- Research Article
- Open Access
© 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.
- Hayman WK: Meromorphic Functions, Oxford Mathematical Monographs. Clarendon Press, Oxford, UK; 1964:xiv+191.Google Scholar
- Laine I: Nevanlinna Theory and Complex Differential Equations, de Gruyter Studies in Mathematics. Volume 15. W. de Gruyter, Berlin, Germany; 1993:viii+341.View ArticleGoogle Scholar
- Amemiya I, Ozawa M: Nonexistence of finite order solutions of . Hokkaido Mathematical Journal 1981, 10: 1–17.MathSciNetView ArticleMATHGoogle Scholar
- Frei M: Über die subnormalen Lösungen der Differentialgleichung (Konst.) . Commentarii Mathematici Helvetici 1961, 36: 1–8.MathSciNetView ArticleMATHGoogle Scholar
- Langley JK: On complex oscillation and a problem of Ozawa. Kodai Mathematical Journal 1986, 9(3):430–439. 10.2996/kmj/1138037272MathSciNetView ArticleMATHGoogle Scholar
- Ozawa M: On a solution of . Kodai Mathematical Journal 1980, 3(2):295–309. 10.2996/kmj/1138036197MathSciNetView ArticleMATHGoogle Scholar
- Gundersen GG: On the question of whether can admit a solution of finite order. Proceedings of the Royal Society of Edinburgh. Section A 1986, 102(1–2):9–17. 10.1017/S0308210500014451MathSciNetView ArticleMATHGoogle Scholar
- Wang J, Laine I: Growth of solutions of second order linear differential equations. Journal of Mathematical Analysis and Applications 2008, 342(1):39–51.MathSciNetView ArticleMATHGoogle Scholar
- Hayman WK: On the characteristic of functions meromorphic in the plane and of their integrals. Proceedings of the London Mathematical Society. Third Series 1965, 3(14):93–128.MathSciNetView ArticleMATHGoogle Scholar
- Edrei A, Fuchs WHJ: Bounds for the number of deficient values of certain classes of meromorphic functions. Proceedings of the London Mathematical Society. Third Series 1962, 12: 315–344. 10.1112/plms/s3-12.1.315MathSciNetView ArticleMATHGoogle Scholar
- Bergweiler W, Langley JK: Zeros of differences of meromorphic functions. Mathematical Proceedings of the Cambridge Philosophical Society 2007, 142(1):133–147. 10.1017/S0305004106009777MathSciNetView ArticleMATHGoogle Scholar
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