Operators preserving the completely regular growth

3.12.

OPERATORS PRESERVING THE COMPLETELY REGULAR GROWTH*

We shall assume that the concepts and facts regarding the theory of entire functions of completely regular growth (c.r.g.) found in Chaps. II and III of [I] are known. In [2] one has considered the derivatives and the primitive functions of entire functions f of cor.g, and one has obtained the following result: I) the derivative f' has c.r.g. on all the rays arg z = 0, except possibly on those for which~ hf(0) = 0; 2) the primitive function

~(Z)=f~Ct)~t

has c.r.g,

in the entire plane.

Now, instead of the differentiation operator

~) = d/dz, we shall consider a more gen-

eral operator of the form ~ ( ~ , where ~ is an entire function of exponential type. Is there an analog of the result of [2] in this case? For entire functions f of c.r.g, of order P, P < I, the answer is given by the following theorem. THEOREM. tion

@(~)~

Let f be an entire function of c.r.g, of order P, P < I.

Then:

has e.r.g, on all the rays except possibly on those for which hf(0)

solution ~ of the equation in the entire plane.

@(~))~=~

= 0; 2) every

in the class of entire functions of order P$ has c.r.g.

This theorem follows directly from the above-formulated ing lemma.

result

[2] and from the follow-

LEMMA. Let f be an entire function of order p, P < I, and let ~ of exponential type. We have the asymptotic equality

C,D) (z) = where m is the multiplicity

I) the func-

be an entire function

[e~(0), o(1)] ~ o, oo,

of the root of the function

cp at the point z = 0, while the sym-

0

bol z § ~ means that z § ~ outside some C~ For the proof of the lemma one needs a statement which is a simple consequence of Theorem 2 of [4]: let g be a meromorphic function of order 0, O < I, and let R be a fixed number, R > 0; we have, uniformly with respect to q, lq] < R, the relation g'(z + q)/g(z + q) § 0 0

(z § ~), where the exclusive C~

does not depend on q.

F r o m this statement and from the equality

o

it follows that we have, uniformly with respect to q,

iqJ < R,

q,CZ+'~)I~(z)----~ CZ-S-'*oo). (Basically,

(I)

this relation has been obtained also in [4, p. 414].)

Assume that f and ~ satisfy the assumptions of the lemma. We denote by ~ the function associated to ~ according to Borel and by F we denote an arbitrary circumference, containing the set of the singularities of the function ~. From the equalities *I. V. OSTROVSKII. Physicotechnical Institute of Low Temperatures, Academy of Sciences of the Ukrainian SSR, Pr. Lenina 47, Khar'kov 164, 310164, USSR. tin [2] there is given a more precise, exhaustive characterization of the set of rays on which the c.r.g, may be lost. SThe set of these solutions is not empty by virtue of A. O. Gel'fond's well-known theorem [3], p. 359.

2289

~Pr and from the fact that

~k)(o)=0

= ~

~,...)

q2(z:)~,~cL< ck = o, t,

(k = 0, I ..... m -

(2)

I), it follows easily that

p

where

~ m-o!

]/

(z). Since 4

0

C(z)

'

applying (I) [with g(z) = f(m)(z)], we obtain that Gm(z , ~) § ~m/m! (z ~ ~), uniformly relative to ~ ~ F. Taking into account (2), we obtain the assertion of the lemma. For entire functions f of order p, p ~ that of [2] will be discovered as simply as the following example. Let ~(t) = tn -- I, type o, o < I/2, which is not of c.r.g. We (~)6

=~

I, one in the n ~ 3; denote

cannot expect that a result similar to case O < I. This can be seen already from let g be an entire function of exponential by G the solution of the equation

in the class of the entire functions of exponential type o ( s u c h a s o l u t i o n exists

even by T h e o r e m 4 of

[3], p. 36 ).

We set

q n

k=O

function of exponential

type and of c.r.g, with a positive

~

.

Then f is an entire

~

indicator, while the function

~(~)~ = ~ does not have c.r.g. For functions f of exponential

type the following conjecture

is likely to hold.

Conjecture. Let f be an entire function of exponential type and of e.r.g, and assume that ~ does not have zeros at those points of the boundary of the conjugate diagram of the function f (if these exist) which are common endpoints of two segments lying on the boundary of the diagram. Then the function ~(~))~ is a function of c.r.g. In particular, if the function ~. does not have zeros at the boundary of the conjugate diagram of f, then ~(~)~ is of c.r.g. For functions f which have a growth greater than of exponential type, the situation must be more complicated since, in particular, the solutions of the homogeneous equation @(~))~ = 0 in this class of functions may have an irregular growth. Also the following problem remains unsolved. Let f be an entire function of c.r.g, of order p, p ~ I, and let ~ be an entire function of exponential type. Are there solutions of the equation ~(~))~=~ , which are entire functions of c.r.g, relative to the same proximate order as f? In the case when ~ is a polynomial, it is easy to obtain an affirmative answer with the aid of the integral formula for the solution and of the result in [2]; however, the general case is not obtained from here by a limiting process. LITERATURE CITED I ~

2.

3.

4.

2290

B. Ya. Levin (B. Ja. Levin), Distribution of Zeros of Entire Functions, American Mathematical Society, Providence (1964). A. A. Gol'dberg and I. V. Ostrovskii, "On the derivatives of the primitive functions of entire functions of completely regular growth," Teor. Funkts. Funkts. Anal. Prilozhn., No. 18, 70-81 (1973). A. O. Gel'fond, Calculus of Finite Differences [in Russian], Nauka, Moscow (1967). A. J. Macintyre and R. Wilson, "The logarithmic derivative and flat regions of analytic functions," Proc.. London Math. Soc., 47, 404-435 (1942).