1. Let f(z) be an entire function of order p (0p as r->c°, (1.3) rp'(r) log r->0 as r-»oo, (1.4) log Af(r,/)gfW forr^ro = rp(r) for a sequence of values of r. S. M. Shah [2] has proved the existence of a function X(r) for an entire function of lower order X (0^X<) analogous to p(r), having the following properties: (2.1) \(r) is a non-negative, continuous function of r for r = r0. (2.2) X(r) is differentiable except at isolated points at which X'(r-O) and X'(r+0) exist. (2.3) X(r)->Xasr->«.

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... ) r\'(r) log r-K) as r->co. (2.5) log M(r,f)^r^iorr^ra = rx(r) for a sequence of values of r. 3. In this note we prove a number of results applying the properties of X(r) and p(r). In what follows we shall take 0 0 for r = r". dr With the usual notations of log M(r, /), n(r, a) and N(r, a) we prove the following theorems: