No abstract

We prove two uniqueness theorems for entire functions of finite order that share one finite value with one or two of their derivatives. ᮊ

In 1933 Henri Cartan proved a fundamental theorem in Nevanlinna theory, namely a generalization of Nevanlinna's second fundamental theorem. Cartan's theorem works very well for certain kinds of problems. Unfortunately, it seems that Cartan's theorem, its proof, and its usefulness, are not as widely known as they deserve to be. To help give wider exposure to Cartan's theorem, the simple and general forms of the theorem are stated here. A proof of the general form is given, as well as several applications of the theorem.

No abstract

We prove two sharp inequalities for the growth of solutions of certain linear differential equations in the unit disk. For the proofs of these inequalities, we use the method of successive approximations and sharp estimates for the logarithmic derivatives of finite order meromorphic functions in the unit disk. These techniques can also be used to give an alternate proof of a well‐known result in the plane. The sharp logarithmic derivative estimates are a corollary of general estimates, and all these estimates have independent interest. 2000 Mathematics Subject Classification 34M10, 30D35.