UNBOUNDED FUNCTIONS IN THE UNIT DISC

A survey is made of results related to the value distribution of functions which are meromorphlc or analytic in the unit disc and have unbounded growth according to some specific growth indicator. KEF WORDS AND PHRASES. Value dtribution, unbounded funct2n, evanlinna theory. 1980 THEMATICS SUBJECT CLASSIFICATION CODE. 30?35 i. INTRODUCTION We shall consider functions which are meromorphic or analytic in the unit disc D {z zl < I} and have unbounded growth according to some specific growth indica- tor. We give a survey of known results related to value distribution. Initially we shall consider functions whose growth in the disc makes them correspond to meromorphlc functions of finite order in the plane. Then we shall proceed to functions with slo- wer growth. We assume the notation of the elementary theory of Nevanlinna theory as it appears in Chapter One of Hayman's book (i). 2. SECTION TWO First we consider functions which have their growth determined by the Nevanllnna characteristic function. Let f be a meromorphlc function in D. We define the order u of f by


i. INTRODUCTION
We shall consider functions which are meromorphic or analytic in the unit disc D {z zl < I} and have unbounded growth according to some specific growth indica- tor.We give a survey of known results related to value distribution.Initially we shall consider functions whose growth in the disc makes them correspond to meromorphlc functions of finite order in the plane.Then we shall proceed to functions with slo- wer growth.We assume the notation of the elementary theory of Nevanlinna theory as it appears in Chapter One of Hayman's book [i].

SECTION TWO
First we consider functions which have their growth determined by the Nevanllnna characteristic function.
Let f be a meromorphlc function in D. We define the order u of f by l...oE T(r,f_) llm sup r-l -lg(Ilr) (2.1) where T(r,f) is the Nevanllnna characteristic of f at r.We note that 0-<u < +.
Further we define the lower order of f by + rl -log(l-r) (2.2)  and observe that 0 < E -< u < +.
In this case we have the following relationship between the order of a function and the order of its derivative function.
THEOREM 2.1, Let f be a meromorphic function in D with order defined by (2.1).Let u' be the order of f' defined by (2.1), and let and ' be the lower orders of f and f' respectively defined by (2,2), Thus u u (TsuJl [2] and [3,p.228]) and ' < (Sons [4]).
If f is a meromorphic function in D for which defined by (2.1) is posi- tlve, we ask if f must assume every value on the Riemann sphere and how frequently it may assume such values, An omitted value is called a Picard THEOREM 2. 2. Let f be a meromorphic function in D such that T(r ,f) r/l Let g be a meromorphic function in D with (r,s) 0 (), (r / ).
Then f(z) g(z) has infinitely many zero points in D with two possible exceptions for g.More preclsely, lira sup .N(r i/(.f-g)) r/l -log(l-r) with two possible exceptions for g (Tsuji [5] and [3,p,297]).
Exceptions can indeed occur, For k > i we define the functions fk in D by fk(z) exp((l.-z)-k).
We note that fk has order k-i and fk omits the values zero and infinity.
A closer relationship between the growth of the function and the values assumed is given below.
T!.EORF 2.3.Let f be a meromorphic function in D with finite order > 0 defined by (2.1).Let {a k} be a set of a-values for f where a is in E u {(R)}.for each e > 0 with two possible exceptions for a; and 1 f) lr)l (ili) when f T(r, dr = , we have Z(l]ak]) u+l with two possible exceptions for a (TsuJi [3, p,204 and 293] and [5]).
It can be further shown that if f is a meromorphic function in D, then the convergence of ;i T(r,f) (l-r) 7I dr, ( for some 7 > 0, implies the finiteness of the three quantities N(r,a)(l-r) Y-I dr, n(r,a)(l-r) Y dr, E(l-lakl) 7+I (2.4) where a is in u {=} and {a k} is the set of a-values for f.In fact, it can be shown that the three quantities in (2.4) converge or diverge simultaneously regardless of the known behavior of (2.3) (see Nevanlinna [6, Ch.X]).
If the {a k} are the non-zero zero points of f, it is convenient to define the convergence exponent > 0 of the {a k} as follows.
k If E(l-lakl) , then is that number such that E(l-lakl)+le..
k k and Z(l-lakl)+l+e.. < k for any e > 0. From THEOREM 2.3 we see 0 <] < if a is the order of f.We assume u is finite.
Let p >i be a positive integer such that [(l-lakl) p and (l-lakl) p+I We observe that P is analytic in D and P(a k) 0 for k i, 2, 3, Further, when P is defined by (2.5) we see IPCz) < i, (z D).
We then have two useful theorems.THEOREM 2.4.Let f be a meromorphic function in D which has finite order Let {a k} be the non-zero zero points of f, and let P be the canonical product formed with {ak}.Let s 0 be the order of P. Then where .V is the convergence exponent of the {a k} (Tsuji [3, p.227]), THEOREM 2.5, Let f be a meromorphlc function with finite order u in D.
Then f(z) can be expressed in the form f(z) (fl(z))/(f2(z))' (z D), where fl and f2 are analytic functions in D with order < e (Tsuji [3, p.227]) Using an extended notion of the characteristic function M. M. Dzhrbashyan [7] has introduced another factorization for functions meromorphic in D. M. A. Girnyk has studied relationships between the growth of products of the form (2.6) and their value distribution.In [8] he has determined the asymptotic behavior of the logarithm of the modulus of products of the form (2.6) assuming the zeros are all positive and have an asymptotic behavior which is essentially n(r,i/f) c(1-r)for 0 < C < , 0 > 0. In [9] Girnyk discusses the asymptotic behavior of the Nevan- linna characteristic and the logarithm of the modulus of products of the form (2.6) in the case when the zeros possess an angular density.He gives an example to show that even in the case of nonintegral order, the angular density of the zeros does not uni- quely determine the asymptotics (so his theorems require some additional restrictions).
The parts (i) and (ii) of THEOREM 2,3 and the relationship among the quantities in (2.4) lead us to define Borel exceptional value as follows.
Let f be a meromorphic function in D and define for each a in by IoE N(r ,a) By the First Fundamental Theorem of Nevanlinna theory we have N(r,a) < T(r,f) + 0(i), < =.
is defined similarly.If e < +, we say a is a (r / i), and so 0 < a Define N(r,a) in a similar manner to N(r,a) where we consider only the dis- tinct a-values of f.Let log N(r,a) lira sup ,,log(iH5 r+l a THEOREM 2.7.Let f be a meromorphic function in D such that f has order < Then for every a in K u {} with at most two exceptions, a (c.f.Sons [I0]).
In the same sphere of ideas as THEOREMS 2.6 and 2.7 results may be obtained (c.f.Sons [i0]) concerning the counting function for simple zeros, and also one may be ob- tained involving the counting function for distinct simple and double zeros.
-3/2 Another type of measure of the values a function assumes is the Nevanlinna defici- ency which is defined using a ratio of the counting function N to the Nevanlinna characteristic T. A transitional result in this direction is below, THEOREM 2.8.Let {a k} be a sequence of non-zero complex numbers with moduli less than one such that l(z-I%1)= , k but the convergence exponent of the {a k} satisfies < +, Let P be a canoni- L.R. SONS cal product formed with {a k}, and let be the order of P. Then (1) 0 ; (ll) when p # , there is a positive constant c such that lim sup > c r+l T(r,P) where c depends only on s and p (Sons [II]), If f is a meromorphic function in D and a is in u {}., then the First Fundamental Theorem of Nevanlinna theory says N(r,a) + m(r,a) T(r,f) + 0(I), (r I).
This theorem relating the behavior of re(r ,a) to T(r,f) also can give information about value distribution.
THEOREM 2.9.Let f be a meromorphic function in D such that lira T(r,f) +. r+l Then for almost all a, m(r,a) < 1/2 llm sup og T(r-fY The above theorem was proved by J.E. Littlewood [12].This theorem and some re- fated results of L. Ahlfors [13] are stated in W.K. Hayman [14] where examples are given to show their best possible character. If f is a meromorphlc function in D and a is in u {} we define:  exp{(1 z) }, then (O,f) (=,f) I, so THEOREM 2.10 has two as a best possible bound.
It is natural to ask whether functions exist which have an "arbitrary" assignment of deficiencies at an arbitrary sequence of complex numbers subject only to the condi- tions of THEOREM 2.10.We state two results in this direction, THEORE 2.11.Let be a non-negative real number, and let {k be a se- quence of positive numbers less than or equal to one satisfying Let {a k} be a sequence of distinct complex numbers.Then there is a meromorphic function f in D with order such that (ak,f) > (k)/4, k i, 2, 3 (Krutin [15]).
THEOREM 2.12.Let {k be a sequence of positive numbers such that [ kl.
k=l Let {a k} be a sequence of distinct complex numbers.Then there is a function f which is analytic in D such that (ak'f) k and (b,f) 0 for b E and b {aklki, 2, ...} (Girnyk [16]).
There are some theorems which relate how a number of specific values are assumed when the function grows rapidly enough.
THEOREM 2. 13.Let f be a meromorphic function in D. Let p(r) be the total number of roots in {z Izl < r} of the equations f(z) ak, k I, 2, ..., q where the a are q distinct elements of u {} and q > 3.
If k (l-r) logM(r f) lim inf -----Jl0g (ir > 7 (2,7)  where y is some finite positive number, then lira sup (l-r)p(r) > b r+l where b y(q.-l) or y(q-2) according as the a k are all finite or not.If f is an analytic function in D and (2,7) is replaced by (l-r) logM(r, f) lira sup -log(i-r) r+l the same conclusion is obtained, and the theorem is best possible (}layman [17]).
Relationships can be formulated between the deficiencies of a function and the deficiencies of the derivatives of the function at zero.THEOREM 2.14.Let f be a meromorphic function in D for which > 0 where is defined by (2.2).Then I Ca.,f) < (0,fCn) n+l j a.
If f is an analytic function in D for which > 0, then (a ,f) -< (0,f(n)), for n > I.
Analogous results for functions meromorphic in the plane or for entire functions appear in H. Wittich [18].
If the functions grow rapidly enough, some conclusions can also be made about the value distribution of the derivative functions or combinations of the function and its derivative functions.The next two theorems give important growth interrelations used to obtain such results.(i 0, i, u) is a meromorphic function in D such that i T(r,a i) o(T(r,f)), (r + I).Then m(r,/f) o(T(r,f)), (r i), and TCr,) < (+l)TCr,f) + oCT(r,f)), (r + i), THEOREM 2. 16.Let f be a meromorphic function in D, and assume (2.8) holds.
Let be as in THEORE} 2.13, and assume # is not constant, Then ,) where in N (r,i/') only zeros of not corresponding to the repeated roots of o (z) 1 are to be considered.
We proceed now to theorems about the values derivative functions assume THEOREM 2.17.Let f be a meromorphic function in D such that (2.8) holds.

Suppose
and Let be defined as in THEOREM 2. 15.Then is identically constant or assumes every finite complex value except possibly zero infinitely often.In fact, in the latter case, has no finite deficient values except possibly zero.

Let be a positive integer Then
In particular, f () I )(a, f()) < i + l-!- a# +i assumes every finite value with at most one exception infinitely often.
THEOREM 2. 19.Let f be a meromorphlc function in D such that (2.8) holds.
If is a positive integer, then T(r f) Consequently, either f assumes every finite value infinitely often or f () assumes every finite value except possibly zero infinitely often.
Meromorphic functions with order e < + in D have been considered as coeffi- cients of certain differential equations in D. G. Valiron [20], S, Bank [21,22,23], and A.A. Gol'dberg [24,25] have studied restrictions on the growth of the analytic or meromorphic functions in D which are solutions of some differential equations for which the coefficients are analytic or meromorphic functions of finite order in D. From the definitions of Nevanllnna and Vallron deficiency it is clear that each Nevanllnna deficient value is also a Vallron deficient value.How are Borel excep- tlonal values related to deficient values?THEOREM 2.21.Let f be a meromorphlc function in D which has finite order and lower order (1) If a is a Borel exceptional value for f, then ( (li) If a is a Borel exceptional value for f and , then (a,f) i.
Another type of function behavior of special interest is the presence of radial limits, or asymptotic values in general.A function such as f defined in D by /Z+z has radial limits 1 and +(R) as Izl / 1 along the negative axis and positive axis respectively.However, we have the following theorem.and Collingwood [28,29] ).
The exceptional values in the theorem can occur even in such a way that the two values cluster only to one boundary point (Barth and Schneider [30]).
K. Barth  [31] introduced the class A of meromorphic functions in D which have m asymptotic values at a dense set on {zllz i}.If f is a meromorphlc function in D with order u < 2, then the following theorem gives a sufficient condition for f to be in A m If THEOREM 2. 24.Let f be a meromorphlc function in D which is not constant.
and for some a in E v {(R)} 1 (1-r)T(r,f)dr < 0 N(r,a) 0(i), (t-+l)., then f is in A (Barth [31]). m There are examples (Barth [31]) to show condition (2.9) cannot be relaxed to (a,f) Let f be a non-constant meromorphic function in D. An .a.symptot.i_ctrac____t {D(e),al associated with the finite complex number a is a set of non-vold domains D(e), one for each e > O, such that  K is defined to be the en_d of the tract.If K is a point, then the tract will be termed a polnt-tract; if K is an arc, then the tract will be called an arc-tract.Barth  Another interesting question concerning A is whether the derivative of a func- tion in the class is again in the class.We now consider only analytic functions in D with growth measured by the maxi- mum modulus of the function on circles centered at the origin, For such functions there are some types of theorems which are not available for meromorphic functions.
Further, sometimes one can make added assertions for analytic functions, We define the M-order of such functions below, Let f be an analytic function in the unit disk D. Let O lira sup -log(l-r) where M(r,f) is the maxlmummodulus of f on {zllz -r}, We say p is the M-order of f and observe that 0 -< 0 < +.We define the lower M-order A of f by According to M-order it can be shown that a function and its derivative function behave similarly.We have THEOREM 3.1.Let f be an analytic function in D such that f has M-order and lower M-order .If the derivative function f' has M-order ' and lower M- order X' then p p' and X X' (c f Boas [33, p 13] and Sons [4]) For an analytic function in D it is possible to relate the growth of the func tion to the growth of the coefficients in its power series expansion about zero, THEOREM 3.2.Let f be an analytic function in D with finite M-order 0. If fCz) Kapoor [35] and Kapoor and JuneJa [36]).
A theorem of a slightly different nature comes by considering the maximum term and central index in the power series expansion.-log(l-r) (Sons [37]).
The function defined in D by ekjz (k')3 2 where k is 2 and kj+1 (kj) shows equality need not hold fn the last statement o A well known inequality relates the Nevanllnna characteristic of a function to the maximum modulus.If f is an analytic function in D, then T(r,f) < log+M(r,f) R + r T(R,f), (0<r<R<l) (3 4) Rr THEOREM 3.5 is a direct consequence of this inequality.THEOREM 3.5.Let f be an analytic function in D with finite M-order p.If a is the order of f in D as defined by (2.1), then a < p < a + i.
Using the notation of THEOREM 3.5 we observe that the functions fk defined in D for k > i by fk(z) exp((l-z) -k) have M-order p k and order a k-1.To see that other posslbilltles allowed by the theorem can be assumed, we give two theorems.Then there exists a function f analytic in D such that T(r,f) S(r,i/f) log M(r,f) @(r), (r-l), (Shea[38]).
THEOREM 3. 7. Let and be a non-negatlve, increasing functions which are defined for r in 0 r < i, which are convex with respect to log r in 0 < r < I, and lira "log(i--r')' +' lim -ig(l-r +' (t)dt o((r)), (r+l), 0 and $' (r)(l-r) is monotonically increasing for r in the interval (0,1).Then there is a function f analytic in D such that (Linden [39]).
log M(r,f) ,(r) and T(r,f) (r), (r+l), Turning to a consideration of values assumed by our functions, we see: THEOREM 3.8.If f is an analytic function in D with M-order 0 > i, then THEOREM 2.2 applies and f omits at most one value.If f is an analytic function in D with M-order p I, then f may omit infinitely many values.
1 (l-r) in inequality The first part of THEOREM 3.8  To see the second part we note that g defined in D by has M-order p 1 and Ig(z) > 1 for z in D. Translating THEOREM 2.6 fo= an analytic function f in D shows that such an f can have at most one Borel exceptional value.In fact, f defined by f(z) exp ((l-z) -3/2) has zero as a Borel exceptional value and M-order O 3/2.This func- tion also shows that the M-order of a function analytic in D with a Borel exception- al value need not be an integer.Indeed it can be shown that analytic functions in D with a Borel exceptional value need not have regular growth according to either order or M-order.THEOREM 3.9.There exist analytic functions f in D which have order lower order , M-order 0, and lower M-order such that f(z) exp(s(z)), for z in D where g is an analytic function in D, and both (Sons [lO]).
By using a canonical product which is essentially that of M. TsuJl in SECTION TWO we obtain a factorlzatlon of functions analytic in D when the M-order is large enoush.is analytic in D and of M-order at most O (Linden [40]).where p is of the form (3.5) and formed using the zeros of f, and both p and q are analytic and of M-order at most p in D, (Linden [40]), Making additional hypotheses concerning the zeros of f, we obtain a theorem concerning functions of smaller }.order growth, THEOREM 3.12, Let f be an analytic function in D which has finite H-order O. Let {a be the zeros of f in D, and define n(r,8,f) to be the number of n these zeros satisfying r< la n (l+r) and 18arg a -< w(lr)  p(z)q(z), (zED), where p(z) is of the form (3.5) with s [y] and both p and q are analytic functions in D with M-order at most max(p,y) (Linden [40]).
Using Jensen's formula it can be seen that if f is an analytic function in D and If(O) logIR/r i If f has finite M-order p, then taking R --1(l+r) we find for e > 0 there is an r such that o n(r,f) < (l-r) "l--e (3,6)   for r < r < i.
If a function has nearly all the zeros this allows, the zeros will o need to be fairly uniformly distributed near {z IzI i} as THEOREM 3.13 shows.
THEOREM 3. 13.Let f be an analytic function which has M-order p in D where I -< p < (R).Then for each positive s and any 8 (0<8<2), there is a number r o such that the number of zeros of f in i -< Izl -<7 zl is at most (l-r) "p'e for r < r < i (Linden [40]), o For smaller M-order we have: THEOREH 3,14, Let f be an analytic function in D which has M-.order p < i.
Suppose p < n, Then there is a constant A such that the number of zeros of f in is less than A/(l-r) whenever 0 <r < I and 0 < 8 < 2 (Linden [41]), If restrictions are made regarding the location of the zeros of a function, the bound in (3.6) can be made better.THEORI24 3.15 gives an example.
THEOREM 3,15.Let f be an analytic function in D which has finite M-order Suppose that the zeros of f all lie on a finite number of radii of .zzl i}.
Then if e > 0 and q max(p+e, I), there is a constant C such that n(r,f) < C(l-r) -q whenever 0 < r < I (Linden [41]).
In general for functions with order less than one, it is not possible to improve (r/l) (l-r) { log(l-r) } (c.f.Linden [41]).
Turning our attention to values assumed by the derivative function, we shall give two theorems.THEOREM 3. 16.If f is an anlaytic function in D with lira sup (l-r)log M(r,f) += r+l then either f assumes every complex number infinitely often, or every derivative of f assumes every complex number except possibly zero infinitely often (c.f.Hayman [17]   and [19]).
The exceptional position of zero in the above theorem cannot be eliminated by assuming a stronger growth condition.Functions f defined in D by f(z) exp(| exp g(z)dz) where g is an arbitrary function which is analytic in D satisfy f(z) # 0 and f' (z) # 0 for any z in D. So for the first derivative one cannot strengthen THEOREM 3. 16.However, one can prove: THEOREM 3.17, If f is an analytic function in D with sup (1-r)log log M(r,f) + r+l then either f or f" assumes every complex number infinitely often (Hayman [19]), Suppose f is an analytic function defined in D by where T is a positive number.If E is a fixed bounded set and a fixed posirive integer, then lira (l-r)log M(r,f) y and if 7 is sufficiently large, f(z), fV(z),,.,,f()(z)for z in V assume no value in E. Thus the growth hypothesis in THEOREM 3,16 cannot be sharpened (}layman [19]).
To see the growth hypothesis in THEOREM 3.17  expCa(l_ + -----!----1 )) (1.z) I/2 and a is a positive constant.Clearly f is an analytic function in D with fz) # 0 and f' (z) 0 for z in D. For all sufficiently large a, one can show also f"(z) # 0 for z in D (Hayman [19]).
The situation concerning asymptotic values for functlons analytic in D is differ- ent from that for meromorphlc functions.We give three theorems concerning the exls- tence of such values.
THEOREM 3.20.Let f be an unbounded analytic function in D. Then f need not have as an asymptotic value (MacLane [34, p.71]).
At a point on the boundary of D an analytic function may have more than one asym- totic value, but it is possible to determine the size of the set of such points.
THEOREM 3.21.Let f be an analytic function in D, Then the set of points E on {z zl i} at which f has more than one asymptotic value is at most countable (Bagemihl [44] and Helns [45]).
G. R. MacLane introduced the class A of non-constant analytic functions in D to be those functions which have asymptotic values at a dense set on {z zl i}.THEOREM 3.22 gives the limitations on the growth of such functions.
THEOREM 3. 22.Let f be a non-constant analytic function in D. If 0 then f e A. In partlcular, if f has finite M-order, then f A. Further, for c > 0. there exists a non-constant function f which is analytic in D such that f A and for some number r with 0 < r < i, o o e (r <r<l) (Hornblower [46]).
M(r,f) < exp exp ('l'-r) (-iog(l.-)Yo (G.R. MacLane [47] and P. J. Rippon [48] both give different discussions of THEOREM 3.22)   MacLane [47] also introduces the classes B and [ of non-constant analytic functions in D. B is the class such that there is a set of Jordan arcs r in D, each ending at a point of {z zlffil} and such that the end points are dense on {z Izlffil} and such that on each r either f tends to infinity or f is bounded.
To define t we first need the notion of a set ending at points of the unit clr- cle.
Let S be a subset of D. For each r with 0 < r < 1, label the componente of S (zJr<Jzl<l) by Skit) here k is a non-negative integer.Let dkr) be the dimaeter of S k(r), and set d(r) sup d k(r) k with d(r) 0 if there are no components Sk(r).We say S ends at points of (zlJzl i) if and only if d(r) decreases to zero as r approaches one.
l is then the class of non, constant analytic functions f in D for ich every level set (z[ lf(z) }, 0, ends at points of (z lzl LcLane's startllng result is that A=B=Lp so a non-constant analytlc function f in D with finite H-order by THEOREH 3.22 is also in B and in (In [31] K.F.Barth defined the corresponding classes and of non- m m constant eromorphlc functions in D, but the classes turn out to be distinct from each other and from A as defined in SECTION TO).m Actually, a function which is analytic in D may have infinitely many asymptotic values at a point on {z Izl i}.Such a function was noted in [34] by considering the entire function F of W. Grosz [49] for which every complex number is an asymp- totic value.If the plane is sllt along one path on which F / =, then the sllt plane can be mapped onto D. The function f defined in D by means of the composition of the map and F is a function in A which has every complex number as an asymptotic value at one point z with z i.
o o THEOREM 3. 23.tells us that the number of asymptotic tracts at a point for a func- tion analytic in D is related to the order of growth of the function.
THEOREM 3. 23.Let f be a non-constant analytic function in D. Suppose there is a number y with y > 0 such that i (l-r)71og M(r,f)dr < 0 Let z be a complex number such that z i, and let nz'n(R)) denote the number o o of distinct asymptotic tracts of f associated with z for which the asymptotic o values are finite (infinite).Drasin [50] and IfcLane [34,   p.54]).
THEOREM 3.23 is best possible.To see this is true for y 0, consider f de- fined in D by 1 + z l+z f(z) for which n_ I and n 2 at z I. To see the result is best possible when O 0 < y < i, we consider f defined in D by where 0 < < y < i for which n. THEOREM 3.24.Let f be an analytic function in D. If f has only finitely many tracts for and the ends of the arc tracts of f. for do not cover {z Izl i}, then the Lebesgue measure of the set A is positive where A is the set of with II i for which there is a complex number a such that f has the asymptotic value a at (McMillan [51]).
Another growth behavior of interest is how the minimum modulus of a function on a circle about the origin acts.
THEOREM 3. 25.Let f be an analytic function in D with finite M-order where p > i.Then there exists a constant K such that log L(r,f) > -K(log M(r,f))(iog log M(r,f)) for a sequence of numbers r increasing to one where (Linden [52,53]), The above theorem is best possible (c.f.Linden [52,53]).For functions of lesser growth there is THEOREM 3.26.
THEOREM 3.26.Let f be a non-constant analytic function in D which has order p where p < I. Then there exist positive constants K and C and a number s o with 0 < s < i such that if s satisfies s < s < l, the interval (s,l%-(l+s)) o o contains a set of values r of measure at least C(ls) for which K lOg(l.lr)log L(r,f) > i where L(r,f) is defined by (3.7).The constant C need not be less than 1/4 (Linden [54]).
Finally, we shall look at the subset of D in which a given function has its maximum modulus greater than one and observe how this set intersects {z Izl r} for 0 < r < i.First we introduce some notation.
Let f be an analytic function in D, and let {z If(z) > i and z D}.
For 0 < r < i let (r) be the arcs of {zl zl r} which are contained in , and let r8 k(r) be their lengths.For 0 < r < i, if {zllzl r} lles wholly in D, define 8(r) +=; otherwise set 8(r) max 8k(r).IfCz)l -< =p (cC-Izl>-n>, (z where C is a positive constant and 0 < n < .If f is an analytic function in D, f is said to be in B p (0 < p < ) if and only if I (.og+lf(z)[) p dx dy < =,.
zl <i E. Belier [56] points out that these classes are related by the following inclusions: B-n (l/n) + for all > 0 and 0 < n < ; Beller [56,57] has proved the following theorem concerning the zeros of functions in these classes.THEOREM 3.29. (i) If f is an analytic function in D which is in -n where 0 < n < and {a k} are the zeros of f in D, then for e > O, [ (-[akl) "++ < k:l (ii) If f is an analytic function in D which is in B p where i < p < and the {a k} are the zeros of f in D, then for e > 0, I (1-1:1) 1+1/p+ k=l For B I we get further information by taking 0 in the theorem of A.
THEOREM 3.30.If f is an analytic function in D with ] (log+[f(z),)(1-1z,) dx dy<=, for some real number > 0 and {a k} are the zeros of f, then k=l If 0 < u < i, the condition is sufficient, H,S.Shapiro and A,L, Shields [59] showed that for some n, if f is a function in B -n and {a k} are the zeros of f on a single radius of the unit circle, then I (l-lakl) < Their theorem was extended by W.K, Hayman and B. Korenblum [60]: k=l THEOREq! 3. 31.If is a positive, continuous non.decreaslng function of r on [0,i) such that (r) approaches infinity as r approaches one, and if f is an analytic function in D for which loglf(z) < (Izl), (zD) then those zeros {a k} of f which lie on a single radius of the unit circle satisfy C.N. Linden [61] has found additional conditions on in the previous theorem besides (3.9) to conclude that (3.8) holds for the zeros of f in certain tangential regions in D. Bruce Hanson [62] has shown further relations between the growth of an analytic function in D and the behavior of its zeros which lie on a single radius of the unit circle and also has some interesting xamples concerning their best-posslble nature.He shows THEOREM 3.32.Let f be an analytic function in D, and let {a k} be the zeros of f which lle on a single radius of the unit clrcl.e.where the sequence {n k} satisfies nk+l >-q > i, ( for some real number q. (3.11) (i) If then f assumes every complex number infinitely often.In fact, f assumes every complex number infinitely often in every sector in D of the form {-.Io, <  For smaller gaps in the series representation for the function, there are also value distribution results ones relating the gap size and the growth of the func- tion.
THEOREM 3. 35.Let f be a non-constant analytic function in D which has Morder 0 and is defined by the power series representation (3.10)where the sequence {} satisfies lira inf lg(nk+l -_nk)_ > I ) k log nk+I ((3.12)Then (i) for each complex number a, (a,f) 0; (ii) where L(r,f) is defined by (3.7) and (iii) f has no finite asymptotic values (Wiman [66]).
In [67] the conclusion (i) of THEOREM 3.35 is obtained for some functions with less restrictive gaps than (3.12).A result where the density of the sequence {n k} and the growth of the function is linked follows.THEOREM 3.36.Let f be a non-constant analytic function in D defined by the power series representation (3.10).Let (t) be the number of n k not greater than t, and suppose that, for some fixed 8(0<8<1), (t) 0(tl-8), (t-).
The best possible nature of (3,13)  In this section we give some theorems concerning functions meromorphic in D with growth more restrictive than that in SECTION TWO, Define a and by lira sup (r,f) where T(r,f) is the Nevalinm characteristic of f at r. Then 0 N N N +.
It is possible to obtain some restriction on the growth of the derivative of a function in this class in terms of the growth of the function.r/l (Sons [4]).
The growth of the function places some restriction on the number of a-values and placement in the disc where f may assume a-values.(iii) for each e > 0 there exists a constant K such that (l-r)l+l+(i/IP(r)J), K.A restriction on the possible number of zeros (or a-values) is Riven bel.
TflEOR/{ 4.3.Let f be a meromorDhlc function in D for which 0 < a < 4o with defined by (4,1), Then (l-r)n(r) r/l (Sons [4]>. We now consider the number of values a function in our class can omit and number of times values are assued relative to the growth of the function.(r/l), and so 0 < Ua THEOREM 4.4.Let f be a meromorphlc function in D for which with defined by (4.1).Then for every a in u {(R)} with at most q-i excep- tions, where q > 3, > (l/q) ((q-2)i). (i0]).a In SECTION TWO we introduced the Nevanlinna deficiency and the index of multipli- city.Using the notation established there, we have the following theorem related to THEOREM 2. i0.where p is an integer larger than 2, then f cannot have p values taken only a finite number of tlmes.
In fact, it is true that the above theorem holds if is replaced by .We where p is an integer larger than 2, then f cannot have p values taken only a finite number of times (Tsujl [3, p.215]), THEOREMS 4.6 and 4.6' rely on the fact that when such f take on a value a only a finite number of times, then (a)=l.These theorems are best possible.For each integer p with p -> 3, we Consider an automorphic function g which maps the unit disc onto the universal converlng surface of the plane punctured at the p For g we have i/(p-2), (Nevanlinna [6, p, 272] points a I, a a 3,...,a THEOREM 4. 7. Let f be a meromorphic function in D such that 0 < < + where e is defined by (4.1).Then f cannot omit an infinite number of values.
The techniques used in proving THEORE_S 2.17 and 2.18 make it possible to estab- llsh some theorems concerning the value distribution of derivatives of functions in our class.
If one is willlng to take e larger, slmilar concluslons can be made for hlgher order derivatives.

L.R. SONS
One interesting property of normal functions is how asymptotic values at points of the unit circle are related to angular limits at such points.
THEORE 4.11.If f is a normal meromorphic function in D and has the asymptotic value a along an arc in D ending at , II-i, then f has the angular limit a at (Lehto and Virtanen [27]).
For normal analytic functions another value distribution result can be stated.
THEOREM 4.12.Let f be a normal analytic function in D, Then either (-i) f has finite radial limits on a dense subset of {z Izl =i}; or (ii) f assumes every complex number infinitely often (Sons [73]), Other interesting examples of functions in F can be found in [42] and [59], M. Tsuji [74] has shown that certain automorphic functions are in F. Also the deri- vatlves of some functions of bounded Nevanlinna characteristic are not of bounded char- acteristic but are in F (with e < I) (c.f.Hayman [75]). 5. SECTION FIVE.
We consider theorems concerning functions analytic in D with rowth more restric- tive than that in SECTION THREE.where M(r,f) is the maximum modulus of f on {zllzl=r}.We have 0 < ), < p _<..I-.
Subsets of the class here include univalent functions (the classical Koebe func- tion has 0 2), p-valent functions of various types, and functions in H p classes (0 < p < ) (c,f.Duren [76], Hayman [77], and Pommerenke [78]), Some restriction is obtained on the growth of the derivative of the function in terms of the growth of the function, THEOREM 5.1.Let f be an analytic function in D for which 0 is defined by For functions in A p the followlng can be proved, THEORKM 5.6, If f is an analytic function in D which is also in A p (0 < p < (R)) and if {a k} are the zeros of f arranged so that lal <-la21 < la31-<.Further, those results are sharp (Belier [57] and Horwitz 0]).
In ['81] B. Korenblum has given a necessary condition for a subset of D to be a zero set of A-n; he has also given a sufficient condition for a subset of D to be a zero set of A -n and a necessary and sufficient condition for a subset of D to be a zero set of u . .for some n).
There are functions defined in D by gap power series wlth gaps smaller than (3.11) and gr#th not as rapid as that implied by (3.13) for xhlch conclusions of the nature of THEOREM 3.36 can be made.For example, THEOREM 5.7.Let f be an analytic function in D defined by the power series representation (3.10)where the sequence {n k} satisfies (Anderson, Clunie, and Pommerenke [83] and Offord [84]), In recent years annular functions have been studied concerning their value dis- tribution (c.f.Bagemlhl and Erds [85] Bonar [86] and Bonar and Carroll [87] ).An analytic function f in D is said to be annular if there exists a sequence {J n of Jordan curves in D such that (i) J lies in the interior of Jn+l n (ii) for each e > 0 there exists a number n O such that, for n > no, Jn lies in the region {zIl-e<Izl <I}, and (ill) (J min{If(z) llzEJ + as n -.
n n THEOREM 6.2.Let f be an annular function in D with defining curves {J n as above.Let {r be a sequence of numbers increasing to one such that J is n n contained in {zlIzl < r }, Then n N(r l/f) n > i +/-if tog (S (Nicholls and Sons [68]).
A number of properties of m 2(r,f) and some value distribution theorems involving m2(r,f) appear in [88].In [89] some additional results are obtained with special consideration given to the case where m2(r,f) approaches infinity as r approaches one, but T(r,f) is bounded.

Then
l-a k)  .In any case, k p-l_< u <u.Now we define the _canonical product P(z) formed with {a k} asbelow.If    Z(l-lakl) .A(k,z))exp(A(k,z)+(A(k,z))2+,,.+ (A(k,z)) p)(2 is called the .Nvanlinna_ deficien..cy of a and 8(a) is called the index of ultiplicity of a.It is clear from the definition of g(a) that 0 < (a) < I, and if a is a Picard value, then 6(a) I, Some functions may have a number of deficient values, but for the class of functions under consideration there are some restrictions on the number and their size.

THEOPo-! 2 ,
10, Let f be a meromorphic function in D such that T (rf) lira sup Log(i-r' Then the set of values a in u {} for which ()(a) > 0 is countable, and summing over all such values a, one gets .{(a)+ 8(a)} < (a) a function defined in D by f(z)

L
.R. SONS Another gauge of values assumed is the Vallron deficiency.If f is a meromor- phic function in D and a is in u {(R)) we define is called the Valiron deficiency of a.The next theorem gives some measure of the size of the set of Valiron deficient values.THEOREM 2.20.Let f be a meromorphic function in D Assume Then lira T(r,f) + =. rl N(r,a) i= T (r,) z, r+l except for at most a set of a-values of vanishing inner capacity.Thus the Vallron deficiency A(a,f) is positive at most for a set of a-values of inner capacity zero, (Nevanlinna [6, p.280] ).

THEOREM 2 . 22 ,
There exist functions meromorphic in D which have no asymptotic values, finite or infinite (and hence no radial limits).Further, the Nevanlinna characterlsitc of such functions may have arbitrarily sl growth.(MmcLane[26]), (0. Lehto and K. Virtanen[27] have shown that there even exist meromorphic func- tions f in D without asymptotic values which are also normal functions in D that is, they satisfy for z in D and some constant C)0 Meromorphic functions in D which have no asymptotic values possess interesting value distribution properties, THEOREM 2.23.A function meromorphic in D which has no asymptotic values assumes every value infinitely often.Further, in every neighborhood of each point o.I-Ian perhaps two values (c.f.Cartwright (i) D(e) is a component of the open set {zlzDu{z] (ii) 0 < e I < e 2 implies D(el) c D(e2) and replaced by the only change is to replace If(z)-al<e in (i) by > lYne set >0 is a non-void, connected, closed subset of {zl [31] has given a suffi clent condition for the nonexistence of arc-tracts for functions in A m THEOREM 2.25.Let f be a meromorphlc function in D such that f is in A m If there exist a and b in u {} with a b such that N(r,a) 0(i), and N(r,b) 0(i), (r+l), then f has no arc-tracts.

THEOREM 2 . 26 .
Let f be a meromorphlc function inD.If    N(r, ) 0(i), (r+l), and I (l-r)T(r,f)dr < 0 then for each positive integer n, f(n) is in A or f(n) m and Schneider[32]), is a constant (Barth 3. SECTION THREE. THEOREM 3.4.Let f be an analytic function in D with M-order p such that 0 < O < and lower M-order A. Suppose f(z) has the form(3.3).Let B(r) be the maximum term in (3.3) for {z [z[ r} and v(r) be its central index.
follows from setting R r + (3.4) to get the implication log T(r l llm sup liog(l_r > O.r/l L.R.SONS

THEOREM 3 .
10. Let f be an analytic function in D with M-order where I < < and let {a be the zeros of f in D. If s is an integer not less n than [0], the function p defined by an(an-Z)

THEOREM 3 .
11. Let f be an analytic function in D with finite M-order p where p I. Then f(z) p(z)q(z),

( 3 . 6 )
much.In particular, if we let an analytic function in D which has M-order zero and n(r,1/B) 1 2 of tracts for is restricted for an analytic function in D, it is possible to say more about the set where the function has finite asymptotic values.

28 .
The following two theorems k follow from a line of reasoning used by K. Arima in[55].THEOREM 3.27.Let f be an analytic function in D which has finite MLet f be an analytic function in D which has finite positive order defined by (i.I).If there exists a finite M-order have received special study.If f is an analytic function in D, f is said to be in B -n if and only if r, f) dr < Another interesting class of functions in D is those defined by gap power series.If the gaps are large and regular, the conclusions on value distribution are especially nice even for functions which grow slowly: THEOREM 3.33.Let f be a non-constant analytic function in D defined by the

THEOREM 3 . 34 .
Let f be a non-constant analytic function in D defined by the power series representation(3.10where the sequence {n k} satisfies(3.11).Then (i)f is in A; (ii) if llm sup ..[Ck[ > 0k then f has no finite asymptotic values (Murai[65]).
to obtain (li) is s[own for n k of the form k p where p is an integer not less than 2 by examples in [70] L.R. SONS 4. SECTION FOUR.

THEOREM 4. 2 .
Let f be a meromorphic function in D with defined by (4.1) and < 4. Let {ak} be the set of a-values for f where a is in E u{}.
Let f be a meromorphic function in D, and define for each a in by a N (r ,a) llm sup r/l -log (l-r) a By the First Fundamental Theorem of Nevanlinna theory we have N(r,a) < T(r,f) + 0(i), -< B(R) is defined similarly.

TEOREM 4. 5 .
Let f be a meromorphic function in D such that where is defined by (4.2).Then the set of values a in u {(R)} for which)(a) > 0 is countable, [ {(a)+0(a)} < [O(a) < 2 +! (Sons [4]) aAn immediate corollary to THEORKM 4.5 presents a limitation on the number of values such a function can take only a finite number of times.THEOREMS.4.6.Let f be a meromorphic function in D such that where is defined by (4,2), If > /(p-2), have IEORK._'4.6'.Let f be a mero:rphic function in D such that where is defined by (4.1).If u > i/(p-2), For an analytic function f in D define 0 and by log+M(r ,f) lira sup -log(iLr) .Borel exceptional valu for f if a < " If , we say a is a Borel exceptional ,v for f if a < " THEOREM 2.6.Let f be a meromorphic function in D such that f has order Then for every a in u {} with at most two exceptions, u e (c.f, a Sons[i0]).