Abstract. Let f (z) be an entire function of order less than 1/2. We consider an analogue of the Wiman-Valiron theory rewriting power series of f (z) into binomial series. As an application, it is shown that if a transcendental entire solution f (z) of a linear difference equation is of order χ < 1/2, then we have log M (r, f ) = Lr χ (1 + o (1)) with a constant L > 0.

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ABSTRACT. It is shown that transcendental meromorphic solutions f(z) of thewhere 0 < |c| < 1 is a complex number and aj(Z), j = 0,1,..., n, and Q(z) are rational functions with ao(z) ^ 0, an(z) = 1, satisfy T(r,f) = 0( (logr) 2 ) and (logr) 2 = 0{T(r,f)). Moreover, in the case n = 2 and Q(z) = 0, necessary and sufficient conditions for the existence of solutions are given.

We consider the second order equation f'+(e^{P_{1}(z)}+e^{P_{2}(z)}+Q(z))f=0 , where P_{1}(z)=\zeta_{1}z^{n}+\ldots , P_{2}(z)=\zeta_{2}z^{n}+\ldots , are non-constant polynomials, Q(z) is an entire function and the order of Q is less than n . Bank, Laine and Langley studied the cases when Q(z) is a polynomial and \xi_{2}/\xi_{1} is either non-real or real negative, while the author and Tohge studied the cases when \xi_{1}=\xi_{2} or \xi_{2}/\xi_{1} is non-real. In this paper we treat the case when \zeta_{2}/\zeta_{1} is real and positive.