Series I: COM MENTATION ES MATHEMATICAE XXIX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXIX (1990)
H. S. Ka s a n a (Roorkee, India)
The generalized type of entire functions with index-pair (p, q)
Abstract. The generalized (p,q)-type of an entire function f(z) = £*= i anzkn (0 ^ < ... < Xn < Àn + l. . . , À„ -> oo as n —►oo) is defined and its coefficient characterisation is obtained. As a supplement, a result on Hadamard composition of entire gap power series is also given.
0. Introduction. Let f( z ) = i anzXn (0 ^ < . . . Я„ < Àn+t < ..., Я„
-► oo as n -*■ oo) be an entire function. Set M(r) = M (r,/) = шаХ|г| =г \f(z)\ and ц(г) = n ( r , f ) = maxn>1 {\an\rXn}, M(r) and p(r) are called maximum modulus and maximum term of f(z), respectively.
The concept of (p , g)-order g#d (p, <?)-type of f{z) having index-pair (p, q), p ^ q ^ 1, was introduced by Juneja et al. ([1], [2]). Thus f(z) is said to be of {p, g)-order q if it is of index-pair (p, q) and
log IriM(r)
bmsup = ^(P> 4) = в .
and the function f(z) having (p, g)-order g (b < q < oo) is said to be of {p, <?)-type T if
log[p_ 11M (r)
s“p (iogf-Чг)» = r (P> «) s r - 0 « T ^ œ,
where log[p] x stands for the pth iterate of log x {x is large enough such that 0 < log[p]x < oo) and b = l if p = q and b = 0 for p > q.
The growth of a function/(z) can be studied in terms of its (p, <?)-order and (p, <?)-type, but these concepts are inadequate to compare the growth of
1980 Mathematics Subject Classification. Primary 30D15.
K ey words and phrases. Proximate order, generalized type.
Research supported by University Grants Commission, New Delhi.
21 6 H. S. K a s a n a
those functions which are of the same (p, g)-order and of infinite ip, g)-type.
For a refinement of the above scale, we may utilize the concept of proximate order ([3]) defined as
De f i n i t i o n. A p o s i t i v e f u n c t i o n ç(r) d e f i n e d o n [ r 0 , oo), r0 > e x p [ 9 - 1 ] 1 , i s c a l l e d a proximate order o f a n e n t i r e f u n c t i o n o f (p, g ) - o r d e r q {b < q < oo) i f
(i) Q(r) dip*
q)
= Q as r -► oo;(ii) Л[4?](г) £>'(r)-+0 as r -+ oo, where e'(r) denotes the derivative of g(r) and, for convenience, A[q] (x) = fl?= 0 log™ x.
We now define generalized (p, q)-type T* of f(z) as loglp-1i M (r)
(0.1) Urnsup^ gi,--TT-j^7 = T*(P> «) = T*’ 0 « T* < “ •
The comparison function q(r) is called proximate order of the given entire function /(z), if the quantity T* is nonzero and finite. Clearly, the proximate order and the corresponding generalized (p, <?)-type of the given function are not uniquely determined. For example, if we add c/logl4] r (0 < c < oo, q = 1, 2, 3 ...) to g{r), we obtain a new proximate order for the given function and the generalized ip, q)-type T* is divided by ec.
In the present paper we obtain generalized ip, q)-type T* in terms of coefficients and exponents of the series Jÿ* t an zXn. We have also shown that an entire gap power series with index-pair ip, q) and its Hadamard composition have the same ip, ^-growth parameters and comparison function.
1. Auxiliary results. In this section we give some auxiliary results in the form of lemmas which will be used in the sequel.
Lemma 1. Let g(r) be a proximate order of an entire function of ip, q)-order g ib < q < oo). Then (log[<*~1]r)t?(r)_'4 is a monotonically increasing function of r for r > r0, where A = 1 if ip, q) = (2, 2) and zero otherwise.
P ro o f. We have
— [(log[« -1] г)<Кг,~л] = (log[«~1] r)e(r)~A dr
For given e > 0, using the properties of the proximate order, it follows that for sufficiently large values of r
d По2 1 < ,-1 ]гУ?(г) -а
— [(log[e A~\ > (g(r) — A — e)--- ---—----> 0, 0 < е + Л < (). ■
dr Aiq-i](r)
Now, we define a function 4>(t), assumed to have a unique solution for t > t0, such that
(1.1)
g'(r)logMr+ Д 1.
A \a- 1] 0*)J
t = (\oglq~1]r)e(r)~A log19 1]г = Ф(г).
Le m m a 2. Let Ф(Г) be a function defined as in (1.1). Then
(1.2) lim 0 < ц < oo.
*(*)
P ro o f. By simple differentiation, we observe that
dl\og0(t)'] d [log1*1 r] _________ 1_________
d l logt] <f[{e(r)-4}logtolr] {е(г)-Л} + Л[9](г)р'(г).
Passing to limits, we get
( 1 .3 )
</[ 1оё Ф(р] 1 d llogt] <?-Л’
After little calculation and taking limits, (1.2) follows. ■
Le m m a 3 . Letf{z) b e an entire function with index-pair (p, g ) and T* be the generalized {p, q)-type. Then
( 1 .4 ) lim sup
r~* 00
logfp 1]p(r) (log[9_1]r)<?(r)_ y*
Using a result due to Valiron ([4], Theorem 11, p. 32), this lemma can easily be proved. So for conciseness the proof is omitted.
2. Main results
Th e o r e m 1. Let f(z) = ! an zXn be an entire function with index-pair (p, Q)i (p, q)-order q (b < q< oo) and proximate order g(r). Then generalized (p, q)-type T* of f(z) is given by
(2.1) lim sup
n~*00
Ф(1оё[р~2] Xn) log[*-2] {—(1/Л„) log|a„|}
Г* • лГ’
where A has the same meaning as in Lemma 1 and N is defined by
N = N(p, q) =
t e - i r 7 e c if(p ,q ) = ( 2,2),
1 /(Qe) if (p, q) = ( 2 , 1 ) ,
1 for all other index-pairs (p, q).
P ro o f. From (0.1), for a given s > 0 and r > r'(e), log M (r) < explp~ 21 {(Г* 4- г) (log1* - 11 r ) ^ } . where exp[p] x is the pth iterate of e*.
Using Cauchy’s estimate, we have for sufficiently large r, (2.2) log|a„| < expfp-2] {(T* + e)(log[*-11r)e(r)} — A„logr.
218 H. S. K a s a n a
Choose r such that
(2.3) (log*-11!-)*’- '4 = — ■!— -logl'’" 21^ . v & 1 (Г* + е) & q
For (p, q) Ф (2,1) and (2,2), the expression (2.3) implies (1о„й - " r)»w = — - logt'- 21
' 6 (T*+e) g
if and only if
Ф So (2.2) is reduced to
Ф а о ^ - ^ я .)
log1* 21{(l/À„)log|a„| *}< Ф О о ^ Л ,,)
$,(W^logIP_2,7){1+o(1»
Passing to limits and using (1.2), we have (2.4) lim sup <paogü,‘ 21^ )
log1* 11 |a„| 1;1"
For (p, q) = (2, 2), (2.3) becomes (lo g r ) ^ " 1 = Hence (2.2) is reduced to
( T * + e ) g
^ T * , 3.
log r = Ф
q(T* + s)
j-log|flJ < )logr, or
ф { К )
< Q*{K)
I O ! k r '"
On taking limits and applying Lemma 2, it follows that
(2.5) lim sup
log \an\ 1
^ qT*
Ш -
Again, for (p, q) = (2, 1), (2.3) yields
;
Q(r) _ _____
r ' =
(T*+£)Q <=> r = Ф к
(T * + s)q
Thus, in view of (2.2), we may similarly have
(2.6) lim sup H K ) ^ geT*.
\ a n \ 1 / A "
Combining (2.4), (2.5) and (2.6), we conclude that for any index-pair
<P(log^-2U„
(2.7) lim sup
q — A J , *
« 1 Г log[« !] |a j
To prove the reverse inequality, let (3 be defined by the equation lim sup Ф(1оё[р-2,Л„)
loglq 2]\ - ~ \ o g \ a n\
q-A I N ‘
Then, for every /?г > f3 and for all n > n0 (e), logjaj < -A„exp[« 21
h ) Ф (1 о ^ -2и„) From above (in view of Lemma 2), it follows that
(2.8) log n (r) ^ max 1
For (p, q) Ф (2, 1) and (2, 2), using (1.3) it can easily be seen that the maximum value on the right-hand side is attained for
К = expь - v U jlogt"- log
A„ log r — Xn exp[9 2] Ф \ ~ 1 о ^ - 2Ч„
Thus, for r sufficiently large, we get from (2.8),
(2.9) log[p 13 n(r) < [log[q 2i( g ( l+ g) Mogr)]*00 (log[«"1]r)e(r) "" 1 (log[<J~1] r)e(r)
Consider when (p, q) = (2, 1). Let Xn — ($±{ге lle)e{r)/N, equation (2.8) is then reduced to
(2.10) logp(r) /?. e
reir) qN
Again, if (p, q) = (2, 2), in order to get the maximum value on the right hand side of (2.8), Xn is given by
220 H. S. K a s a n a
which reduces (2.8) to
(2.11) log^(r) ^ / g - i V <r) 1 (logr)e(r) Nq\ Q J
Taking limits in (2.9), (2.10) and (2.11) separately and combining the outcomes so available (in view of (1.4)), we get
T * ^ Pi-
Since the inequality (2.12) holds for every pt > ft, it follows that T* ^ p.
This with (2.7) proves the theorem. ■
Co r o l l a r y 1. Let f(z) = £®=1 anzXn be an entire function of index-pair (p, q). I f 0 < V < oo, the function f{z) is of {p, q)-order q (b < q < оо) and (P, qftype T (0 < T < со) if and only if T = NV, where V is given by
loafp-2] j
<2ЛЗ) U^ P ( l o g - ‘> |a J-‘' V - = ViP’ q) " K
For g(r) = g and <P(t) = tlliQ~A), this corollary is the immediate con
sequence of Theorem 1. This corollary is stated as Theorem 1 in a paper of Juneja et al. ([2]).
Th e o r e m 2. Let f (z) = x an zXn be an entire function with index-pair (p, q) and {bn} be a complex sequence satisfying lim„_cc (log \bn\)/Xn = 0. Then the Hadamard composition g (z) = i an bn zXn is an entire function with the same (p, q)-order and generalized (p, qftype as that of f(z).
P ro o f. Since \b„\~llXn -> 1 as n->oo, it follows that
lim inf|a„bJ_1/An ^ lim in f|aJ~ 1/AM im inf|bn|_1/An = oo.
n-> 00 00 n~+ QC
Thus g(z) is an entire function.
It is known ([1], Theorem 1, pp. 61-62) that f{z) is of (p, q)-order gf if, for a pair of integers p ^ 2, q ^ 1,
(2.14) where
Qf = Qf (P, q) = Pi(L(p, q)),
r Pi(L{p, q)) = <
L(P, q)
1+L(p, q) m ax(l, L(p, q)) _oo
if q < p < oo, if p = q = 2 ,
if 3 < q = p < oo, if p = q = oo;
L(p, q) = limsup n-»oo
log1' - 11 A, log1’ *4 “ logW
Hence to claim that/(z) and g{z) have same (p, q)-order, it is sufficient to show that L(p, q) is same for both entire functions.
For given e > 0 and n > п0(в),
(2.15) ~ E A n< log \Ьп\ < &X„.
Also,
(2.16) -lo g |a ,A I 1 lo g kl 1 [ 1 log k l 4 К 1 lo g k r M By using (2.15) in (2.16), we get for sufficiently large n,
logw K , r 1/A" m log lq]\anbn\~ilAn logM|a „ r 1M‘
l o g " '11 A, ° ( , < lo g " -‘U„ < log"-4A„ + o(l).
On taking limits in above, we find that L{p, q) is same for /(z) and g{z).
Let T f(p , q) and Tg*(p, q) be the generalized (p, ^f)-types of/(z) and g{z), respectively. Again, we observe that
(2.17) (1—£)|а„|~1/я" •< \a„bn\~1/Àn < (1 +е)|а„|~1/Яп.
Since <P(t) is increasing, and taking (2.17) into consideration, we find that for sufficiently large n,
log[<* 11 [flj 1M" log[g l]\a„b„\ llXn \oglq l]\a„\ 1/A'
<P(log[p_2U„) ° < <P(\oglp~24 n) < 0(logIp-2]A„)
Considering (2.1) for /(z) and g(z) and taking limits in above, we get
v ) = v w
which implies Tg*(p, q) = 7}*(p, q). ■
Co r o l l a r y 2. The functions f( z ) and g{z) as defined in Theorem 2 have the same proximate order and (p, q)-type.
In view of Corollary 1, it follows that (p, q)-types of/(z) and g(z) are same and thus comparison functions of /(z) and g{z) coincide.
Acknowledgements. The author is thankful to the referee for making substantial improvements in the paper.
2 2 2 H. S. K a s a n a
References
[1] O. P. J u n e ja , G. P. K a p o o r , S. K. B a jp a i, On the (p , q)-order and lower (p , q)-order o f an entire function, J. Reine Angew. Math. 282 (1976), 53-67.
[2] —, —, —, On the {p, q)-type and lower {p, q)-type of an entire function, J. Reine Angew. Math.
290 (1977), 180-190.
[3] K. N a n d a n , R. P. D o h e r e y , R. S. L. S r iv a s ta v a , Proximate order o f an entire function with index-pair (p, q), Indian J. Pure Appl. Math. 1, 11 (1980), 33-39.
[4] G. V a lir o n , Lectures on the General Theory o f Integral Functions, Chelsea Publ. Co., New York 1949.
DEPARTMENT O F MATHEMATICS, UNIVERSITY O F ROORKEE ROORKEE, INDIA
