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In this article, we prove two results on the value distribution of meromorphic functions. Using the theorem of Yamanoi, the first result gives a precise estimation of the relationship between the characteristic function of a meromorphic function and its kkth derivative in a concise form. This result extends and improves some results of Shan, Singh, Gopalakrishna, Edrei, Weitsman, Yang, Wu and Wu, etc. The second result answers a conjecture posed by C. C. Yang. This conjecture turned to be false by a counter-example, but it will be true with an additional condition.

We prove that if f is a transcendental meromorphic function of finite hyperorder such that Σ∈C{∞} δ(a, f) = 2, then f is of regular growth and K(f(k)) = 2k(1-δ (∞,f))/1+k-kδ(∞,f), where ... This result improves Theorem 1 [3] and the corresponding result in Singh and Kulkarni [8]. Moreover, by proving the main result of this paper, we will correct the a defective reasoning in the proof of Theorem 1.3 [9], which was resulted from an incorrect result of Lemma 8 [9]. An example is also provided to show that the conclusion of Lemma 8 [9] is incorrect. [ABSTRACT FROM AUTHOR]