© Abdullah Alotaibi. 2010
Received: 1 July 2010
Accepted: 12 September 2010
Published: 15 September 2010
We observe that all the above results concern the order of growth only. In this paper, we are going to prove the following theorem which concerns the exponent of convergence.
has zeros with infinite exponent of convergence.
3. Some Lemmas
Throughout this paper we need the following lemmas. In 1965, Hayman  proved the following lemma.
In 1962, Edrei and Fuchs  proved the following lemma.
In 2007, Bergweiler and Langley  proved the following lemma.
Let be a transcendental entire function of order . For large define to be the length of the longest arc of the circle on which , with if the minimum modulus satisfies . Then at least one of the following is true:
We deduce the following.
4. Proof of Theorem 2.1
Lemma 4.1 is proved.
Let be small and positive and suppose that or is a polynomial. Let be large with and . Since is small it follows from (4.2) and (4.8) that . Choose with such that the intersection of with the ray given by is bounded. Applying Lemma 3.4 to the function , with a subpath of , gives , but may be chosen arbitrarily small, and this contradicts (4.7).
Let and be as in Lemma 4.3. Choose such that the ray has bounded intersection with the -set . Let be the union of the ray and the arcs , where is the set chosen following (4.9). Then one of the following holds:
by (4.8), and so (4.17) follows using (4.7). This proves Lemma 4.5.
To finish the proof suppose first that conclusion (ii) of Lemma 4.4 holds. Then Lemma 3.4 implies that has order at least . Since may be chosen arbitrarily small, this contradicts Lemma 4.6. The same contradiction, with replaced by , arises if conclusion (i) of Lemma 4.4 holds, and the proof of the theorem is complete.
The author thanks Professor J. K. Langley for the invaluable discussions on the results of this paper during his visit in summer 2008 and summer 2010 to the University of Nottingham in the U.K.
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