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Abstract. In this paper, by using the q-difference analogue of lemma on the logarithmic derivative lemma to re-establish some estimates of Nevanlinna characteristics of f (qz), we deal with the value distribution and uniqueness of certain types of q-difference polynomials.

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In this paper, we investigate the delay differential equations of Malmquist type of formwhere R(z, w(z)) is an irreducible rational function in w(z) with rational coefficients and a(z) is a rational function. We characterize all reduced forms when the equation ( * ) admits a transcendental entire solutions with hyper-order less than one. When we compare with the results obtained by Halburd and Korhonen[Proc.Amer.Math.Soc., forcoming],we obtain the reduced forms without the assumptions that the denominator of rational function R(z, w(z)) has roots that are nonzero rational functions in z. The growth order and value distribution of transcendental entire solutions for the reduced forms are also investigated.