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We say that an entire function $f\left(z\right)={\sum }_{k=0}{a}_k{z}^{n_k}(0=n_0\<n_1\<n_2\<...)$ has Fejér gaps if ${\sum }_{k=1}^{\infty }1/n_k\<\infty .$ The main result of this paper is as follows: An entire function with Fejér gaps has no finite deficient value.

The purpose of this paper is to establish a theorem which answers a conjecture of Paley on the distribution of values of Hadamard lacunary series and which is useful to study the Peano curve property of such series.