BOUNDED INDEX , ENTIRE SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS AND SUMMABILITY METHODS

A brief survey of recent results on functions of bounded index and bounded index summability methods is given. Theorems on entire solutions of ordinary differential equations with polynomial coefficients are included.

The least such integer N is called the index of f (see Lepson [30], Shah [40]). DEFINITION 1.2. An entire function f is said to be of bounded value distribution (b.v.d.) if for every r > 0, there exists a fixed integer P(r) > 0 such that the eouation f(z) w has never more that P(r) roots in any disc of radius r and for any w e (see Bayman [16,17]).
A survey of the properties of functions of b.i., and of b.v.d., and a list of references published up to 1975 and some up to lO76, are given in [40].
In Section 2 we ive some extensions of these concepts to meromorphic functions [3]. If an entire function is of b.i. N, then its growth is (I; N + I) ( [17], [37], [13]). In Section 3 we study extensions of (I.i) suitable for entire functions of finite order. Here we show, following Bennekemper [19], that  It is known that if f is of b.i., then it is of exponential type ( [17], [13]) but Functions of exponential type need not be of b.i. In fact there exist functions of exponential type and having simple zeros and of unbounded index [39].
The followin theorem gives a necessary and sufficient condition for an entire function of exponential type to be o. b.i. THEOREM 2.1. (ricke [i0]). Let f be an entire function o exponential type. (If f is entire we may consider R(z) to be constant.) For entire functions and for R(z) C, Hayman [17] showed that the above condition is equivalent to bounded index. DEFINITION 2.5. A function f meromorphic on is said to be L.D.I. (logarith- for all z where D is the set of zeros and poles of f and L(z) is a decreasing function with respect to the distance of z to and L(z) > i.
Using the above definitions, Beauchamp [3]). Lt f be D.I. Then f is of order not exceeding 2 and finite type.
In [3] Beauchamp also examines and obtains similar results for functions meromorphic on the unit disk. Here R(z) and L(z) depend not only on the distance to the zeros, respectively zeros and poles, but also on the distance to the boundary of the unit disk.

BOUNDED INDEX CONCEPT FOR FUNCTIONS OF FINITE ORDER.
It is known that if f, entire on , is of order > i or of order i and maximal type then the growth rate of the derivative may be larger than that of the function (Shah [36], Vijayaraghavan [46], KSvari [28]) and so inequalities of the type (i.i) This definition is an extension of (I.I) to entire functions of finite order.
If f is of b.i. N then f is of growth (i, N+I) ( [40]). Here we have THEOREM 3.2. (Beauchamp [3]). If f is y-b.i, satisfying (3.1) then f is of N+I growth (y + i, ).
Another extension of (i.i) is as follows: [19]). An entire function f is said to be of bounded m-index N if R and N is the smallest integer such that for all n, This definition is a slight variation of the one given by G. Frank [6], and is used by Hennekemper to prove Theorem 3.4 below.
(ii) The following example shows that the hypothesis, in Theorem 4.9, that all solutions are entire is necessary. zw" + (z2 z 1/2)w' (z )w 0 Here (4.7) is satisfied. One solution Wl(Z) e z is entire but the second solution is not entire, and the conclusion of Theorem (4.9) does not hold.

BOUNDED INDEX AND SUMMABILITY [ETHODS.
We begin with definitions and notations. A sequence X  [12]). If [12]). Let f be of b.i.  [12]). Let f be of b.i. If either A(f,z i) or A'(f,z i) is anmethod then A'(f,z i) is an E-E method.
In a recent paper and its corregendum ( [43]) Sridhar further examines the A(f,z i) matrix transformation and obtains results which can be summarized as follows.
THEOREM 5.5. (Sridhar [43]). Let f be of b.i. Then the following are equivalent.  n Azpeitia [i] considers entire functions f(s) and proves that if f(s) is of bounded index N, then it reduces to an exponential polynomial. Bajpai [2] replaces the condition of b.i. of f(s) by four conditions and proves that if any one of these four conditions is satisfied then f(s) reduces to an exponential polynomial. Gross [15] and Shah and Sisarcick [41] have considered similar conditions for functions defined by Taylor series.
He also obtains a necessary and sufficient condition for f(z) to be of b.i.
A similar theorem for a function of one variable is due to Fricke [9].