TH E P R O X IM A T E T Y P E A N D ITS REFER E N C E TO A N A L Y T IC FU N CTIO N S

H.S. K a s a n a

(Received November 1987, revised June 1988)

1. IntroductionTo obtain a more refined measure of growth than is given by the

proximate order [4] we consider a positive function T(r) in 0 < r < 1 having the properties:

where T '(r) denotes the derivative of T (r). Such a function T(r) is called the proximate type.

The outlines of this paper are: We start with proving an approxima-

of this section are some more asymptotic properties of these compari­son functions. In the next section proximate type is constructed for a class of analytic functions. Lastly, we obtain lower and upper bounds for (1—r)7v(r )/T (r ) in reference to growth parameters of analytic functions.

2. A sym ptotic Properties

Theorem 1. For every continuously differentiable proximate type T(r), there exists a twice continuously differentiable proximate type S(r) such

(i) T (r) —► T as r —► 1, 0 < T < oo;

tion theorem for arbitrary proximate types and the remaining results

that(2.1)

andT (r )~ S (r ) as r —♦ 1. (2.2)

P roof. Let us assume that S(r) be a proximate type and coincide with T(r) on the sequence {rn} in [0,1) as

T (r„) = S (r„), rn = 1 — n = 0 , l ,2 , . . . (2.3)

Research was supported by U.G.C., New Delhi, India.

Math. Chronicle 19 (1990), 35-43

Indeed, in this case, for r lying in the intervals |r„,rn+i)

log T(r)S(r) -V..

- !£•(£) =0(l0gr^ ) =0

r M s '(x )T (x) S(z)

dx\

}dx

( 1 ) as 1.

Hence T(r) = 5 (r)c 0̂ 1̂ , which implies (2.2).

Thus it suffices to construct a twice continuously differentiable prox­imate type S(r) satisfying the conditions (2.1) and (2.3). Define the functions on the interval [0, f ] :

0 < t < l

? < < < * .34 >

and1K * )= / ^(0 ^ -

Jo

Since ^(f) is continuous on [0, f ], it follows that is continuously differentiable on this interval. We also note:

(a) 0 = V(0) = V ( f ) = ^ ( 0) = V»' ( f ) -(b) 0 <

(c) |V»V)I < \-

(d) l } W x )d* = 6 > 0.Consider a sequence {en} defined by

_ lo g (T (r „+ 1) /T ( r „ ) ) -6

Since

it follows that

r ( r )

T([ r ) _ J 1 \ r) V l - r J ’

or

l° g i f e y i= 0 ( l0 g 4 ) -

Hence en —♦ 0 as n —♦ oo (choose 6 = log 4). Finally, we define

log S(r) = log r (rn) + J ' v ( i r i r ) * <2 4)

on the interval [rn,rn+i].The verification of the properties (i) and (ii) and derivation of (2.1)

for the positive function S(r) in (2.4) is left to the reader.

Theorem 2. Let T (r) be a proximate type. Then, for p(0 < p < oo) and T(0 < T < oo),

(a) exp{(l — r)“ pT (r)} is monotonically increasing for r > ro.(b) exp{(l - r - p)T(r + p) + pT } ^ exp{(l - r )T (r)} as r -* 1.

P roof, (a) We have

For T > 0, the condition (ii) may be replaced by (1 - r)T '(r) —► 0 as r —► 1 (Actually, for finite and positive T these conditions are equiv­alent). Thus, we obtain asymptotically

A [ e x p { ( l - r ) - 'T ( r ) } ] = exp{(l — r) pT (r)}(1 - r)H*

(pT(r) + ( l - r ) r ( r ) ] .

In case, T = 0, (1 — r) lT(r) —► oo as r —► 1 and

(b) For T > 0, consider the function

L(r) = exp{(l - r)(T (r) - T )}. (2.5)

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Thus, for all values of r sufficiently close to 1,

which further implies

^ - 0(1) L(r) ~ ° (1)'

f t k,g £ w : i = 0- (2 6 )

In case, T = 0, (2.5) is reduced to

L

i (' ( r) T , r ) f ( l - r>r(r> l l^ ) - T ( r )\ ^ F ) ‘ j -

In view of properties (i) and (ii), again L'(r)/L(r) —► 0 as r —» 1 and ultimatly, (2.6) is available which means

L(r + p) — L(r) as r —* 1.

This immediately corresponds to (b).

3. Construction and BoundsLet f (z ) be a function, analytic in U — {z €C : \z\ < 1} having order

P(0 < p < oo) [5] and type T(0 < T < oo) [3] and satisfying in addition (i) and (ii). Then for a given /?(0 < (3 < oo), T(r) satisfies also:

(iii) T(r) is continuous and piecewise differentiable for r > ro; and(iv) limr_ i sup M (rtf ) = max|,j=r \f(z)\.

Now T '(r) in (ii) can be interpreted as T ;(r+ ) or T ^r—) whenever these are unequal and the comparison function T (r) is called the prox­imate type of the given analytic function f (z ) . Obviously, proximate type of an analytic function is not uniquely determined. For example, if we add 7 /(1 — r)~p, 0 < 7 < 00 in the proximate type T(r) we, again obtain a new proximate type for the same analytic function and the corresponding value of /9 is divided by e7.

In this section we investigate a problem proposed by Juneja and Kapoor [1, page 66] in a more generalized context than they enunci­ated. Lastly, upper and lower bounds for (1 — r)T,(r)fT (r) are obtained which forecast the sufficient conditions for the existence of (ii) for a class of analytic functions.

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By the Hadamard three circle theorem, we know, if f ( z ) is analytic in U, log A /(r ,/) is an increasing convex function of logr in 0 < r < 1. Moreover, it has the representation

logAf(r,/ ) = logAf(r0, / ) + f — — t/z, 0 < r0 < r < 1, (3.1)Jto x

where w (x ,f ) is a positive,continuous and piecewise differentiable func­tion of x.

To prove the main results we need the following:

Lemma 1. For a function f , analytic in U and having order p and lower order we have

lim inf ^ ^ ^ P ^ SUP "̂i— (3-2)r—*i r logA /(r ,/ ) “ r-*i r lo g A /(r ,/)

P roof. Let be the set of extended positive real numbers. Then, for A € R+U {0), we define

u (r ,f ) _ r'-̂ i K r (l — r)

For A = 0, p = 0. On differentiation (3.1) givesM'(r, f ) _ u;(r, f )M (r ,/) r

The expression (3.3) in conjunction with (3.4) is rewritten as(1 - r )M '(r , /)

lim SUP T77— 771---- TTt—r- 1 M (r, / ) log M (r, / )For given e > 0 and r > r0(e),

M '(r , / ) < ^ + 6

lim sup V = A. (3.3)* — f* 1 *

(3.4)

A /(r , /) lo g M (r ,/) 1 - r Integrating above inequality, we get

log log Af(r, / ) < 0(1) + (A + e)log(l - r )” 1. Passing to limits, we have

_ < Jim sun ( l j “ r M r . / )— r—l r log AT(r, / ) ’

which obvious holds for A — oo. Likewise, for lower order A,

— r— l r log M (r ,/)Combining these two inequalities (3.2) is immediate.

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Lemma 2. / / f (z ) is a function, analytic in U having order />(0 < p < oo), type T and lower type t then

lim inf ~r < pt < pT < lim sup , —r>/ " . . (3.5)r-*i r (l - r) - ^ - 1 ~ r ~ r r-»i r (l — r)“ _̂ l v '

P roof. The proof of this lemma proceeds exactly on the lines of Lemma 1 and hence details are omitted.

For the definitions and related work on lower order and lower type we refer to Kapoor [2] and Kapoor and Gopal [3].

Definition. An analytic function f ( z ) is said to be of regular growth if 0 < A = p < oo and further, it is of perfectly regular growth if t = T.

Functions which are not of regular growth are called of irregular growth.

Lemma 3. The lower type of an analytic function of irregular growth is zero.

Proof. If f (z ) is of irregular growth than p > A > 0. We know

lim in f! E E ^ M = A.r—i log( 1 - r) " 1

Since A /(r) - + o o a s r - > 1, log+ may be replaced by log. FYom above definition, for given € > 0 and r > r„(e),

l o g M ( r , / ) > ( l - r ) - * + ‘ , (3 .6 )

whereas for a sequence of values of r tending to infinity

l o g M ( r , / ) < ( l - r ) - A-* . (3.7)

Dividing (3.6) and (3.7) by (1 — r)~p and then proceeding to limit the argument shows that

lim inf - rr— = 0. r- 1 (1 — r)~P

From Lemma 3, we conclude that t > 0 is only limited to the study of analytic functions of regular growth. In such case we define

log M (r, / ) ( i _ r)lim inf \L\

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The quantity t\ is termed as A-type of the function /(* ) . It is sig­nificant to mention that there exist analytic functions for which t\ is nonzero and finite. For such analytic functions, we shall utilise the com­parison function analogous to proximate type as A-proximate type 5a(r). The occurrrence of S\(r) is justified in Theorem 4.

Theorem 3. Let f ( z ) be a function analytic in U having order p(0 < p < oo) and type T (0 < T < oo) such that limits in (3.2) and (3.5) exist. Then, for a positive real number 0, log(/?- 1A f(r , / ) ) / ( l — r)~p is a proximate type of f (z ) .

Proof. For a given constant (3(0 < (3 < oo), let

w n ’ y . y (3.8)

Since log M (r, f ) is positive, continuous and increasing function of r for r > ra > 0,which is differentiable in adjacent open intervals, it follows that Sp(r) satisfies (iii). Existence of limits in (3.5) implies that f ( z ) is of perfectly regular growth and moreover, Sp(r) —► T as r —» 1.

Differentiating (3.8), we have

_ M '(r , / ) p

so that

5 ,(r ) M (rtf ) log(/?“ 1M (r, / ) ) 1 - r ’

(1 - r)5^(r) = ( 1 - r M r , / ) _ 5/»(r) r log (P~1M (r ,f ) )

Again, limis in (3.2) exist by assumption. Hence

—► 0 as r —► 1.(1 — r)5p(r)

Sp(r)

Thus 5p(r) satisfies the assertion (ii).From (3.8), the assertion (iv) is readily obtained. In this way all the

assertions for Sp(r) to be a proximate type of f (z ) are satisfied and hence the theorem.

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Theorem 4. Let f (z ) be a function, analytic in U and having order p, lower order A(0 < A < p < oo), type T and lower type t.

Then

* < lim sup — + P < y, (310)

wherelim 8“ P (3.11)r—l inf r (l — r)~p~l d

Moreover, if f (z ) is of irregular growth then

- o o < lim inf < - - A, (3.12)“ r-*i S(r) ~ tx

where S\(r) is a function in (3.8) corresponding to A and t\ is the A-type o f f (z ) .

Proof. In view of (3.1) and the definition of type T and lower type t we observe that

lim 8“ P — (3.13)r —»I inf (1 - r ) - ' Jr. * <

Similarly, for analytic function of irregular growth,

(»•“ )

Fix r0 6 [0, oo) such that /? = log M (rot / ) . Hence

l o g ( r ‘ M ( r , / ) ) = r ^ r ^ d x .J r 0 X

Dividing by (1 — r)~p and then differentiating with respect to r, we get for almost all values of r > r0,

S'p(r ) _ u ( r , f ) __________ p _

Sf ( f ) r £ : f e £ l , i x 1 - r

and this, on simplification, gives (3.9).Consequently, proceeding to limts in (3.9) and making use of (3.11)

and (3.13), the inequalities in (3.10) follow at once.In case p > A, we have

o - I # 1# . / ) ) _ 1 rSAr) - (1 - r)-A - ( T T T P X .

By the parallel arguments and making use of (3.14), (3.12) can be disposed of.

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R eferen ces

1. O.P. Juneja and G.P. Kapoor, Analytic Functions-Growth Aspects, Research Notes in Mathematics:, Pitman Advanced Publishing Pro­gram, Boston, London and Melbourne, 1985.

2. G.P. Kapoor, On the lower order of functions analytic in a unit disc, Math. Japon. No.l, 17 (1972), 49-54.

3. G.P. Kapoor and K. Gopal, Decomposition theorems for analytic functions having slow rates of growth in a finite disc, J. Math. Anal. Appl. No.2, 74 (1980), 446-455.

4. H.S. Kasana, Existence theorem for proximate order of analytic functions, Ukrain. Mat. Zh. 40 (1988), 117-121.

5. G.R. Maclane, Asymptotic Values of Holomorphic Functions, Rice University Studies, Houston(Texas), 1963.

H.S. Kasana, University Roorkee, Roorkee-247667(U .P.), INDIA.

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