On Power Series Analytic in the Open Unit Disk with Finite Doble-Logarithmic Order

Transcription

1 Applied Mathematical Sciences, Vol 2, 2008, no 32, On Power Series Analytic in the Open Unit Disk with Finite Doble-Logarithmic Order Mehmet Açıkgöz University of Gaziantep, Faculty of Science and Arts Department of Mathematics, 2730 Gaziantep, Turkey Peter Tien-Yu Chern I-Shou University, Department of Applied Mathematics Ta-Hsu, Kaohsiung 840, Taiwan Abstract By introducing the concept of double-logarithmic order, we deal with intrinsic properties between maximum term and central index for a class of analytic functions with slow growth in the open unit disk Mathematics Subject Classification: Primary 30B0 Keywords: Analytic in a disk, maximum term, central index, jump points, double-logarithmic order Introduction and results Let f z a n z n be a power series expansion which is analytic in the n0 open unit disk D, and let M r, fmax z r f z denote the maximum modules of f, μr, fmax n<+ a n r n denote the maximum term of f z for z r, and we denote the rank of μ r, fbyν r, f which is also called the central index for the power series expansion f There are many interesting problems that concern the intrinsinc properties of an analytic function itself G Valiron in [2] established such a basic

2 550 M Açıkgöz and P Tien-Yu Chern relationship between log μ r, f and ν r, f, say log μ r, f log μ r 0,f+ r r 0 ν t, f dt t In this article we study the intrinsic properties for a class of power series expansion which is analytic in D with slow growth A positive unbounded increasing function S r, 0 <r<, is said to be of double-logarithmic order λ if log S r 2 lim sup r log λ r We develop an integral characterization for increasing function S r tobe of finite double-logarithmic order see Theorem In section 4, two growth indexes of jump points r j } j and the central index ν r, f, say ρ f and λ f, for a given power series expansion f analytic in D are introduced Then we have ρ f λ f see Theorem 4 We give relations between series in terms of r j and among integrals in terms of log μ r, f and ν r, f in section 5, and it follows that log μ r, f has double-logarithmic order λ f see Theorem 6 In Theorem 5 we obtain that if f z isa power series analytic in D and if log + μ r, f is of finite double-logarithmic order λ, then for every μ>0, the series r j 3 μ, j r j and integrals 4 and ν t, f μ dt, t log μ t, f 5 t log t μ+ dt t are all convergent for μ > λ and all divergent for μ < λ For μ λ in theorem 5, above expressions can be either all convergent or all divergent see Theorem 8 In section 7, we obtain that if log μ r, f is of finite positive double-logarithmic order λ, then ν r, f r log r r 6 lim sup λ r log μ r, f Our proof see section 9 on the above inequality is an application of the L Hopital s rule for the proximate double-logarithmic order λ r The proof

3 Functions with double-logarithmic order in a disk 55 of the existence of a proximate double-logarithmic order λ r for a given S r with finite double-logarithmic order λ is given in section 8 2 An integral criterion for a positive increasing function to be of finite double-logarithmic order in a finite interval A positive unbounded increasing function S r, defined for r 0,, is said to be of double-logarithmic order λ if log S r 2 lim sup r log λ r If S r is of finite positive double-logarithmic order λ, we say S r isof maximum, mean or minimum double-logarithmic type according as the upper limit S r 22 lim sup r λ r is infinite, finite positive, or zero If S r is of minimum double-logarithmic type with double-logarithmic order λ, we say S r isof the convergence class or of the divergence class according as the convergence or divergence of the integral 23 r log r S r r λ+ The function S r λ r log μ r, for example, is of double-logarithmic order λ for each real μ, and of maximum double-logarithmic type for μ > 0, of mean double-logarithmic type for μ 0 and of minimum double-logarithmic type for μ<0; it belongs to the divergence class for μ 0, to the convergence class for μ< Theorem An integral criterion Let S r be an unbounded positive increasing function defined for r 0, Then S r is of double-logarithmic order λ 0 <λ<+ if and only if the integral S r 24 r log r μ+ r is convergence for μ>λand divergent for μ<λ To prove Theorem we need a lemma as follows Lemma If S r is an unbounded positive increasing function defined for r 0, such that the integral 24 is convergent, then S r is of doublelogarithmic order not exceeding μ

4 552 M Açıkgöz and P Tien-Yu Chern Proof Since S r is positive and decreasing, we see that 25 + > > Sr r log r S r S r μ+ d r r r μ+ d t μ+ r t S r μ r holds for r> which implies that S r is of double-logarithmic order not e e exceeding μ Proof of Theorem We prove this theorem in two steps Step I Let S r be of finite double-logarithmic order λ For μ λ + ε > λ, there exists a positive number r ε such that S r < r for r>r ε > It follows that e e 26 e e r ε e e + r ε < O + S r r log r μ+ r S r r log r μ+ r S r r log r μ+ r r ε O + lim r r log r r rɛ x ε 2 r + dx < + ε + 2 μ λ+ ε 2 On the other hand, for μ<λ,the integral 24 must be divergent Indeed, if 24 is convergent, it follows from Lemma that S r is of double-logarithmic order μ < λthis contradicts the fact that S r has double-logarithmic order λ Therefore the integral 24 is convergent for μ>λand divergent for μ<λ

5 Functions with double-logarithmic order in a disk 553 Step II We assume that the integral 24 is convergent for μ>λand divergent for μ<λby lemma, the double-logarithmic order of S r λ+ɛ, for any positive number ɛ, hence the double-logarithmic order of S r λ Suppose S r has double-logarithmic order λ 2ɛ for some positive number ɛ By Step I, we see that the integral 24 is convergent for μ λ 2ɛ+ɛ λ ɛ This contradicts the assumption of Step II Hence the double-logarithmic order of S r λ Results in Step I and Step II complete the proof of Theorem 3 The double -logarithmic order for analytic functions in a disk Definition A function f z analytic in the open unit disk D is of the same double-logarithmic order, and of the same double-logarithmic type as log M r, f Theorem 2 Let f z be analytic in D and with unbounded μ r, f then log μ r, f is of double-logarithmic order λ 0 <λ<+, if and only if the integral log μ r, f 3 r log r μ+ r is convergent for μ>λ,and divergent for μ<λ Proof of Theorem 2 Theorem 2 is an immediate consequence of Theorem, since log μ r, f is an unbounded increasing function in r 0, Definition 2 A non negative continuous function λ r :0, 0, + is called a proximate double-logarithmic order of S r, if λ r satisfies the following properties: i lim r λ r λ ii λ r exists everywhere in 0, except possibly in a countably set where the right derivative λ r + and the left derivative λ r both exist Furthermore, if we use λ r + orλ r insteasd of λ r in the exceptional set, then have lim r log r r r log 32 r λ r 0 iii Let U r r λr, we have S r U r for sufficiently near and 33 S r lim sup r U r

6 554 M Açıkgöz and P Tien-Yu Chern Above U r associated with S r is called a double-logarithmic type function of S r Theorem 3 The existence Theorem of proximate double-logarithmic order If S r : 0, 0, + is an unbounded increasing function and of finite positive double-logarithmic order λ, then there exists a proximate double-logarithmic order of S r 4 Two growth indexes of the central indexes of a power series expansion in the open unit disk Let f z a n z n be a power series expansion analytic in the open unit n0 disk D and log μ r, f be of finite double-logarithmic order We introduce two growth indexes of ν r, f as follows: Let r j } + j be the sequence of jump points of ν r, f with 0 r 0 <r < < r j <r j+ < < } 4 ρ f inf μ μ>0, g μ r j < +, 42 where g μ t λ f inf μ μ>0, t t 0 with t 0 r 2 j μ ν t, f dt < + t, t 0 μ ds + g μ 0 s t 0 μ ds 0, as r ; s Theorem 4 If f z is a function analytic in D and if log μ r, f is of finite positive double-logarithmic order in D, then we have 43 ρ f λ f the expression 43 holds including cases where either λ f 0 or ρ f 0 Proof of Theorem 4 For μ>0, we have the identity 44 J j 0 g μ r j R t 0 g μ t dν t, f ν t, f g μ t R t 0 + R t 0 ν t, f μ dt t

7 Functions with double-logarithmic order in a disk 555 Let ɛ>0, put μ λ f+ɛ, by the meaning of λ f, we have μ 45 ν t, f dt < + t t 0 For each δ>0, there is a R 0 <, such that if r R 0,, then μ δ > ν t, f dt t 46 r ν r, f r μ dt ν r, f g μ r t Combining 44, 45 and 46, we deduce the series + g μ r j < +, since ɛ > 0 is arbitrary, so λ f ρ f On the other hand, if the series + 47 g μ r j < + j Then by 47 and the identity 44, it follows J + > g μ r j +ν t 0,f g μ t 0 j ν t, f 48 ν R, f g μ R+ μ dt t t 0 Since the two terms of the right hand side of 48 are non-negative, we have μ 49 ν t, f dt < +, t t 0 so ρ λ f This completes the proof of Theorem 4 5 Basic properties between the growth of the maximum terms and the central indexes We first need to prove a Lemma which gives relations among the series of g μ r j and of r j and among the integrals of the logarithm of the maximum term function log μ r, f and the central index function ν r, f, and it shall play a very useful role for further study Since it leads to a useful information on the magnitude of ν r, f R j

8 556 M Açıkgöz and P Tien-Yu Chern Lemma 2 If f z is a power series expansion which is analytic function in D and if log + μ r, f is of finite logarithmic order, then for every μ>0, the series + 5 g μ r j, and 52 and integrals 53 and + j j r j μ, r j μ ν t, f dt, t log μ t, f 54 t log t μ+ dt t are either simultaneously convergent or simultaneously divergent, where r j } + j is the set of jump points of the central index ν r, f of the power series expansion f Proof of Lemma 2 From the identity 44 and the inequality 46 we see that the series 5 and the integral 53 are either simultaneously divergent Next, by the Stieltjes integral and integration by parts, we have the following identity: J 55 r j a> e e R e e r j μ r j r μ dν r, f r r ν r, f μ R e r e + R e e ν r, f μ + o r } μ r By using above identity 55, we deduce that the series 52 and the integral 53 are either simultaneously convergent or simultaneously divergent Finally,

9 from the identity 56 R r 0 Functions with double-logarithmic order in a disk 557 ν t, f t μ dt t μ log μ R, f log μ r 0,f R r 0 R log μ t, f +μ μ+ dt, r 0 t log t t μ and repeating a similar procedure as in the proof of Theorem 4, we deduce that the integrals 53 and 54 are either simultaneously convergent or simultaneously divergent This completes the proof of Lemma 2 By the meaning of ρ f, then applying Theorem 4 and above Lemma 2 we have the following: Theorem 5 If f is a power series expansion which is analytic and log μ r, f is of finite positive double-logarithmic order λ in D, then the series 5 and 52 and the integrals 53 and 54 are all convergent for every μ>λ,and all divergent for every μ<λ Remark 2 Results in Theorem 5 are the best possible in the sense that there exists an example such that the series 5 and 52 and the integrals 53 and 54 are all convergent for μ λ, there also exists another example such that the series 5 and 52 and the integrals 53 and 54 are all divergent for μ λ see examples in section 5 of this article Theorem 6 If f is a power series expansion which is analytic and log μ r, f is of finite positive double-logarithmic order in D, then log μ r, f has doublelogarithmic order λ f Proof of Theorem 6 By above Theorem 5 the integral 54 is convergent for μ>λ f, divergent for μ<λ f Then since log μ r, f is increasing, applying Theorem, we deduce that log μ r, f has doublelogarithmic order λ f Theorem 7 If f is a function a power series expansion which is analytic in D and log μ r, f is of finite positive double-logarithmic order λ in D, then ν r, f g μ r o, for μ>λ as r, and for any ɛ>0, we have ν r, f r log r lim inf r λ +ɛ 0 Proof of Theorem 7 We first prove 57 It follows from Theorem 42, λ λ f, so we have 35 then using 36 we deduce 57 r

10 558 M Açıkgöz and P Tien-Yu Chern We next prove 58 Suppose there exists a positive number K, such that ν r, f r log r 57 lim inf r λ +ɛ K>0, r then by taking μ, such that λ<μ<λ+ ɛ, 58 R r ɛ ν t, f K ɛ K ɛ R r ɛ μ dt t t log t log μ ɛ dt t r rɛ s μ ɛ ds +, as R Since μ>λ f, by Theorem 5 and Theorem the integral 54 is convergent, then applying Lemma 2 the integral 43 is convergent, this contradicts to 56 Thus we complete the proof of theorem 7 6 Constructive Examples of functions in a disk with finite double logarithmic order Theorem 8 For any positive number λ, there exists a power series expansion function f which is analytic in D such that log + μ r, f is of doublelogarithmic order λ and such that the series 5, 52, and the integrals 53 and 54 are all convergent for μ λ There also exists a function f analytic in D such that log + μ r, f is of double-logarithmic order λ but the series 5, 52 and the integrals 53 and 54 are all divergent for μ λ Proof of Theorem 8 Let λ be a positive number Given a real number k, let r j be the solution of the equation λ k 6 r log log j r r r where j, 2, Let a n r r n ; n, 2, 3, Put 62 f k z n r r n z n

11 Functions with double-logarithmic order in a disk 559 It follows from [] page 6 that f z is a function analytic in D and has the central index [ λ ] k 63 ν r, f k r log log, r r r where [x] denotes the Gauss integer of x, obviously f has the set of jump points r j } + j, further 64 ν r, f k j for r [r j,r j+ and hence ν r, f k log r μ μ λ+ r r log r r 65 lim R R log r r k μ λ+ d r The integral 65 is convergent for μ>λand divergent for μ < λ By Lemma 2, the integral in 54 is convergent for μ > λ and divergent for μ<λsince log μ r, f k is apositive increasing function in r, it follows from the integral criterion Theorem that, log μ r, f k has double-logarithmic order λfor μ λ, it follows from 65 that 66 ν r, f k μ lim R r R x k dt 66 is convergent for k< and divergent for k> Let f f 2, it follows from 66, the integral 53 is convergent for μ λ and hence by Lemma 2, the series 5, 52 and the integrals 53 and 54 are all convergent for μ λ Let g f 0, it follows from 66, the integral 66 is divergent for μ λ It follows from Lemma 4 that the series 5, 52 and the integrals 53 and 54 are all divergent for μ λ and for the function gthis completes the proof of Theorem 8 7 Basic asymptotic behavior between the central indexes and the logarithmic maximum term of an analytic expansion in the open unit disk Theorem 9 Let f z + a n z n be a power series expansion which is n0 analytic in D If log μ r, f is of finite positive double-logarithmic order λ,

12 560 M Açıkgöz and P Tien-Yu Chern then 7 and ν r, f r log r r lim sup r log μ r, f λ, 72 ν r, f r log log r r r lim sup λ r log μ r, f μ r, f To prove Theorem 9 we need the following Lemma as a preliminary Lemma 3 L Hopital s Rule Let a r,br be positive continuous functions defined for r 0, Assume that a r and b r both exist and are positive continuous except a set E of isolated points in 0, If lim r b r +, then a r a 73 r lim sup lim sup r b r r b r Proof of the Lemma 3 Let a 74 r lim sup r b r μ<+ For any ɛ>0, there is a r ɛ, 0 <r ɛ <, such that for any x r ɛ, E, we have 75 a x μ + ɛ b x For any r, r ɛ r<, the inequality r 76 a t dt μ + ɛ b t dt r ɛ r ɛ holds and for any r r ɛ, E, we have 77 a r a r ɛ μ + ɛb r b r ɛ Since lim r b r +, this leads to ar br 78 Since ar br 79 lim sup r,r 0, E a r b r lim r μ + ɛ for r sufficiently near, so sup r,r 0, E a r b r is continuous for r 0,, and E is a set of isolated points, so a r lim sup r b r lim sup r, r 0, E a r b r The inequality 79 implies the inequality 73 This completes the proof of the Lemma 3 The proof of Theorem 9 will be given in section 9

13 Functions with double-logarithmic order in a disk 56 8 The proof of Theorem 3 We prove this theorem in three cases Case I If the inequality S r > r λ holds for a sequence of values of r tending to, then the function χ :, 0, + e ee log S x 8 χ r : max x [r, log x is continuous nonnegative and χ r λ as r Let } log Sr M r χ r then sup M, and for each r M, S r > log r λ and hence χ r >λfor r M Pick an r r M, with r > Let t min t t is a positive integer, t >r, with χ t <χr } We now consider two curves y x χ r log 5 x + log 5 t, x t, y 2 x χ x, x t, where log 5 x : log x Since y t χ r >χ t y 2 t,y x, and y 2 x λ, curves y and y 2 must intersect in the one dimensional interval t, Let a be the abscissa of the first point in the intersection of curves y and y 2 Put r 2 min M [a,, and let t 2 min t t is a positive integer, >r t 2, with χ <χr2 } t Then we consider two curves y 2 x χ r 2 log 5 x + log 5 t 2, x, t 2 y 22 x χ x, x t 2 Since y 2 t 2 χ r 2 >χ t2 y 22 t 2, y 2 x, and y 22 x λ, curves y 2 and y 22 must intersect t 2, Let a 2 be the abscissa of the first point in the intersection of curves y 2 and y 22 Inductive Hypothesis: Suppose the numbers r i,t i,a i, and curves y i and y i2 have been already defined We put r i+ min M [a i,, and let t i+ min t t is a positive integer, t >r i+, with χ } <χr i+ t e ee

14 562 M Açıkgöz and P Tien-Yu Chern Then we consider two curves y i+ x χ r i+ log 5 x + log 5 t i+, x, t i+ y i+ 2 x χ x, x t i+ Since y i+ t i+ χ r i+ >χ t i+ y i+ 2 t i+,y i+ x, y i+ 2 x λ Curves y i+ and y i+ 2 must intersect in t i+, Let a i+ be the abscissa of the first point in the intersection of curves y i+ and y i+ 2 Repeating the above procedure; by Mathematics induction, we obtain three sequences r j }, t j } and a j } with r j < t j <a j <r j+, since t j, so lim j + r j Put d j t j Noticing that χ r :χ r i+ for r [a i,r i+ ], this makes that we can define a function λ :[0, ] [λ, χ r ] 82 λ x : χ r x [0,d ] χ r log 5 x + log 5 t x [d,a ] χ r 2 x a,d 2 ] χ r 2 log 5 x + log 5 t 2 x [d 2,a 2 ] χ r i log 5 x + log 5 t i x [d i,a i ] χ r i+ x a i,d i+ ] such that λ r is nonnegative piece wise continuous in [0,, its derivative function λ r has the following properties: λ r exists except the exceptional set E a i } i d i} i, 83 λ r 0 for r 0,d i a i,d i+, 84 where λ r ξ r for r i d i,a i, ξ r log r r log r r r

15 Functions with double-logarithmic order in a disk 563 At numbers in the exceptional set E, we have λ a i ξ ai, λ 85 a + i 0, and λ d + i ξ di, λ 86 d i 0 If we combine the result of 83, 84, 85, and 86 and if we use λ r +, λ r replacing λ r for numbers in the exceptional set E, then λ r satisfies the expression 32 Based on the selection of the sequence r i } and through a careful checking, we deduce that for r r,, 87 λ r χ r Hence S r 88 λ r log log for r r Since r j a j, tj M, so 89 and implies 8 for r sufficiently near 82 λ r j χ r j log lim j + λ r jλ U r S r, 89 implies U r j Sr j From 8 and 82 we can deduce S r j log r j,,, r 83 S r lim sup r U r Case II The inequality S r r λ holds for r near and if λ there is a sequence r j } with r j, Sr j r j, we may adapt λ r λ, obviously λ r is a proximate logarithmic order of S r Case III The inequality S r < λ for r>ρ r 0 > eee, then the function log + S x 84 Ψr : max x [ρ 0,r] log x

16 564 M Açıkgöz and P Tien-Yu Chern is continuous with Ψ r λ as r Let L r Ψr } log+ Sr, log r then sup L There exists a ρ sufficiently near to with ρ 0 <ρ < such that if b is the maximum abscissa of points in the intersection of the following curves: Y x : λ + log 5 x Y 2 x : Ψx, log 5 ρ then we have ρ 0 <b <ρ and L [ρ 0,b ] being nonempty Putting τ max L [ρ 0,b ], we deduce that ρ 0 τ <b <ρ Furthermore, there exists ρ 2 > +ρ 2, such that if b2 is the maximum abscissa of points in the intersection of the following curves: Y 2 x : λ + log 5 log x 5 ρ 2 Y 22 x : Ψx, then we have ρ <b 2 <ρ 2 and L [ρ,b 2 ] being nonempty Putting τ 2 max L [ρ,b 2 ] Pick up a c [b,ρ ] with λ + log 5 log c 5 Ψτ 2 ρ Inductive Hypothesis: Suppose the numbers ρ i,b i,τ i and c i have been defined There exists ρ i+ > +ρ i 2 such that if bi+ is the maximum abscissa of points in the intersection of the curves: Y i+ x : λ + log 5 log x 5 ρ i+ Y i+ 2 x : Ψx, then we have ρ i <b i+ <ρ i+ and L [ρ i,b i+ ] being nonempty Putting τ i+ max L [ρ i,b i+ ] Pick up a c i [b i,p i ] with λ + log 5 log c 5 Ψτ i+ i ρ i Repeating the above procedures, by Mathematics induction, we obtain four sequences ρ i }, τ i }, b i } and c i } with the inequalities e ee <ρ 0 b c τ 2 b 2 <

17 Functions with double-logarithmic order in a disk 565 Noticing that Ψ r Ψτ i for r [τ i,b i ], we can therefore define a function λ :[0, ] [Ψ τ,λ] as follows: λ r : Ψτ r [0,b ] λ + log 5 log r 5 r [b,c ] ρ Ψτ 2 r c,b 2 ] λ + log 5 log r 5 r [b i,c i ] ρ i Ψτ i+ r c i,b i+ ] 85 Then λ r is continuous and λ r λ as r λ r the derivative of λ r has the following properties: λ r exists in 0, except in exceptional set b i,c i } + i and 86 λ r 0 r c i,b i+ 87 λ r ξ r r b i+,c i+ for r b i,c i } we have λ b i 0, λ 88 b + i ξ bi and λ c i ξ ci, λ 89 c + i 0 Combining above results of λ r and replacing λ r + orλ r for r b i,c i } + i, then λ r satisfies the expression 33 Through a careful checking, we deduce that for r r and 820 lead to λ r Ψr S r 82 λ r log log r for r ρ Since τ j c j,b j L, so λ τ j Ψτ j log S r j 822 log r j and hence 823 lim λ r j lim Ψr jλ j + j + This completes the proof of Theorem 3

18 566 M Açıkgöz and P Tien-Yu Chern 9 The proof of Theorem 9 Proof of Theorem 9 Since log μ r, f is of finite positive doublelogarithmic order λ, by Theorem 3 there exists a proximate double-logarithmic order λ r of log μ r, f We succeed the same notations and symbols in the proof of Theorem 3 of this article Put A 0,d + i a i,d i+, B + i d i,a i, C a i } + i, D d i} + i, E C D; G 0,b + i c i,b i+, H + i b i,c i, I c i } + i, J b i } + i, where a i,b i,c i and d i are given in the proof of Theorem 3 In this article, let U r λr r exp λ r log r For each r>0, U r exists, we have } 9 U r U r λ r log r + λ r r log In the following we will estimate d log μr,f U r Case I If log μ r, f > r λ holds for a sequence of values of r tending to, then λ r exists everywhere of 0, E Case I-i r A, it follows from 82 that 92 d log μr,f U r Case I-ii r B, by 82 λ r ξ r we have d log μr,f U r r ν r, f r log r r ru r λ r r log r r ru r λr r log r log r ν r, f r λr r r r + } r ν r, f log r r 93 ru r λ r Case I-iii For r C, r a i for some i, it follows from 82 that λ r ξr, λ r + 0, so we have 94 d log μr +,f U r r ν r, f log r r ru r λ r

19 and 95 d log μr,f U r Functions with double-logarithmic order in a disk 567 ru r λr r log r ν r, f r r ν r, f log r r ru r λ r λr r + } Case I-iv For r D, r d i for some i, it follows from 82 that λ r + ξr, λ r 0, so we have 96 and 97 d log μr+,f U r d log μr,f U r ru r λr r log r ν r, f r r ν r, f log r r, ru r λ r λr r ν r, f log r r ru r λ r r + } Case II log μ r, f r λ for near and there exists a sequence r j with r j, such that log μ r j,f log λ r in this case λ r λ, λ r 0 obviously we have 98 d log μr,f U r r ν r, f log r r ru r λ r Case III log μ r, f r λ for r near Case III-i For r G, it follows from 85 that λ r 0, we have 99 d log μr,f U r r ν r, f log r r ru r λ r Case III-ii For r H, it follows from 85 that 90 d log μr+,f U r ru r λr r log r ν r, f r r ν r, f log r r ru r λ r λr r + } Case III-iii For r I, r c i for some i, by 86 λ r + 0, by 87 λ r ξ r, so we have

20 568 M Açıkgöz and P Tien-Yu Chern 9 and 92 d log μr,f U r d log μr +,f U r ru r r ν r, f log r r ru r λ r λr r log r ν r, f r λr r + } r ν r, f log r r ru r λ r Case III-iv For r J, r b i for some i, by 85 λ r λ b i 0, λ r + λ b + i ξ bi, so we have 93 and 94 d log μr+,f U r d log μr,f U r ru r λr r log r ν r, f r r ν r, f log r r ru r λ r λr r ν r, f log r r ru r λ r r + } Noticing that 0, + A B C D G H I J, combining above d log μr,f results on, taking lim sup U r r for them, and applying the L Hopital s Rule see Lemma 6 of this article by putting a r log μ r, f, and b r U r, we obtain log μ r, f r ν r, f log r r 95 lim sup lim sup r U r r ru r λ r Since λ r λ, above inequality implies the inequality 7 We next prove the inequality of 72 r ν r, f log r lim sup r log μ r, f r ν r, f log r lim sup r log μ r, f r ν r, f log r lim sup r log μ r, f r r r log r μ r, f log r lim inf r μ r, f lim sup r μr,f log r 96 λ λ

21 Functions with double-logarithmic order in a disk 569 This completes the proof of Theorem 9 References [] G Polya and G Szego, Problems and Theorems in Analysis II, Springer-Verlag Translation by D Apeppli 972 [2] G Valiron, Lectures on the general theory of integral functions, Toulouse, 923 Received: November 25, 2007