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ETNA Electronic Transactions on Numerical Analysis. Volume 27, pp. 94-112, 2007. Copyright  2007, Kent State University. ISSN 1068-9613. Kent State University etna@mcs.kent.edu PICK FUNCTIONS RELATED TO ENTIRE FUNCTIONS HAVING NEGATIVE ZEROS HENRIK L. PEDERSEN Abstract. For any sequence satisfying and representation of the function .!"$#&%('*),+ 0/ 21 %(' Const we find the Stieltjes where ) denotes the canonical product of genus 1 having .3 as its zero set. We also find conditions on the zeros (e.g. 425 1 768:9 for ;8 ) in order that the function #&%('<),+ .- 6 #&%('<),+ 8 !" =/ %(' be a Pick function. We find the corresponding representation in terms of a positive density on the negative axis. We thereby generalize earlier results about the > -function. We also show that another related function is a Pick function. Key words. pick function, canonical product, integral representation AMS subject classifications. 30E20, 30D15, 30E15, 33B15 as 1. Introduction. The ? -dimensional volume @*A of the unit ball in B A can be expressed F AHGJI KLMN ?POQHRTS @CAED K where is Euler’s gamma function. In [3] the asymptotic behaviour of the ?VUXWZY!? ’th root M of @CA was studied (the limit as ? tends to infinity is seen to be O[ \ , by applying Stirlings formula). This initiated an investigation of monotonicity properties of the function ] L_^ R`D UaWHY ^ K`L_^bNcM ^ UaWHY ^edgf R S S see [2], [1, Kapitel 2], [7] and [4]. In [5] we proved that a holomorphic extension of cut plane h DjiEkPlPmon f S l ] to the S is a Pick function. We also found the corresponding integral representation in terms of a positive density on the negative axis. The proof consisted in applying a Phragmén-Lindelöf argument to the harmonic function pbqrtsvu UaWHY K`L p NwM p!x W Y Z df R p y in the upper half-plane z{D}| p~ i s,p . In order to do so it was necessary to investigate the growth at infinity and the boundary behaviour on the real line of this harmonic function. These investigations depended heavily on the functional equation KL p NwM R!D p KL p R Received April 25, 2003. Accepted for publication June 8, 2003. Recommended by F. Marcellán. Department of Mathematics and Physics, The Royal Veterinary and Agricultural University, Thorvaldsensvej 40, DK-1871, Denmark (henrikp@dina.kvl.dk, http://www.matfys.kvl.dk/˜henrikp/). This work was supported by the Carlsberg Foundation. 94 ETNA Kent State University etna@mcs.kent.edu 95 PICK FUNCTIONS RELATED TO ENTIRE FUNCTIONS K of the -function. K In this paper we generalize the results about the -function to a class of entire functions having negative zeros. K The reciprocal of the -function is an entire function having negative zeros. Indeed, the K Weierstrass factorization of the -function states that M KL p NwM where have R LMN R ( L p Dw p L O=R p m O=R S is Euler’s constant. If denotes the infinite product on the right hand side then we UaWHY K`L p NjM L p RDmUXWZY`P p Rm DmUaWHY!3 Lp R N p UaWHY`P LM R( Therefore it seems natural to consider functions of the form L p mUaWHY` N R p x ! p WZY p LM UaWHY` R S where denotes an infinite product having negative zeros. We recall that if | is any sequence of complex numbers ( D converges then pqr (P LM p m f ) such that I Lp O RT. O R defines an entire function. It is commonly denoted the canonical product of genus 1 associated with the sequence | , or having zeros at . (The genus is defined as the smallest integer f ¡ such that converges.) *¢Z Throughout this paper |£ denotes a sequence satisfying ¤¥£ §¦ (1.1) £ I ¦ . and (1.2) for all L_¨ ? ¨2d¥f R ¦ Const ¨ S . Here ? is the counting function associated with the sequence |£ : L¨ ? ¨H R,Dc©ª|«£ ¦ We shall consider the canonical product of genus 1 having zeros at m¬£ , ¡ function is defined because of (1.2). Since all its zeros are negative we may define UXWZY` L p R!D h (P x WZY LMN p O£ Rm p O£ M . This S for p®~ . (Here x WHY denotes the principal logarithm, defined in terms of the principal L argument ¯¬°Y ). This is the unique branch of the logarithm of p R that is real on the positive axis. Its imaginary part is given as ± °JY! L p R!D ¯¬°Y P ( LMNwL_^bNo²´³ RO£ R7m ³ O£ S ETNA Kent State University etna@mcs.kent.edu 96 HENRIK L. PEDERSEN h ^2No²´³ ~ . for p D We recall that a Pick function is a holomorphic function µ in the upper half-plane z with L p s ¡ f µ R for pª~ z . Pick functions are extended by reflection to holomorphic functions in i¶k,B and they have the following integral representation (1.3) L p µ p N R!Dc£ Ng· M u ¸ ¸ p m m L¸ ¸ I NwM y¹Zº R S f where £ ¡ , ~ B and º is a non-negative Borel measure on B satisfying · See e.g. [6]. It is known that £D»¾Ua¿ ¼X½ µ L²À³ RJO L_²À³ R L¸ R ¹Hº ¸ I NwM s L_² DwÁµ S ¤jnw R DÂ¾.UX¿ ¼X½ º S L ¸ N¶²À³ µ F (Ã R ¹ ¸ S h where the last limit is in the vague topology, and finally that µ has a holomorphic extension L f to if and only if ÄJÅ== º RÆÇlCmon l. S We describe our main results. M T HEOREM 1.1. If |£ satisfies (1.1) and bmÈ£ ¦ Const for ¡ , we have mUaWHY` p`x p WZY where É ^ ÉeL_^ ~¶Ë R ^ ~¶Ë LM UXWZY` M m p R N ÉeL ¸ · ¸ m¬£ (ÌP S m¬£ l and ¡ M m¬£ m p R ¹ ¸ S L^ N ^ mUaWHY§ R UaWHY§ , ^ LJL ^ N I F I UXWZYÊ R R R`D and ÉeL^ for MN D is defined as (1.4) for Lp R`D L_^ m UXWZYÊ Ç R . ^ LJL ^ I N F I UXWZYÊ R R f l. L L S Theorem 1.1 fournishes the Stieltjes representation of UaWHY` p RJO px WHY p R on a half-line. The representing real-valued measure has density w.r.t. Lebesgue measure on the negative line and has a point mass at 1. We could equally well have given the representation of mUXWZY` L p R p x ! N p WHY p UXWZY` LM R in terms of a density on the negative half-line (there is no support of the measure on the positive half-line, since the function has a removable singularity at 1). In this general setup there is no particular reason for this density to be positive. However, when ÍDÎ! is the K canonical product in the Weierstrass factorization of the -function above, the corresponding density is positive. This leads to the question of finding conditions on the distribution of the zeros in order that the corresponding density be positive, or, what amounts to the same, that ETNA Kent State University etna@mcs.kent.edu PICK FUNCTIONS RELATED TO ENTIRE FUNCTIONS 97 the function above is a Pick function. Our second result (Theorem 1.2) gives such a condition and is a generalization of the main result of [5]. NjM M T HEOREM 1.2. If |£ satisfies (1.1) and if ¦ £ Ï¦ for ¡ then Ð Lp L p mUXWZY` RÑ R p x ` N p p WZY LM UXWZY` R is a Pick function. It has the representation Ð Lp · MN RD ^ ¹ ~ÓË for m¬£ ÌP S m¬£ l L^ mUaWHY R`D ^ ~ÓË m¬£ M and ¡ ¹ for L_^ L^ ^ R LJL N R m ¸ p ¹ S where the positive density ¹ is defined as (1.5) L¸ ¹ ¸ LM b ^ N ^ UaWHY` R UaWHY§ ^ N I F I Ò UXWZYÊ R R and R`D L^ N LM ^ mÇUXWZY R. UXWZY R ^ LJL ^ I N F I UXWZYÊ R R f l. We notice that Theorem 1.2 can be generalized, by shifting the zeros to the left; see Theorem 5.7. In terms of the counting function associated with the zeros, the positivity of the density ¹ can be expressed more compactly as S UXWZY§ ^ L^ R.mÈUXWZY` f LM R ^ ¦ ? L m ^ RTUXWZYÊ L_^ ^ S for ¤ . This gives an upper bound on R. on the negative axis. We shall describe the ^ L^ asymptotic behaviour of the maximum of UXWZY§ R. as tends to m§n . To prove our main results we shall use a Phragmén-Lindelöf argument, and need to find K new arguments (avoiding the functional equation of the -function) in order to investigate the growth at infinity and the boundary behaviour on the real line. 2. Growth estimates. Throughout this section denotes the canonical product of genus 1 having negative zeros m¬£ , where |£ satisfies (1.1) and (1.2). We shall estimate the growth of the holomorphic function p2qr UXWZY` p!x L p WHY p R in the upper half-plane. To do this we need some preliminary results about the growth of the canonical product in the half-plane. L EMMA 2.1. Suppose that |£ satisfies (1.1) and (1.2). Then UaWHYÊ L p R. ¦ Const p : UXWZY p for all large values of p . The proof is relatively straight forward. We have included it here for the readers conveL L nience and also in order to illustrate the use of p R m p R . ETNA Kent State University etna@mcs.kent.edu 98 HENRIK L. PEDERSEN Proof. We get, by partial integration, Lp UaWHY` · R!D L x p · Dm LMN WZY L¸ ? p R ¸ ¸ O p Rm M u ¸ O L¸ R ¹ ? M ¸ N m ¸ R ¸ p ye¹ S so that Lp UXWZY§ ^ · R.Dm N§³ I · ^ f ¡ R ¸ M u L¸ ? If L¸ ? L^N ¸ ¸ I N¶³ I ye¹ R M R ¸ ^bN L^bN m ¸ ¸ ¸ I N¶³ I ¹ R ¸ then ^bN L^bN ¸ M ¦ ¸ I N¶³ I R ^N M ¦ ¸ ¸ S so that the first term in the relation above comes out negative. Hence, by the assumption (1.2), L p UXWZYÊ R ¦ ³ · L¸ ? R ¸ ³ ¸ ¸ I No³ IC¹ R L_^bN Const p Ô ¦ We conclude that ª is of finite exponential growth in the right half-plane. To estimate the growth in the left half-plane we use the identity Lp It is easily seen that p m ] Lp RD p I u M P ( R7Ñ m I y £ is an entire function of finite exponential type. We thus get L UaWHY§ L p ] L R p m L _L ^ R . Const p mÈUXWZY§ p .R ¦ Const p mUXWZYÊ R ¦ L^ ¡ S R R. . Therefore, a lower bound on ª in the right half-plane will give since also us an upper bound on ª in the left half-plane; we thus consider (2.1) for ^ L_^ ¡ mUXWZY! f d¥f . We find, with Õ ^ · ? L¸ ¸ R R`D M ¸ ? L¸ ¸ R u M M m ¸ ^bN ¸ ye¹ ¸ m ^N ¸ y ¹ ¸ ¦ ^ Ö Const UaWHY ¦ p ^ · Const D p M u M ^N m ¸ LMN¶^ OÕZR p Const :UXWZY S for large. The lemma is proved. We also need an estimate of ± °JY! . L EMMA 2.2. Suppose that |£ satisfies (1.1) and (1.2). Then ± S smaller than £ , M u ^Ê· °JY! L p R ¦ N Const p UaWHY§ p Const ¸ y ¹ ¸ ETNA Kent State University etna@mcs.kent.edu 99 PICK FUNCTIONS RELATED TO ENTIRE FUNCTIONS h . for all p~ Proof. We may assume that p D ± (2.2) L^bNo²À³ °JY ^bNo²À³ ± Rm ~ L_²´³ °Y` . We have z ·Ø× RD ± ^ Ù Ù L²À³ L ¸ No²´³ °Y` R ¹ ¸ L^NÓ²À³ ± We shall now find estimates of ± °Y! R and of Ú J° Y R . From these estimates the _L ^ÛNg²À³ ± Ú × relation above can be used to find estimates of °` Y R in z . A simple computation shows that ± ^ Ù L_^bNo²´³ °JY! ³ · R!Dm LÝÜN¶^ I No³ IC¹ ? R LÝÜ R( Ù We also find that ± L_²´³ °Y` ³ R ¦ L³ÏNjM Const UXWZY R( This relation is straight forward to verify, e.g. by using partial integration. ^ f Suppose first that ¡ . Then, by (2.2), ± L^bNo²À³ °Y! R. ¦ ± L_²´³ °Y` LÝÜN Ng· Const p : UXWZY p ¦ ³ NÞ·Ø× · R × · L´Ü,N Ü I N¶³ I0¹ ? LÝÜ ¸ I N¶³ I ¹ R ³ No^ · Const p : UXWZY p ¦ ¸ I No³ I ¹ ? R ³ L´Ü ¸ R ¹ ¸ L´Ü ¹ ? R R Here, by integration by parts, ³ ^Ê· ^ Suppose next that ± LÝÜ Ü I N¶³ I ¹ ? °Y` f ¤ L^bNo²À³ R`DcQ ^Ê· LÝÜ ? ³ Ü I Ü ¦ Ü I No³ I Ü I No³ I ¹ R Ü ^ Const . In this situation, R. ¦ ± L²À³ °Y! ³ × · · N R. LÝÜ ¸ I No³ I0¹ ? R m L´Ü ¸ R ¹ S and we turn to estimate the double integral on the right hand side. We write it as ³ × · · L´Ü m ¸ I No³ IC¹ ? R LÝÜ R ¹ ¸ × · · D ³ L´Ü × · · N ÌP × N ³ I=¹ ? ¸ I Ø R ³ m L´Ü P Ì m × LÝÜ ¸ R ¹ LÝÜ ¸ I N¶³ IC¹ ? R ¸ R ¹ S and we estimate these integrals separately. We find × · · × ÌP ³ L´Ü m ¸ I No³ IC¹ ? R LÝÜ R ¹ ÌP × · ¸ D D × · ¦ F ÌP ? L m × · · ^bNw M ³ L´Ü LÝÜ R ¦ m ¸ I No³ IC¹ R ³ N ³ I ¹ ¸ I ¶ R ^ Const m ¸ ¸ ¹ ? ¹ ? L´Ü LÝÜ R R ETNA Kent State University etna@mcs.kent.edu 100 HENRIK L. PEDERSEN ¸ ~ÞË f The second integral is more delicate to estimate: first of all, if integration, · ³ ÌP LÝÜ m × LÝÜ ¸ I No³ IC¹ ? R L R!Dm? · N ? P Ì × S , we get by partial l ³ ^2NjM m ^ m R L ¸ No^ LÝÜ Ü R L´Ü Q LJLÝÜ m m M I N¶³ I Ü ³ ¸ Ü m R ¸ I N¶³ I C I ¹ R R S R so that × · · ³ ÌP × L´Ü LÝÜ ¸ I No³ I ¹ ? R m × · · ¸ ¦ R ¹ ÌP × LÝÜ ? R Ü LJLÝÜ LÝÜ Q Ü ³ ¸ Ü ¸ R o N ³ I I ¹ ¹ ¸ I R R m m This integral we write again as · × · P Ì × Ü qr Ü ÝL Ü ¸ O m R L´Ü ? Ü R Ü Ü ¸ LÝÜ m L´Ü Q m and, using that is decreasing for that this integral is bounded by ¦ D D ÜÏd ¸ Ü ¸ I No³ I0¹ R ¹ ¸ S and the assumed bound on ? ^2NjM m Q b ^ NjM ¸ LÝÜ ¸ Ì m m m R × ^2NwM · · × m ^bNwM LÝÜ Const ¸ m m m ^2NwM · × m ¸ ^bNwM Const ¸ ¹ m m L ^bNjM L ^bNjM Const m RTUXWZY m R( · Const ³ ¸ I R ¸ I No³ I ÝL Ü R m m × · ³ I No³ I0¹ Q Ü ¸ I N¶³ I0¹ R R , we see ¸ ¹ ³ LÝÜ Ü ¹ ¸ The lemma follows by combining all these estimates. P ROPOSITION 2.3. If |£ satisfies (1.1) and (1.2) there exist a constant ß ¨ quence | A tending to infinity such that à à and a se- à à à à h UaWHY` p!x à Lp WZY p R à ¦ ß ¨ for all p~ of absolute value A and all ? . Proof. A classical result, going back to Littlewood, see [8], states that for some sequence ¨ r A n , ã ä¼aã á Òâ åæ UXWZY§ L p R. ¡ m Const ã äJÄ ã Å=3 å æ UaWHY§ L p R Therefore, by Lemma 2.1, :UXWZYÊ for all p satisfying p D ¨ A L p Ra ¦ Const p : UXWZY p and all ? . The proposition now follows from Lemma 2.2. ETNA Kent State University etna@mcs.kent.edu 101 PICK FUNCTIONS RELATED TO ENTIRE FUNCTIONS 3. An auxiliary Pick function. In this section we suppose that the sequence |£ satisfies (1.1) and also £ m¶Ò ¦ (3.1) Const This is equivalent to (3.2) L¨ ? ¨ Rm ¦ Const The main result of this section is the following theorem. T HEOREM 3.1. For any given sequence |£ satisfying (1.1) and (3.1) there exists a LM ¦ real constant ç UXWZY R such that ÐZè L p Lp mÇUXWZY p!x R!D N R p WHY p ç is a Pick function. d LM R EMARK 3.2. If ç UaWHY` R then the function in the theorem above is certainly not LM a Pick function (its imaginary part tends to m§n as p tends to 1). Thus çD®UXWZY R yields LM the strongest result. In many cases the constant ç should be taken smaller than UXWZY` R in ÐZè order that be a Pick function. As an example, consider Lp R!D LM,N p L RT.T LMN Ré (Òê p m p L OZ=RT.T p m OZ=R S M which has zeros at m më m¬ì . . In this case, computer experiments indicate that there ^ M L^ ^ LM d ^ S S are points ~cË më m S l such that UXWZY§ R.Tm UaWHY` R UXWZYÊ . This implies (see the S proof of Theorem 3.1 below) that the imaginary part of L p mUXWZY` N R p x ` p p WZY LM UXWZY` R in the upper half plane has some negative boundary values and hence that the function cannot NØM be a Pick function. However, in í 5 we show that if ¦ £ ¦ then we may take ç equal LM R . to UaWHY` To prove the theorem we need some lemmas. L EMMA 3.3. For the canonical product of genus 1 associated with a sequence |£ ¦ satisfying (1.1) and (3.1) we have (with Õ £ ) L¸ mÈUaWHY` for LMN UaWHY ¸ ¸ OÕR ¦ Const ¸ dÞf . Proof. From (2.1) we have (with Õ ¦ £ ) mUaWHY` for ¸ R7m ¸ dgf L¸ R!D ¸ · Ö ? mÈUXWZY Rm ¸ UXWZY LM,N Ü R M u M ÜÛm Ü,N Ü ¸ ye¹ S à à . Therefore, and by (3.2), L¸ L´Ü à à ¸ à à à mÇUXWZY! à OÕZR.D L¸ à D à ¸ · Ö u ? R7m ¸ · LÝÜ Ü R m u Ö M y M u Ü m ÜN M Üm à M ¸ y M ÜN ¸ ye¹ à ¹ Ü Ü à à à à ETNA Kent State University etna@mcs.kent.edu 102 HENRIK L. PEDERSEN ¦ ¦ M ¸ · Const Ü Ü I LÝÜN Ü,N m ¸ Ö ¸ ¦ M Ö ¸ · Const M u Ü ¸ ¹ R ¸ y ¹ Ü Ü Const The lemma is proved. L EMMA 3.4. For any given sequence |£ satisfying (1.1) and (3.1) there exists a real L_^ ^ ¦ f ^ f constant ç such that UXWZYÊ for all ~ÓË m¬£ l and R.mÈç L_^ UXWZYÊ ^ R.ZmÓç ^ ¦ ,UaWHY§ S ^ M for all ~ÓË m¬£ ÌP m¬£ l and all ¡ . ^ S f f L^ M R ¦ Proof. For ~jË m¬£ l we have ¦ , so the first assertion of the lemma is f S evidently true. Given any bounded interval of the form Ë m¬î l , it is possible to choose ç f S such that the asserted inequalities hold on any interval Ë m¬£ (ÌP m¬£ l`Æ Ë m¬î l . Hence we ^ ¦ M ^ ~ÓË ^ ¸ S S ¡ M may assume that m . If and m¬£ (Ì m¬£ l , we put Dm and find L^ mUaWHY§ ^ RmÓç DwUXWZY§ S ] L¸ L¸ RmUXWZY, N R ç ¸ ] L LM I p I m O £ R is (as used before) an entire function of exponential type. Here, p RVDðï ( Hence, by Lemma 3.3, L^ mUXWZY§ ^ R.mÓç ¸ N ¦ LM,N ¸ ¸ Const UaWHY O ÕZR ¦ Const ¸ N ¸ aU WHY ¸ N ç ¸ N ¸ ç ¸ ~ Ë We get, for Ó £ £ (ÌP l , S L^ ¸ UXWZY ¸ ¸ ¦ m¶UaWHY ^ ¸ ¦ £ (ÌP m¶ÒUaWHY ¸ N ¸ N Const UXWZY ¸ ¸ S ¦ Const so that mUXWZY§ R.ñm«ç ç UXWZY . From this relation we see that it is possible ^ ~ÓË L^ ^ ¦ ^ to choose ç such that mUXWZY§ m¬£ (ÌP m¬£ l . This R.mªç UaWHY§ for all and all S completes the proof. Proof of of Theorem 3.1. We consider the harmonic function L p @ Lp u mÇUXWZY px s R!D ¸ ~ for all have B . Indeed, if ¸ dcf Ð L p where ö M is holomorphic at p D @ M ¸ and D L p then @ Lp Lp in z . We claim that r R LM y f ¡ R çjmÈUaWHY` M p m R!D f ¡ p ç p WHY in the upper half-plane. Our goal is to show that @ ¼X½Þ ¼Xá ä Ua¿« ôâ õ @ òÀó ä. N R @ L¸ R NØöL p f R R¬D S as pªr . Thus, L R7Dm çwmUXWZY` M I p m LM RJR ³ N s öL p R S ¸ . for p near 1 we ETNA Kent State University etna@mcs.kent.edu 103 PICK FUNCTIONS RELATED TO ENTIRE FUNCTIONS L f LM R ). Since mUXWZY! (and is 0 if çcDjUXWZY` so Ua¼X½g¼aáâ ä ¿ @ p R ¡ holomorphic in a neighbourhood of the origin) has a zero there, L p @ as pr f Const R. ¦ x p WZY L ¸ within z . If ~ m¬£ (ÌP m¬£ R for some ¡ S @ Lp rtsvu mUXWZY ¸ L R,m F ¸ LJL D L¸ ² F R.m ¸ No² F UaWHYÊ R m ¸ I N UXWZYÊ R L FÒI f R M N ç ç p (which is actually , y L¸ L N ¸ mUXWZY ¸ R f r m Lp N R. ç ¸ N ¸ ,UXWZY§ R f (and a similar estimate holds for ED and ~ m¬£ R ). From the lemma above we see that L ¸ these expressions are non-negative. If D÷m¬£ for Ssome then one sees that @ p R r n . Our claim is verified. The function m@ is a harmonic function in the upper half-plane, which has, as we have just verified, non-positive boundary values on the real line. It is (by Proposition 2.3) bounded from above by some fixed constant on some semicircles, whose radii tend to infinity. The ordinary maximum principle yields that m@ is bounded from above in all of the upper halfplane, and hence, by applying an extended maximum principle, that m@ is non-positive in the upper half-plane (see e.g. [9, p. 23]). The theorem is proved. 4. Integral representation. We shall find an integral representation of functions of the form mUaWHY! px p WZY pqr Lp R S where is the usual canonical product associated with the sequence |Hm¬£ satisfying (1.1) and (3.1). We shall do this by using the Pick functions described in Theorem 3.1. We choose LM ç ¦ UXWZY R such that ÐZè L p Lp mÇUXWZY p!x R!D WHY R N p ç p is a Pick function. We shall find the integral representation of ÐTè as expressed in (1.3). We have L L L EMMA 4.1. We have, with @ è p R!D s`ÐZè p R , M F @ è L ¸ No² O?3R ¹ ¸ m r A ¹ è L ¸ ¸ NcL R ¹ UXWZY LM RmÈçÊRÀÕ L¸ in the vague topology. Here Õ is the point mass at 1 and ¹ è is defined as ¹ è R D ¸ ¡ f , ¹ for ¸ ~ÈË m¬£ (ÌP S m¬£ l è L¸ and ¡ M ¹ f ¸ ~ Ë for È m¬£ l. S mUaWHY ¸ LJL RDm L ¸ R. ¸ N UXWZYV R ç ¸ N I N F I ¸ ,UXWZY§ R and è L¸ R`Dm L¸ N ¸ mUaWHY R. ç L L N ¸ ¸ I F I UXWZYÊ R R S S f for ETNA Kent State University etna@mcs.kent.edu 104 HENRIK L. PEDERSEN Proof. A computation shows that (4.1) p!x p I è L p @ R!D WHY L_^ Lp N ³ ± o Lp p aU WHYÊ R. ° Y! R R¯°JY NbL_³ Lp ^ ± L p p UXWZY§ R. m J° Y R RTUXWZY§ p I p m¬çb ¯¬°Y L Here we notice that ± °Y! p R is bounded on compact subsets of the upper half-plane (Lemma 2.2) L p L ~ UaWHY§ R involves only logarithmic singularities. If ø ß¬ù B!R has its support in and that M Ë m¬î OQl it thus follows that S GI · L¸ ø L ¸ N¶² RUaWHY§ ¸ O?3R ¹ *ú GI · r A m L¸ ø L¸ RTUXWZYV ¸ R. ¹ <ú S and that GJI · L¸ ø ± R L ¸ No² °Y` <ú ¸ O?3R ¹ r m GI · A L¸ ø R ± L¸ °JY` <ú ¸ R ¹ By (4.1) we see that M GI · F ø L¸ è L ¸ No² R@ O?3R ¹ ¸ <ú r m GI · A L¸ ø <ú R ¹ è L¸ R ¹ ¸ S where ¹ è is given as in the statement of the lemma. (We notice that the origin does not L represent any difficulty since, as noted before, @ è p R ¦ Const OC x WZY p for p near zero.) MN ËM ûl , then If ø has its support in mÈû M F S ÌÒü · ü L¸ ø R@ è L ¸ N¶² ¸ O?3R ¹ r m A L m ç¥mUXWZY, LM RRø LM R( This follows from the (already used) fact that L Ð Lp ö LM çjmÈUaWHY` M p m RD RR NÞöL p R S M p where is holomorphic near D . The lemma is proved. We have thus found the integral representation ÐZè L p R!D · u NbL UXWZY M ¸ m LM ¸ m p ¸ I NjM ye¹ RmÈçÊR u è L ¸ M M M m p m ¸ R ¹ Noý è p N¶þ è y Q S where ý è Ð è L²À³ D»¾.UX¿ ¼a½ RJO L²À³ f R7D S and þ è DcÁ ÐZè L_² R!D F Q UXWZY§ L² R.Ô ETNA Kent State University etna@mcs.kent.edu 105 PICK FUNCTIONS RELATED TO ENTIRE FUNCTIONS f On the other hand, we know that çÊO x WZY p is a Pick function (since ç¤ x M m WHY · D p This gives us M u ¸ M ¸ m ¸ I NwM y m p L ¸ I N UaWHY§ R ¸ N F I§¹ ) and that M M M m p m Q mUXWZY` p!x p WHY L p ÐZè L p R D (4.2) Rm · D N ç x M u ¸ ¸ R M u ÉÿL ¸ ¸ I NwM y m p m LM XU WZY p WHY M M m p m N y Q R ¹ F Q ¸ L_² UaWHY§ R. S É where is given in (1.4). É Concerning the growth of L EMMA 4.2. we note: · ¸ ÉeL ¸ R ¸ ¤wnj ¸ I NwM ¹ ] L LM I p I Proof.] We shall again bring in the function p RD ï P m O £ R . We have earlier used that is of exponential type and now we shall also use that · (4.3) UXWZY ] Ì ] L ¸ R. ¸ ¤¥nw ¸ I NwM ¹ This relation follows since is of polynomial growth on the real line (see e.g. [10]; one may also use estimates similar to those in í 5). It is known (see e.g. [9, p. 50]) that the exponential growth together with (4.3) implies · (4.4) ] L¸ :UXWZYÊ R.X ¸ ¤wnj ¸ I NwM ¹ ¸ ~ Ë m¬£ (ÌP m¬£ l , We find, for Ó S L¸ N mUaWHY R. L ¸ I NwM L L ¸ R XU WZYÊ ] L mUXWZY L ¸ I NwM LL ¸ R UXWZYÊ ¸ ÉÿL ¸ R ¸ I NwMvD D f Hence (with £§D R R I N F I R L ¸ N L ¸ UaWHY` m R UXWZY m R L ¸ I NwM JL L ¸ I N F I R UXZ W Y§ R R N R ), · R ¸ ¸ ,UXWZY§ I N F I ¸ ÉÿL ¸ R. ¸ ¦ ¸ I NwM ¹ ] L ¸ :UXWZYÊ R.X L ¸ I NwM · N Ã · M ¸ I N FÒI ¹ UaWHYÊ R L¸ N ¸ :UXWZY` R UaWHY , ¸ L ¸ I NjM LL ¸ I N F I ¹ R UaWHY R R ¸ ] L¸ The integral involving UaWHY§ R.X is, by (4.4), finite. It therefore suffices to estimate the infinite sum in the last line of the relation above. To do this we use Lemma 3.3, from which ¸ we get (for large ) UaWHY` L¸ R N UXWZY ¸ ¦ Const ¸ S ¸ ~ÓË £ S £ (ÌP l S ETNA Kent State University etna@mcs.kent.edu 106 HENRIK L. PEDERSEN where the constant does not depend on . The infinite sum of integrals is therefore bounded from above by a constant times the finite integral · I ¸ ¹ ¸ I UXWZY R ¸ L The lemma is proved. This lemma makes it possible for us to split the integral in (4.2) into a sum of two and we thus get mUXWZY! p`x p WZY Lp R ÉeL ¸ · D ¸ R ¸ N p ¹ Ï m LM UXWZY M m R N p ß S for some constant ß . We may identify ß by noting that × ¸ so that (by Lemma 3.3), ÉeL ¸ · Ua¿ ¼X½ R ^ m ¸ ¹ D × mUXWZY! ^ ^ UXWZY × UX¿ ¼a½ ^ R f D S ßD LM UXWZY M m UX¿ ¼a½ L_^ R M D We conclude: If the sequence |£ satisfies (1.1) and (3.1) we have the representation mUaWHY! p!x p WZY Lp R D ÉÿL ¸ · ¸ p ¹ m ¸ N R LM UXWZY` M m p h R NwM p~ S S É of the canonical product of genus 1 associated with the sequence |m¬£ . The density defined in (1.4). We have proved Theorem 1.1. is 5. Zero distribution and positivity of density. In this section we prove Theorem 1.2: LM We show that Theorem 3.1 holds with çcDjUXWZY` R provided that the sequence |£ satisfies ~¶Ë £ (5.1) NwM S l S M ¡ We begin our investigation by finding estimates of the function ] L p on the real line. For this we need à any ¡ Q , L EMMA 5.1. For à à à Proof. àà For ¡ à à LF ^ Ä¼Xá R M F ^ Q and L F ^ ÄJ¼aá R M F ^ M m ^ I ^ à ^ I m à à OZ à à S à à à à I D à ¦ I NwM ~¶Ë à OZ à M u M P ( RD l p I m M I y £ M OQ ^ ~ Ë M 2m l for ^ ¶ w N S M S ~ Ë ´l for ¶ S à , LF ^ Ä¼Xá F ^ à m m à F F =R à ^L^N I =R ¦ L I N =R ¦ Q M ETNA Kent State University etna@mcs.kent.edu For àà ¡ à ^ and LF ^ ÄJ¼Xá R M F ^ à ~ÓË m M ^ I m . If EDwQ since ¡ O M l àà , à S à à L F ^ à à Ä¼Xá D I F ^ à à à à F m F m à =R ^PL^N à we should check that à à I ¦ =R L M bm I L M R QHÏm M ¦ R S à à LF ^ Ä¼Xá R M F ^ à ^ 107 PICK FUNCTIONS RELATED TO ENTIRE FUNCTIONS à M m à ^ I M M ¦ O for ~wË Ql . This inequality does hold, but we do not give a detailed argument here. The S lemma follows. ] L^ M R ¦ P ROPOSITION ] 5.2. Suppose that |£ satisfies (5.1). For any ¡ Q , for ^ ~¶Ë L^ M ^ ~¶Ë ¦ £ l and R. OZQ for £ l. ^ S S Proof. Suppose that ~ÓË £ àà £ làà , ¡ Q . Since à à M for D S for ¡ . S M 2m S M à m I £ à D à I M m à à à à I £ ^ I M ¦ m à , and ^ I à M m £ Ô RHD D £ ^ I M ¦ I à à à à à à à à à ^ I à M m à à ^ I à à M m à I Ôà à LF ^ à ÄJ¼aá R M F ^ à à à I àà é a 3 à £ à NwM R I L m à ¦ m à ] L^ ^ I M D I , we get ^ I ^ I à M m O m à à à M I £ à é a3 M ^ I ^ I à L àm à à à à à ^ I N M w à I R à I Thus the proposition follows from the lemma above. P ROPOSITION 5.3. Suppose that |£ satisfies (5.1). We have mUaWHY! for ¸ dÞf L¸ R N ¸ UaWHY` LM · R!D LÝÜ ? M Ü R L´Ü,N ¸ L´Ü,NwM R ¹ R Ü é ¸ L¸ M m R S and in particular L¸ mUXWZY` N R ¸ UXWZY! LM R ¦ K`L ¸ NwM UaWHY R S M ¸ for ¡ . Proof. A computation, based on (2.1), yields the identity in the lemma. The counting L´Ü function ? R is, by the assumption on the £ ’s, bounded from above by the counting function ¸ ¡ M ? corresponding to the case of £ Dv . Thus, when , · ? LÝÜ Ü R M LÝÜN ¸ R L´Ü,NwM R ¹ Ü é ¸ L¸ m M R ¦ · ? DmÇUXWZY3 L´Ü Ü L¸ M R R L´ÜN N ¸ ¸ R LÝÜNwM UaWHY`3 LM R R ¹ S Ü é ¸ L¸ m M R ETNA Kent State University etna@mcs.kent.edu 108 HENRIK L. PEDERSEN M mQ . . But then we know that where denotes the canonical product with zeros m L¸ N ¸ LM K`L ¸ NwM S S mUaWHY! R UXWZY` R`DjUaWHY R . The proposition follows. ^ ~ÓË ¡ M ( P Ì m¬£ m¬£ l we have L EMMA 5.4. Let . For L^ UaWHY§ ¸ Proof. If Dm ^ ~¶Ë £ L_^ UXWZYV £ (ÌP l S ^ Rm S ^ Rm LM UXWZY` R ¦ ^ UaWHY§ , ] L ¸ L¸ N ¸ R!DjUXWZYÊ R.mÈUXWZY` R XU WZY` ] L N K L ¸ ¸ NjM ¦ UXWZYÊ R . UXWZY R S LM UaWHY` LM R by the lemma above. It is therefore enough to show that UXWZY§ ] L¸ N R K`L ¸ NwM M UXWZY R ¦ ¸ UaWHY S N M ¥ NjM ¸ ¸ ¦ £ 2¦ £ ÌP ] L ¸ S f Ë for ¡ . Suppose first that ~ÓË £ by Proposition lÒÆ l . Then UaWHY§ R. ¦ NwM K`L ¸ S NwM ¦ ¸ ~ÈË S ¸ ¡ M 5.2, so we should verify that UXWZY for . This last R `UXWZY l and all K S inequality follows e.g. by induction from the functional equation for the -function: UXWZY K`L ¸ N QZRDwUaWHY K`L ¸ NwM ¸ ~ Ë Suppose next that È 5.2, so that NÇM S UXWZYÊ R N L ¸ NjM UXWZY Ë £ (ÌP l3Æ ] L¸ R. N R ¦ NÇM S KL ¸ NjM UaWHY ¸ N ,UXWZY N ] L¸ . Here, UaWHYÊ Ql R ¦ L ¸ NwM UaWHY KL ¸ NjM UXWZY L R ¦ R ¦ NwM mUXWZYQ RTUXWZY L ¸ NjM R( by Proposition R7mÈUaWHYQ NjM N K`L ¸ w N M M ¸ ¸ ¡ M We now claim that UaWHY R mUXWZYQ ¦ UaWHY for ~ÓË Ql and . If ªD K`L ¸ ¦ N j M ¸ Ó S ~ Ë the inequality is true (UaWHY , we note that R UXWZY,Q for Q l ). To go from to S K`L ¸ N K`L ¸ NwM N L ¸ NjM UaWHY QZRmÈUXWZY,Q ¦ UaWHY R UXWZY RmUXWZY,Q N L j N M L NjM L ¸ w N M ¸ ¦ UaWHY ¸ ¦ UXWZY R RTUaWHY R , The lemma is proved. We can now give the proof of Theorem 1.2 (the promised generalization of [5, Theorem 1.2]): LM Lemma 5.4 shows that Theorem 3.1 holds with çÍD÷UaWHY` R . Then Lemma 4.1 and Lemma 4.2 yield the desired representation mUXWZY` L p R p x ! N p UXWZY` p WHY LM R · D ¸ L¸ ¹ m R p¹ ¸ NwM S where the density (defined in (1.5)) is non-negative. By differentiating under the integral sign, we find the following corollary. M C OROLLARY 5.5. We have, for ? ¡ , A L p RD ] L m M R A ÌP · ¹ L´ÜN ? L m p Ü R Ü R Ì ¹ A P where ] L p R!D mUaWHY` Lp R p x ! N p WZY p UaWHY` LM R S S ETNA Kent State University etna@mcs.kent.edu 109 PICK FUNCTIONS RELATED TO ENTIRE FUNCTIONS product associated with a sequence |m¬£ satisfying (1.1) and (5.1). and is the canonical ] f In particular, is completely monotone on l n Ë . L_^ S R. R EMARK 5.6. One may describe the asymptotic behaviour of the maximum of UXWZYÊ on the negative line for a canonical product having negative zeros. As an example we shall study L^ ÄÅ=Ò|UXWZY ^ Rm LM UXWZY` ^ RÒ ~¶Ë m¬£ m¬£ S l ^ ~¶Ë ¸ as tends to infinity. Here |£ satisfies (5.1). We have (with Dm £ L^ UXWZY§ ^ R.m LM UXWZY, ] L¸ R!DjUaWHY§ L¸ R.mUXWZY R N ¸ UaWHY` LM S £ l ) R( K`L ¸ N®M R . To From the proof of Lemma 5.4 we know that this is bounded from above by] aU WHY find a lower bound on the supremum above, we need a lower bound on . Such a lower ¸ ~¶Ë à à à £ àà £ l ): bound can be found by using the inequalities (for à à à à à M ¡ à à à à à 2mÓQ M m ¡ NwM L ] L¸ N £ ¡ M à à à £ à à à ¡ S à à à à NjM m à ¸ à ¸ à ¸ à M m S à à à à LM F L¸ m ÄJ¼aá m¶=R F L¸ mÓCR à ¡ R £ S à à à m NjM à . One obtains We now put D à à M à à à , and £ S à à à ¸ ¸ MN ¡ à ¦ à à , à for MN à M ¦ à à £ for for à ¸ RJOZQ ¸ O£ R M LM ¸ I m m ¸ à O£ R à and note that ~¶Ë £ Ë S £ l M bm OQ S NwM OZQlÀ Therefore ÄJÅ= |T ] L¸ R. ¸ ~¶Ë £ L¸ N ¸ S ¡ £ l ¡ LM à ] L . R à à à à LF Ü à L I ÄJ¼aá R £ mÈ£ R F Ü à à Const ã (ã ¼Xá â GI K`L ¸ N ¸ ¡ R UXWZY` R UaWHY QHRm UXWZYQ for Furthermore, mUXWZY` in Proposition 5.3) and we thus obtain, for a suitable constant ß , UXWZY KL NwM R,m¥,UaWHYQÊm UaWHY ¦ ÄÅ=Ò|UXWZY§ L^ R.m ¦ UaWHY K`L N QHR( N (by an argument as N £ È m £ R ß ^ LM ^ ~ÓË UXWZY, R3 m £ ¬ m £ l ¬ S Q7UXWZY L ¸ dvM ETNA Kent State University etna@mcs.kent.edu 110 HENRIK L. PEDERSEN Let us end this section by finding ways of weakening the assumption (5.1). As indicated in Remark 3.2, removal of some zeros of may destroy the Pick property of Lp mUXWZY! N R p x ` p p WZY UaWHY` LM R It is possible to obtain the following result, dealing with the case of the zeros being shifted to the left. M N M N M T HEOREM 5.7. Let £ ¡ and suppose that £ ~jË £ ªm £ l for eD Q .. . S S S Then L p mÇUXWZY! p2qr N R p x ! p WHY LM UXWZY p R is a Pick function. Here denotes the canonical product of genus 1 having zeros at |m¬£ . Proof. The proof follows the same lines as the proof of Theorem 1.2 and is based on the inequalities L^ UXWZY§ for ^ ~ÓË First M ¡ m¬£ (ÌP m¬£ l and ¸ S of all, for ~ÓË £ à £ àS à à ] L¸ M R.D aP KL I £R ¦ ^ R.m m £ K`L ¸ R ¦ ,UXWZY§ ^ . We shall briefly indicate how to verify these inequalities. M N NcM Ë N £ 2m £ l, (ÌP l3à Æ à à ¸ I à LM UXWZY S I à à à L ¸ mÓ£R K L¸ N ` £R m¶£R à à à I F L¸ Ä¼Xá m £R Ó M F L¸ m mÈ£R à M L¸ à I mÈ£R O I This estimate can be deduced in the same way as the estimate in Proposition 5.2 and using the identity ¸ I u M Ô m L £ M ¸ Secondly, from Proposition 5.3, for ¡ L¸ mUXWZY! where L has its zeros at m mUXWZY` L¸ R N R N £ N ¸ ¸ bm UXWZY` R LM S D KL ¸ N £R I £R K`L m ¸ N £R , LM UXWZY` M KL RI y N R ¦ ¡ L ¸ mÇUXWZY! M R N ¸ LM UXWZY` R S . The relation R!DjUXWZY M K`L ¸ N K`L £TRmUXWZY KL £R7m ¸ UaWHY`£ N £TR is an entire function with zeros at follows either by computation by noting that O p L N M ¡ M m £ ,m R and of order (at most) 1. Thus, by Hadamards factorization theorem, it is L N KL N LM S Lp p RT ç R , where one finds çcDwUXWZY £R and DwUaWHY`£ UXWZY` R . of the form L N M N c N M ¸ ~ÓË Ë Now, suppose that £ £ (ÌP l Æ £ 2m £ lR . We find UaWHY§ L_^ Rm ^ UXWZY` S LM R ¦ ¦ S ] L¸ K`L N ¸ £RmUXWZY £R7m aU WHY`£ UXWZY§ R K`L N KL ¸ NwM ¸ UaWHY £TRm UXWZY`£ XU WZY m £ Ó R N L¸ N L¸ UXWZY mÈ£R R S UaWHY K`L ¸ N ETNA Kent State University etna@mcs.kent.edu 111 PICK FUNCTIONS RELATED TO ENTIRE FUNCTIONS *L ¸ f N M *L ¸ N ¸ N N ¸ NcM ¦ ¦ £ ¦ £ RD Em and R¬D mUaWHYQ for £ ¦ for £ where Ü ¸ mÇ£ we see that the desired inqualities are verified if according to Lemma 5.1. If we put D KL UXWZY for bm M ¦ Ü ¦ UaWHY Ü K`L £R7m LÝÜN N £RTUXWZY`£ K`LÝÜNwM UXWZY N R Ü ¦ UaWHY mÓUaWHYQ ¦ UaWHY L´ÜN £TR and L´Ü,N £ Rm T NM . LÝÜ¬N N £RTUaWHY`£ KLÝÜNjM UXWZY N R Ü UXWZY UXWZY LÝÜN KL S £R S M ¡ To verify these inequalities we use that UXWZY , £R ¦ £UaWHY`£ for £ Ü Ü K ¡ f ¡ , and the inequalities for used in the proof of Lemma 5.4. UXWZY£ £R UaWHY The theorem is proved.N M M ¡ M m If we take £ D for some ¡ then we obtain the following corollary. S ¡ M C OROLLARY 5.8. For any , the function for ¦ ¦ UXWZY «R7mÈp!x UaWHY K`p L « Rm K`L p N UXWZY pqr p UXWZY WHY is a Pick function. M ¡ M R EMARK 5.9. It should be noted that e.g. the assumption £ m¥Ò ¦ OZQ , for ¨ÊN M is not sufficient to produce a Pick function. Even for arithmetic sequences £ D , m ¨bd¥f where is close to zero the result need not be a Pick function. A convergent sum of Pick functions is again a Pick function. As a consequence we mention the following result, the moral being “the more zeros the better”. C OROLLARY 5.10. Let | be a sequence of products of genus 1 and sup canonical N M ~¶Ë N ¡ M £ £ m £ l , for , where and pose that is associated with |Hm¬£ ¡ M S L I R ¤¥n some numbers £ . If ( aP £ then Lp R!D L p mUXWZY! p2qr is a Pick function, where with |m¬£ . L P p R ï N R p x ! p p WHY LM UXWZY R is the canonical product of genus 1 associated 6. A related Pick function. We note the following generalization of [5, Theorem 5.1]. The proof is exactly the same and we shall not repeat it here. f I P ROPOSITION 6.1. If ¤j §¦ I ¦ . and ( ¤jn then pbqr m UaWHY! Lp R p u P D M x LM,N WHY m p O R p y is a Pick function and it has the integral representation m UXWZY! Lp p R Dv where D and is defined as L ¸ R`D f L ¸ R`D · N ¸ for ¡ m ¸ ( u m M u ¸ ¸ p m L¸R ¹ ¸ S ¸ I NjM y m M m ± "!$# ° ± á u M y¬y and ¸ for ~ Àl m (Ì m Ë E D S S M S Q S . ETNA Kent State University etna@mcs.kent.edu 112 HENRIK L. PEDERSEN REFERENCES [1] H. A LZER , Ungleichungen für die Gamma- und Polygammafunktionen, Habilitationsschrift, Bayrischen Julius-Maximilians-Universität, Würzburg, 1999. [2] G. D. A NDERSON AND S.-L. Q IU , A monotonicity property of the gamma function, Proc. Amer. Math. Soc., 125 (1997), pp. 3355–3362. [3] G. D. A NDERSON , M. K. VAMANAMURTHY, AND M. V UORINEN , Special functions of quasiconformal theory, Expo. Math., 7 (1989), pp. 97–136. [4] C. B ERG AND H. L. P EDERSEN , A completely monotone function related to the Gamma function, J. Comp. Appl. Math., 133 (2001), pp. 219–231. [5] C. B ERG AND H. L. P EDERSEN , Pick functions related to the Gamma function, Rocky Mountain J. Math., 32 (2002), pp. 507–525 [6] W. F. D ONOGHUE , J R ., Monotone Matrix Functions and Analytic Continuation, Springer, BerlinHeidelberg-New York, 1974. [7] Á. E LBERT AND A. L AFORGIA , On some properties of the gamma function, Proc. Amer. Math. Soc., 128 (2000), pp. 2667–2673. [8] W. K. H AYMAN , Subharmonic Functions, Volume 2, Academic Press, 1989. [9] P. K OOSIS , The Logarithmic Integral, I, Cambridge University Press, Cambridge, 1988. [10] N. L EVINSON , On a problem of polya, Amer. J. Math., 58 (1936), pp. 791–798.

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