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Alotaibi, Abdullah Mathker
Languages: English
Types: Unknown
Subjects:
In this thesis, we mainly worked in the following areas: value distributions of meromorphic functions, normal families, Bank-Laine functions and complex oscillation theory. In the first chapter we will give an introduction to those areas and some related topics that are needed. In Chapter 2 we will prove that for a meromorphic function f and a positive integer k, the function af(f(k))n -1, n ≥ 2, has infinitely many zeros and then we will prove that it is still true when we replace f(k) by a differential polynomial. In Chapter 3 we will prove that for a merornorphic function f and a positive integer k, the function af f(k) -1 with N1(r, 1/f^((k)) ) = S(r, f) has infinitely many zeros and then we will prove that it is still true when we replace f(k) by a differential polynomial. In Chapter 4 we will apply Bloch's Principle to prove that a family of functions meromorphic on the unit disc B(0, 1), such that f(f1)m≠ 1, m ≠ 2, is normal. Also we will prove that a family of functions meromorphic on B(0,1), such that each f ≠ 0 and f(f(k))m ,k, m ∈N omits the value 1, is normal. In the fifth chapter we will generalise Theorem 5.1.1 for a sequence of distinct complex numbers instead of a sequence of real numbers. Also, we will get very nice new results on Bank-Laine functions and Bank-Laine sequences. In the last chapter we will work on the relationship between the order of growth of A and the exponent of convergence of the solutions y(k) +Ay =0, where A is a transcendental entire function with ρ(A) < 1/2.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • 1. Abdullah Alotaibi, On the zeros of af(J(k))n - 1, n ~ 2, Computational Methods and Function Theory, 4(1):227-235, 2004.
    • 2. Abdullah Alotaibi, On the zeros of af f(k) - 1, Complex Variables: Theory and Application, 49(1):977-989, 2004.
    • 3. Abdullah Alotaibi, On normal families, Arab Journal of Mathematical Sciences, 10(1):33-42, 2004.
    • 4. Abdullah Alotaibi, On Bank-Laine functions, submitted to Journal of Mathematical Analysis and Applications.
    • 5. Abdullah Alotaibi, On complex oscillation theory, submitted to Results in Mathematics.
    • [5] Steven B. Bank and Ilpo Laine. o where A is entire. Transactions 273(1):351-363, 1982.
    • [8] P. D. Barry. On a theorem of Besicovitch. Quart. J. Math. Oxford Ser. (2), 14:293-302, 1963.
    • [9] P. D. Barry. Some theorems related to the cos n p theorem. Proc. London Math. Soc. (3), 21:334-360, 1970.
    • [18] Yong Xing Ku. A criterion for normality of families of meromorphic functions. Sci. Sinica, (Special Issue I on Math.):267-274, 1979.
    • [19] Yung-hsing Ku. Sur les familles normales de fonctions meromorphes. Sinica, 21(4):431-445, 1978.
    • [37] Lawrence Zalcman. Normal families: new perspectives. Soc. (N.S.), 35(3):215-230, 1998.
    • [38] Zhong Fa Zhang and Guo Dong Song. On the zeros of f(f(k))n Chinese Ann. Math. Ser. A, 19(2):275-282, 1998.
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