Criteria of boundedness from above of subharmonic functions of finite order in an angle

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Criteria of boundedness from above of subharmonicfunctions of finite order in an angleVladimir Logvinenko & Nataly NazarovaPublished online: 29 May 2007.

To cite this article: Vladimir Logvinenko & Nataly Nazarova (1998) Criteria of boundedness from above of subharmonicfunctions of finite order in an angle, Complex Variables, Theory and Application: An International Journal, 37:1-4, 395-411

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Criteria of Boundedness from above of Subharmonic Functions of Finite Order in an Angle VLADlMlR LOGVINENKO* and NATALY NAZAROVA

Communicated by D. Drasin

(Received May 1997)

Let r be a sector embedded in A, = ( r E C : 0 < argr c nip), p 2 1. In this paper it is given a complete description of all subsets E of the boundary ar of r with the following property: each subharmonic function of normal or minimal type with respect to order p in A,, which is bounded from above on E, is bounded from above on r.

Keywords: Subharmonic functions; PhragmCn-Lindelof

Classification Categories: 3 1A05

1. INTRODUCTION

By IEl we denote the linear Lebesgue measure of the set E, by cap(E) its logarithmic capacity, and by C various constants.

In 1971 B. Ya. in [Ll] proved the following theorem on subharmonic functions of finite degree in the complex plane (the corresponding statement for entire functions of exponential type was obtained by A. C. Schaeffer [S] earlier).

THEOREM A (Schaeffer-Levin). Let E be a relatively dense subset of the real line R. Then there exists such a constant C = C(E) that for

Corresponding Author: Vldimir Logvinenko, Ph.D., 2752 PONDEROSA Dr. Apt #190 CAMARILLO, CA 93010 USA

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396 V. LOGVINENKO AND N. NAZAROVA

every subharmonicfunction u(z) offinite type a in C the inequality

sup{u(x) : x E R] I Ca + sup{u(e) : { E E} (1)

holds.

We recall several definitions used above. A subharmonic function u(z) is of type a with respect to order p E

(0, CQ) in C if lim supu(z)/lzlP = a.

IzI'CQ

If a = 0, the type is minimal, if a = ca, the type is maximal, and each one of the types a E (0, CQ) is normal. By definition, a subharmonic function u(z) is of at most normal type with respect to order p if for some constants a and b the inequality

is valid for all z E C. We use the same definition of type for subharmonic functions of order p in some angle A provided that the opening of A is at least r / p . If p = 1, a subharmonic function u(z) of normal type is called afinction offinite degree in C or A respectively.

A measurable set E C 08 is called relatively dense (with respect to Lebesgue measure) if there exist positive constants L and 8 (density constants of E) such that

holds for every x E R.' Not long ago, A. Baernstein [Ba] and A. E. Fryntov [Fl proved inde-

pendently that the periodic set

Eo = U [mL - 612, mL + 8/21 me2

realizes the largest value of C(E) in Theorem A among all sets with the same density constants L and S.

Let l(8) be the ray {z = reie E C : r 1 01, and let Qw(h, l(f3)) be a square in the complex plane with center w E l(8), sides parallel or perpendicular to l(@, and sidelength h. A set E is called relatively dense by capacity if there exist positive constants L and S that

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SUBHARMONIC FWCI?ONS 397

for all w E 4(@. Here (1/2L)(E n Qw(L, 4(8))) is the image of E i l Q, (L, l ( 8 ) ) under the linear map

1 w' -+ -(wl - w ) + W .

2L In the paper [LLS] Theorem A was improved to:

THEOREM B (Levin-Logvinenko-Sodin). Let E be a closed regular (in the sense of Dirichlet Problem) subset of the real line. The following statements are equivalent:

(i) E is relatively dense by capacity on the rays L(0) and L(n). (ii) There exists a positive constant C that (1) is valid for every a E

(0 , oo) and every subhannonicfunction u ofjinite degree a in C.

In discussing Theorem A, B. Ja. Levin asked: How does this theorem change if E is a relatively dense subset of the boundary of an angle less than the halfplane? In regard to Theorem B, it seems natural to refor- mulate Levin's question in terms of capacity: How does the assertion of Theorem B change if E is a subset of the boundary ar of an sector r of angular opening less than x which is relatively dense by capacity on each of the boundary rays?

The particular case of the following theorem (when I = 0) answers this question.

THEOREM 1. Let r beasector {z E C : a < argz < 81, @-a < R, let E c ar be a closed regular set, and choose I E [O, 1 ) . Then the following two assertions are equivalent:

( i) There exist such positive constants L and 6 that

(ii) The inequality limsup u(z)/jzlA < oo

ILI-.+W, z E ~

holds for every subharmonicfunction u offinite degree in C whenever this function is boundedfrom above on E.

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398 V. LOGVINENKO AND N. NAZAROVA,

We obtain Theorem 1 from the following statement concerning subharmonic functions defined in an angle.

THEOREM 2. Let p > 1, let h.E [O; I), let r = {z € & : cr < argz < p} be a sector embedded in Ap = {z E C : 0 < argz < xlp}, and let E be a closed regular subset of ar. Then the following two asserrions are equivalent:

( i) There exist such positive constants L and 6 that the inequalities

hold for all sufficiently large w E [(a) and s E l(p). (ii) The inequality

lim sup u(z)/lzlo < w lzl+m. I&

holds for every subhannonicfrinction u(z) in the sector A, satisfying ( 2 ) whenever it is bounded from above on E.

When A = 0, we have the following PhragmCn-Lindelof result:

COROLLARY 1. Let p 2 1 , let r = {z E C : cr < argz < Bj be an angle contained in Ap and let E be a closed regular subset of ar. Then the following two statements are equivalent:

( i) There exist such positive constants L and 6 that the inequalities

hold for all sufficiently large w E -!(a) and s E C(/?). (ii) Every srrbharmonicfirnction in Ap satisfying inequality (2) and boun-

ded from above on E is bounded from above on r.

Theorem 2, in turn, is deduced from Theorem 1 and a result of A. Yu. Rashkovskii and L. I. Ronkin [RR] on continuation of subharmonic functions from an angle onto the whole complex plane with control of growth. Using another result of this paper, we obtain the following theorem.

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SUBHARMONIC FUNCTIONS 399

THEOREM 3. Let A be the halfilane C+ f i , and let E C i3A be a closed regular set such that the inequalities

ls12 cap ( i i - ( ( ~ - i ) n Q,(LISI-', @dl)) t 6

hold for some positive constants L and 6 and all suficiently large w E

l (0) and s E l(n). Then every subhannonic function u(z) of minimal degree in C+ bounded from above on E is bounded from above oh A.

It is natural that there exist more precise versions of the results formulated above for analytic functions. Here we will prove a rather simple Phragmen-Lindelof result on boundedness. Theorems on slow growth are more sophisticated and need the finer technique based on sharp results of P. Malliavin [MI. To formulate our Phragmin-Lindelof theorem, we need a notion introduced by A. Beurling [Beu].

Given a set A C C and a ray l(8), for R > 0 let

nA(R, 8) = min(card(A n [z, z + ~ e ' ~ ] ) : z E l(@}.

The limit (perhaps infinite)

A(A, 8) = lim n,, (R, 8)IR R-+w

exists and is called Beurling's lower uniform density of A on the ray l (8) . In [LN] we prqved a Phragmtn-Lindelof result on entire functions

of exponential type similar to the particular case of Theorem 1 when A = 0.

THEOREM C (Logvinenko-Nazarova [LN]). Let r = { z E Q : a <

arg c p}, - a < n, and let A be a subset of ar such that

min{ A(A, a ) , A(A, j3)} > a/n. (9)

Then every entire function of exponential type at most a bounded on A is borinded on r.

We deduce the following Phragmtn-Lindelof theorem on analytic functions in an angle from Theorem C and the Keldysh Theorem on approximation of analytic functions in an angle by entire ones (see [G,

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Chap. IV, B, Theorem 21 or [R] where a general version of this theorem is proved).

THEOREM 4. Let r = ( z E C. : or < argz c p] be an angle embedded in A,, p > 1, and let A C ar be such a set that

card(A r l [tllPeiB, ( t + ~ ) ' l ~ e ' ~ ) ) ) > AR. (10)

Let also f (z) be an analyticfunction such that for some constants a and b for all z E A, we have

Then i f f (z) is bounded on A, it is bounded on r.

Using another Keldysh-type theorem recently obtained by A. M. Russa- kovskii [R], we prove the following analogue of Theorem 3 for analytic functions:

THEOREM 5. Let A be a subset of the boundary of angle A2 + 1 + i such that

If an analytic function of exponential type in the positive quadrant A2 is bounded on A, then it is bounded on A2 + 1 + i.

It is necessary to emphasize an essential difference between our Phrag- mhlindelof-type results and Theorem C, on the one hand, and Theo- rems A and B, on the other hand. To see that, let us consider Theorem B and the particular case of Corollary 1 where p = 1. We introduce several classes of subharmonic functions. Denote by K,,(E) the class of all subharmonic functions on A, which satisfy inequality (2) for every or > a and are non-positive on E. We write K,(E) instead of K1,,(E). By K ~ ( E ) we denote the subclass of Ku(E) that consists of subharmonic functions of at most degree a in the complex plane.

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SUBHARMONIC FUNCTIONS 40 1

Let E be a set satisfying conditions of Theorem B. By this theorem all functions of K:(E) are bounded from above on the real line by the same value Co where C is an absolute constant. In general, Corollary 1 does not guarantee a similar uniform estimate for the class K,(E), and in fact not even fo; the subclass K:(E). In other words, Corollary 1 does not contradict the possibility that

The following theorems show that this equality is indeed true save in the trivial case.

THEOREM 6. Let r be an angle embedded in A,, and let E be a closed subset of ar such that for some a > 0

sup{u(z) : z E r, u E K,,,(E)} < co.

Then E = ar. One can say more. This theorem holds for the subclass of K, , (E)

generated by functions of the form u = log If 1 , f analytic. Denote by L,,(E) the class of all analytic functions f in A, such that log If 1 E

K,.u.

THEOREM 7. Let r be an angle contained in A,, and let E be a closed subset of ar such that

Then E = ar. We will see that if p = 1, the assertion of this theorem is true for

the subclass of entire functions. Theorem 6 is a simple consequence of Theorem 7. Therefore, we prove only Theorem 7. The following result is slightly more general than Theorem 6: E is not assumed to be closed.

THEOREM 8. Let r be an angle contained in Ap, and let E C ar be a capacitable set. If for some a > 0

then cap(E n I ) > 0 for every interval I.

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The paper consists of three sections. In the first section we prove Theorems 1,2, and 3. The second one contains the proofs of Theorems 4 and 5. In the final section we prove Theorems 7 and 8.

1. PROOFS OF THEOREMS 1,2 AND 3

Proof of Theorem 2. We prove this theorem in several steps, one of which is the proof of Theorem 1. At the beginning let us verify that Theorem 1 implies Theorem 2.

1. This step is very simple. Define v(z) = u(zllP). This function is subharmonic in the upper halfplane C+. By (2), for every z E C+, v satisfies the inequality

vtz) h alzl + b. (11)

The image of r under the map z -+ zP is a wedge rl = {z E C+ : oil c argz < B1] embedded in C+. Mewing logarithmic capacities as transfinite diameters, we can easily verify that E satisfies conditions (5) and (6) if, and only if, its image e under the map z -+ zllp satisfies (3) and (4). Since the relationships

are equivalent, we have reduced the general case to that with p = 1 and A, = C+.

2. At this point we need the following particular case of the general Rashkovskii-Ronkin extension theorem [RR] mentioned above.

THEOREM D (Rashkovskii-Ronkin). Let v(z) be a subhannonicfunction satisfying (11) in C+. Then for every positive E and 6 there exists such a subhannonicfunction offinite degree in C (i.e. 34, B c oo 3 VZ 'z C : w(z) ( Alzl + B) such that w(z) = v(z) on the domain {z E C+ : 6 c arg z < n - 6, I z I > E}.

This theorem allows us to consider only subharmonic functions w of finite degree in C which are bounded from above on E. In other words, we only have to prove Theorem 1.

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3. Now we need an estimate of harmonic measure first obtained for sets that are relatively dense with respect to Lebesgue measure by M. Benedicks [Ben, Lemma 81 and recently improved by M. L. Sodin [S].

THEOREM E (Benedicks-Sodin). Let R > 0, and let E c Qo(R, t(0)) n R be a closed set such that

for some 6 > 0, h << R, and all x, 1x1 5 R - h. Then

where C is an absolute constant and w(z) = o(z, aQo(R, t(O)), Qo(R, l(O))\E) is the harmonic measure of the exterior boundary of Qo(R, t(O))\E with respect to this domain a t the point z.

Instead of the proof that (i) =$ (ii), we will prove a slightly more general statement.

Let r be an angle (z E C : 0 < argz < B} , B < n, and let E c ar be a set satisfying condition (i) of Theorem 1 for some A E [O, 1). Suppose also that for some p E [0, 1 )

limsup u(z)/lzlP < w 1:1-.+00, zEE

where u(z) is subharmonic function of finite degree in C. Then

where v =.max(A, p}. Let the finite degree of u equal a. Then the inequality

holds for some constant C and all z E C. Applying the Benedicks- Sodin estimate to the squares Q,(x, l(0)) and Qf(h, l (0)) where h = L(l + xA), h < c 2r - h, we have that for any sufficiently large x E e(o)

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The same reasoning is valid on the ray l ( p ) . Hence,

limsup u(z)/lzlv < m. IZI-+W. ~ a r

By the PhragmCn-Lindelof Principle the function u(z)/lzlv is bounded from above on r n { z E C : I z I 1 11. Therefore, the statement (ii) of Theorem 1 is true.

4. Let the statement (i) of Theorem 1 be false. Without loss of generality, we can assume that this condition is violated for the ray -!(a) and that a, = 0. By Theorem B, for every T > 0 there exists such a function w,(z) E K:((E n QO)) U l(n)) that

lim sup w,(x)/x" m. x--+ 00

If r is sufficiently small, then

W ( Z ) = w,(z) - ( a - s> Imz- C

belongs to K:(E) for a suitable constant C. Since

lim sup w(x)/x1 = m, x-+m

(ii) is false. Theorems 1 and 2 are proved.

Proof of Theorem 3. At this point we need another continuation theorem of A. Yu. Rashkovskii and L. I. Ronkin [RR].

THEOREM F (Rashkovskii-Ronkin). Let H be a positive number, and let u(z) be a subharmonic function satisfying condition (2) for some p > 0 in the upper halfplane. Then there exists a subharmonicfirnction w(z) in C which coincides with u(z) on the halklane { z E C : Imz > h] and satisfies the inequalities

Using this theorem, we can proceed as follows. Repeating the third step of the proof of Theorem 2, we. deduce from (7) and (8) that the subharmonic extension w(z) is of order 2 and bounded from above on the rays l (n )+i and l(O)+i. Therefore, the function u(z) is also bounded

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from above on these rays. By the Phragmdn-Lindelof Principle, u(z) is bounded from above on A. Theorem 3 is proved.

2. PROOFS OF THEOREMS 4 AND 5

Proof of Theorem 4. We divide the proof into several steps, as in the proof of Theorem 2.

1. At this point we will prove Theorem C for the convenience of the reader. Assume that r = {z E C : 0 = a! < argz < ,B}, ,5 < R. Since the Beurling uniform lower densities of A on the boundary rays of A are larger than u/n, we can select such a uniformly discrete sequence of elements of this set that its usual densities along these rays are also larger than u/n. NOW we need the following theorem of V. Bernstein [Ber] (see also [L2, Chap. IV]).

THEOREM G (Bernstein). Let {An)?=, be a uniformly discrete sequence of points of the ray l(e), and let

lim card(n E N : IAnI 5 r)/r > u/n. r-r m

Then for every entirefunction F of exponential type a t most u its indicator in the direction 8

hF(e) = lim sup log I ~ ( r e " ) l / r r+oo

can be evaluated asfollows:

By virtue of this theorem every entire function of exponential type at most a that is bounded on A has non positive indicators h(0) and h(p) . Indeed, by the trigonometric convexity of the indicator, both of them are equal to 0. If

S U ~ I I ~ ( Z ) I : z E ar) < CQ,

then there is nothing to prove. If this supremum is infinite, we can assume, without loss of generality, that

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Denote by p(x) the least concave majorant of log 1 f (x)l on the positive ray. The following properties of this majorant are obvious:

(i) p(x) 4 w as x 4 co; . (ii) p(x) = o(x) as x +- ca;

(iii) there exists a sequence monotonically increasing to ca such that log I f (6,)I = p(en) for all n;

(iv) for every finite positive R the difference p(x + h) - p(x) tends to 0 uniformly on the segment -R I h I R.

S. Agmon [A] proved a useful version of PhragmCn-Lindelof Principle in terms of a coricave majorant.

THEOREM H (Agmon). Let F(z) be afunction of exponential ope a in the right halfplane. Suppose also that

where the concavefunction p(x) satisfies condition (i), and (ii) holds on the positive ray. Then for every angle r embedded in the right haljjdane, for every number a' > a, and for every number k > 1 there exists such a positive number xo that the estimate

holds a t each point x + iy E r with x 5 xo.

Now define the functions

By Theorem H, the inequality

is valid for every finite R and all sufficiently large n at all points z = x + iy of the disk ( z ( 5 R. Condition (iv) implies that the sequence {F,) is uniformly bounded on each of these disks. Without loss of generality, we can assume that this sequence converges to a certain entire function F(z) uniformly on every compact set in Q. By our previous inequality

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Since the numbers k > 1, d > a are arbitrary, this means that

Vz = x + iy : IF(x + iy)l 5 expalyl.

Therefore, F belongs to S. Bemstein's class Bu, which consists of all entire functions of exponential type at most a bounded on the real line.

We can also assume that the sequence of shifts {A -(,,j,wt, converges in the weak sense to a certain set Ao, i.e. the Frkchet distance between sets (A - (,,) r l [-R, R] and A. fl [-R, R] tends to 0 as n -+ m for every fixed R < m. In [Beu] it is proved that condition (9) implies that A. is a uniqueness set for the class Bu. Since FIA, = 0, this function vanishes identically. This is impossible, because IF(O)I = 1.

2. Let us assume now that f is an analytic function of exponential type in the upper hdfplane only. Assume also that A has the infinite lower uniform densities on the rays C(a) and l(b)(ar = l (a ) U l(B)). We need a particular case of the well-known Keldysh theorem on the approximation of analytic functions in an angle by entire ones [GI, [R].

THEOREM I (Keldysh). Let f be an analyticfunction in the upper halfplane, I f (z)l 5 C exp Clzj. Then Vr], y > 0, VS E (0,7712) there exists an entire function g(z), Ig(z)l 5 C1 exp Cl lzl, such that

If S is sufficiently small, then the approximating function g is bounded on E by Theorem I. Therefore, as we have just proved, it is bounded on r. Applying Theorem I again, we get that f is bounded on r.

3. The general case of our theorem can be easily reduced to the one just considered by the change of variables z + zP. It is sufficient to notice that condition (10) on the set E means that its image e under the map z + zP has infinite lower uniform densities on the boundary rays of the image y of r. Theorem 4 is proved.

Proof of Theorem 5. We need one general Keldysh-type result recently obtained by A. M. Russakovskii [R]. Let u be a real-valued function, and for I- > 0 set

u y z ) = sup u(w) . ( I w - z l - 4

We say that u has the no-oscillating property if

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THEOREM J (Russakovskii). Let w(z) and +(z) 2 0 be subharmonic functions with the no-oscillating property, let the domains

be such that '

and let f (z) be an analyticfunction in Ro such that

for some K = Kf < oo and all z E Ro. Then for each E > 0 and each N > 1 there exists an entirefunction g(z) such that

Here C -c w depends only on R, w, and +. Let us apply Theorem J to the case where E > 0 is given, w(z) = -xy,

+(I) = max(lxl, Iyl), and Ro = {z = x + i y E 6: : x > 0, y > 0). The approximating function g is of at most normal type with respect to order 2 and approximates f tangentially on the quadrant (z = x + i y E C : x > E , y > E } , and, therefore, on Clos(r). By virtue of Theorem 5 and Theorem 3, g is bounded on r , and so is f . Theorem 5 is proved.

3. PROOFS OF THEOREMS 7 AND 8

proof of he or em 7. By means of the change of variables z H z P and by the Keldysh Theorem we can reduce the genenl case to the particular one in which p = 1, r is an angle embedded in the upper halfplane C+, and A c ar - I where I is an interval. We will show that for every a>O

sup([ f (z)l : z E r, f E L:(A)] = w.

Here L:(A) is the class of all entire functions f that log If (z)l E K ~ ( A ) . Fix a > 0 and assume that r = [z E C : 6 < argz < lr-61, 6 E (0, x/2).

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For the weight +(z) = exp a1 Imzl, introduce the weighted space

This space with the norm ljhll = max(~h(z)~/+(z) : z E ar) is a Banach space. M. M. Dzrbashian [Dl proved that the set of algebraic polynomials is dense in this space. Fix an arbitrary point w E I and define a function g(z) = g,(z) E C$(aI') with the following properties:

(i) 0 ( g ( 1 on a r ; (ii) glar-I = 0;

(iii) g(w) = 1.

For each A > 0 let PA(z) be such an algebraic polynomial that

This inequality implies that the modulus of the entire function PA(z) exp(iaz}, which is of exponential type a, is less than 1 on A and larger than A exp(-o Im w) - 1 at the point w. Therefore,

Since A is arbitrary, Theorem 7 is proved.

Remark 1. In assuming that 0 is not a limit point of A we can prove Theorem 6 in another way. Let be the point nearest to the origin of (A - (0) ) fl l (a) , and let zg be the point nearest to the origin of (A - (0)) n l ( p ) . Choose an arbitrary point w of the bisectrix of I' situated beneath tke segment [G, zB]. Denote by B the angle with the vertex at the point w and the sides passing through z, and zg respectively. The opening of this angle is less than rr; we denote it by the same letter B. Let B' be the image of B under the map z w z - w; the opening of B' is also equal to B. Fix a number p E (Bin, 1). There exists an entire function of order p, the indicator of which is strictly negative in B'. The family f ~ ( z ) = zg(A(z + w)) is uniformly bounded on Clos(B) and, therefore, on A. On the other hand, this family is unbounded on the intersection of any neighborhood of the origin and ar.

Remark 2. We have proved more than was formulated. Indeed, if A satisfies the assumptions of Theorem 7, then

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Here L , o ( A ) is the class of all analytic functions of minimal type with respect to the order p in A, that are bounded by 1 on A. By our construc- tion, this is true in the case considered in Remark 1, but in the general case we need only slightly change the reasoning.

Proof of Theorem 8. Without loss of generality, we can assume that p = 1, A, = C+. Let r be an angle embedded in C+, let E be a subset of ar, and let there exist such an interval I c ar that

Define a subharmonic function uo(z) equal to -cm on E n I . It possible to choose uo of order 0. There exists a point w E I such that D = uo(w) > -a. Let f A ( z ) be the family of entire functions defined as in the proof of the previous theorem for A = ar - i and the chosen point w. For suitable constant C the family of subharmonic functions

are of degree a and satisfy the inequalities

where $(A) H ca as A o a. Since A is arbitrary, Theorem 8 is proved.

Acknowledgements

The authors are grateful to A. Yu. Rashkovskii, L. I. Ronkin, A. M. Russa- kovskii, and M. L. Sodin who informed us about their results before publication.

References

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SUBHARMONIC FUNCTIONS 41 1

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Documents

Criteria of boundedness from above of subharmonic functions of finite order in an angle