An extension of Nevanlinna value distribution theory for algebroid functions on annuli is proposed. The main characteristics are one-parameter and possess the same properties as in the classical case. Analogs of the Cartan theorem, the first fundamental theorem, the second fundamental theorem, deficient values, and the uniqueness of algebroid functions on annuli are proved. | Turk J Math (2015) 39: 293 – 312 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article The fundamental theorems of algebroid functions on annuli Yang TAN∗, Qingcai ZHANG School of Information, Renmin University of China, Beijing, . China Received: • Accepted/Published Online: • Printed: Abstract: An extension of Nevanlinna value distribution theory for algebroid functions on annuli is proposed. The main characteristics are one-parameter and possess the same properties as in the classical case. Analogs of the Cartan theorem, the first fundamental theorem, the second fundamental theorem, deficient values, and the uniqueness of algebroid functions on annuli are proved. Key words: Nenanlinna theory, the first fundamental theorem, the second fundamental theorem, the Cartan theorem, deficient values and the uniqueness of algebroid functions on annuli 1. Introduction Several problems lead us to study meromorphic functions in multiply connected domains. In particular, considering the composition f ◦ R , f is transcendental meromorphic in C and a rational function R with n − 1 distinct poles in C. We obtain a meromorphic function in an n -connected domain. Many authors have studied meromorphic functions in multiply connected domains and generalized the Nevanlinna theory. In particular, they have generalized the first fundamental theorem, the second fundamental theorem, and other important theorems in doubly connected domains [2,3] similar to these theories on plane [4–6,8,9,12,14]. As the extension of meromorphic functions, we have many wonderful achievements on algebroid functions [7,10,11,13,15–18,20]. Naturally, we have this question: whether these wonderful achievements of algebroid functions on plane can be generalized to multiply connected domains. We want to answer this question. In this paper, we mainly study doubly connected domains. By the doubly .
