follows immediaely fom (1) ha Ψ(2) Ψ(), as. (1) Ψ(K) Ψ(), as, (11) wheneve K > 1. Fo we may ake K = 2 p, fo p = 1, 2,..., and pove he esul by inducion on p, using (1). Since Ψ is inceasing, (11) hen follows also fo 2 p < K < 2 p+1. We can now sae ou esuls. Theoem 6.1. Suppose ha wih he above hypoheses, Then as. lim sup 2 log log M() Ψ() = α <. (12) ( ) M() d = o(ψ()), (13) m()

24 24 Coollay 6.1. Coollay 6.2. fo θ 2π. lim sup h Ψ (θ) = lim sup log m() Ψ() log f(e iθ ) Ψ() = α. (14) = α, (15) Some peliminay esuls We assume fom now on ha f() = 1. Ohewise we apply ou conclusions o f(z)/(cz p ), whee p is a nonnegaive inege and c a non-zeo consan. This does no affec he elaions (12), (13), (14) and (15). Nex i follows fom (11) ha Ψ() = o( ρ ), as (16) wheneve ρ is posiive, cf. Theoem in Bingham, Goldie and Teugels 8. Now (12) shows ha f has zeo ode. Thus by Hadamad s heoem ) f(z) = (1 zzν, (17) ν=1 whee z ν ae he zeos of f. We deduce ha, fo z =, ( 1 ) ( f(z) 1 + ). z ν z ν Hence We now have ν=1 ν=1 ν=1 log M() m() log + z ν, < <. (18) z ν Lemma 6.1. If n() denoes he numbe of zeos of f in z, hen n() = o(ψ()), as. In fac we have by Jensen s fomula, cf. (7), if K > 1 and ε >, K n()d log M(K) (α + ε)ψ(), > (K, ε),

25 25 using (11) and (12). Hence n() log K K n()d (α + ε)ψ(), >. This yields Lemma 6.1, since K can be chosen as lage as we please. Lemma 6.2. Fo s > log 1 + s 1 s d = π2 2. We fis pu s/ = x so ha ou inegal becomes log 1 + x dx 1 ( ) 1 + x dx 1 x x = log 1 x x + whee we have pu x = 1/y, when x > 1. Also 1 log ( ) 1 + x dx 1 x x = 2 1 m=1 This poves Lemma 6.2. Lemma 6.3. We have z ν >2 We can wie he sum as ( 1 + log 2 1 z ν log ) dn() = z ν 1 x 2m 2 2m 1 dx = 2 m=1 log = o(ψ()), as. [ ( 1 + n() log 1 = n(2) log 3 + o )] { 2 2 ( ) 1 + y dy 1 y y, ( ) 2 1 = π2 2m Ψ()d 2 n()d 2 2 } (19) by Lemma 6.1. Also i follows fom (11) ha, fo p = 1, 2,... and lage, 2 p+1 2 p Ψ()d 2 p+1 Ψ(2)2 p/2 d 2 < 2 2 p = Ψ(2)2p/2 2 p+1 = Ψ(2) 2 2 p/2,

26 26 since fo lage (and p 1), Ψ(22 p ) < 2 1/2 Ψ(22 p 1 ) <... < 2 p/2 Ψ(2) by (1). We ge Ψ()d 2 < Ψ(2) 2 p/2 = O (Ψ()). 2 2 p=1 Thus (19) and Lemma 6.1 yields Lemma 6.3. Poof of he Theoem and is Coollaies We deduce fom Lemma 6.3 ha, fo < < 2, we have log z ν + z ν = o {Ψ()}. Thus 2 z ν >4 Again by Lemma z ν 4 z ν >4 log z ν + d z ν = o(ψ()) log z ν + z ν d n(4) 2 d = o(ψ()). (2) log 1 + d 1 π2 n(4) = o(ψ()). (21) 2 Puing ogehe (18), (2) and (21) we deduce (13) and he heoem is poved. To pove Coollay 6.1, we suppose given a posiive ε and hen choose a lage, such ha log M() > (α ε)ψ(). In view of (11) and he fac ha log M() inceases wih, we deduce ha if is sufficienly lage log M() > (α 2ε)Ψ(), 2. (22) Nex i follows fom (13) ha we can choose, such ha 2 and log m() > log M() εψ() log M() εψ(), if is sufficienly lage. On combining his wih (22) we obain log m() > (α 3ε)Ψ().

27 27 Since ε is abiaily small we obain lim sup Since m() M() we have fom (12) This poves Coollay 6.1. Clealy fo evey θ lim sup log m() Ψ() log m() Ψ() α. α. m() f(e iθ ) M(). Thus (15) follows fom (12) and (14) and Coollay 6.2 is poven. We emak ha fo Ψ() we may ake no only (log ) α, bu (log ) α exp{(log ) β (log log ) γ } ec. povided ha β < 1. The conclusion (14) is clealy false if α =. We may ake f(z) = e z, Ψ() = (log ) 2. Then α = in (12) and and Refeences log m() Ψ() h Ψ (π) = as. 1. N. I. Akhieze, The classical momen poblem and some elaed quesions in analysis, Olive & boyd, Edinbugh, P. D. Bay, The minimum modulus of small inegal and subhamonic funcions, Poc. London Mah. Soc., 12 (1962), C. Beg, Indeeminae momen poblems and he heoy of enie funcions, J. Comp. Appl. Mah., 65 (1995), C. Beg and H. L. Pedesen, On he ode and ype of he enie funcions associaed wih an indeeminae Hambuge momen poblem, Ak. Ma., 32 (1994), C. Beg and H. L. Pedesen, Nevanlinna maices of enie funcions, Mah. Nach., 171 (1995), W. Begweile and W. Hayman, Zeos of soluions of a funcional equaion, Compu. Mehods Func. Theoy, 3 (23), W. Begweile, K. Ishizaki and N. Yanagihaa, Gowh of meomophic soluions of some funcional equaions I, Aequaiones Mah. 63 (22),

28 28 8. N. H. Bingham, C. M. Goldie and J. L. Teugels, Regula vaiaion, Cambidge Univesiy Pess, Cambidge R. P. Boas, Enie funcions, Academic Pess, New Yok, T. S. Chihaa, Indeeminae symmeic momen poblems, Mah. Anal. Appl., 85 (1982), J. S. Chisiansen, Indeeminae momen poblems wihin he Askey - scheme, Ph.D. hesis, Depamen of Mahemaics, Univesiy of Copenhagen (24). 12. G. Gaspe and M. Rahman, Basic Hypegeomeic Seies, Cambidge Univesiy Pess, Cambidge 199, second ediion A. F. Gishin, Übe Funkionen, die im Innen eines Winkels holomoph sind und do nulle Odnung haben. (Russian), Teo. Funks., Funks. Anal. Pilozh. 1 (1965), M. E. H. Ismail, Classical and Quanum Ohogonal Polynomials in One Vaiable, Cambidge Univesiy Pess, Cambidge M. E. H. Ismail and D. R. Masson, q-hemie polynomials, biohogonal aional funcions and q-bea inegals, Tans. Ame. Mah. Soc. 346 (1994), O. P. Juneja, G. P. Kapoo and S. K. Bajpai, On he (p, q)-ode and he lowe (p, q)-ode of an enie funcion, J. Reine Angew. Mah., 282 (1976), O. P. Juneja, G. P. Kapoo and S. K. Bajpai, On he (p, q)-ype and lowe (p, q)-ype of an enie funcion, J. Reine Angew. Mah., 29 (1977), R. Koekoek and R. F. Swaouw, The Askey-scheme of hypegeomeic ohogonal polynomials and is q-analogue, Repo no , TU-Delf, M. G. Kein, On he indeeminae case in he bounday value poblem fo a Sum-Liouville equaion in he ineval (, ), Izv. Akad. Nauk SSSR Se. Ma., 16 (1952) (in Russian). 2. M. G. Kein and A. Nudelman, The Makov momen poblem and exemal poblems, Ameican Mahemaical Sociey, Povidence, Rhode Island, B. Ya. Levin, Lecues on enie funcions Ameican Mahemaical Sociey, Povidence, R.I., B. Ya. Levin, Nullsellenveeilung ganze Funkionen Akademie Velag Belin, H. L. Pedesen, The Nevanlinna maix of enie funcions associaed wih a shifed indeeminae Hambuge momen poblem, Mah. Scand., 74 (1994), H. L. Pedesen, Sieljes momen poblems and he Fiedichs exension of a posiive definie opeao, J. Appox. Theoy, 83 (1995), J.-P. Ramis, Abou he gowh of enie funcions soluions of linea algebaic q-diffeence equaions, Ann. Fac. Sci. Toulouse Mah., Se. 6 1 (1992), J. Shoha and J. Tamakin, The poblem of momens, Ameican Mahemaical Sociey, Povidence Rhode Island, ev. ed., S. N. Shah and M. Ishaq, On he maximum modulus and he coefficiens of

29 29 an enie seies, J. Indian Mah. Soc., 16 (1952), G. Valion, Lecues on he Geneal Theoy of Inegal Funcions, Chelsea Publishing Company, New Yok, C. Beg, Depamen of Mahemaics, Univesiy of Copenhagen, Univesiespaken 5, DK-21, Copenhagen, Denmak. Fax: H. L. Pedesen, Depamen of Naual Sciences, Royal Veeinay and Agiculual Univesiy, Thovaldsensvej 4, DK-1871, Copenhagen, Denmak. Fax: W. K. Hayman, Depamen of Mahemaics, Impeial College London, Souh Kensingon campus, London SW7 2AZ, UK. Fax: +44 ()