On the Cardinality of Unique Range Sets with Weight One
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ON THE CARDINALITY OF UNIQUE RANGE SETS WITH WEIGHT ONE B. Chakraborty1,2 and S. Chakraborty3
UDC 517.9
Two meromorphic functions f and g are said to share a set S ⇢ C[{1} with weight l 2 N[{0}[{1} if Ef (S, l) = Eg (S, l), where Ef (S, l) =
[� � (z, t) 2 C ⇥ N � f (z) = a with multiplicity p ,
a2S
provided that t = p for p l and t = p + 1 for p > l. We improve and supplement the result by L. W. Liao and C. C. Yang [Indian J. Pure Appl. Math., 31, No. 4, 431–440 (2000)] by showing that there exists a finite set S with 13 elements such that Ef (S, 1) = Eg (S, 1) implies that f ⌘ g.
1. Introduction and Definitions By C and N we denote the set of complex numbers and the set of natural numbers, respectively. By meromorphic function, we mean an analytic function defined on C, except possibly isolated singularities, each of which is a pole. The tool used in the present paper is the Nevanlinna theory. For the standard notations of the Nevanlinna theory, one can go through the Hayman monograph [8]. It is convenient to assume that E denotes any set of positive real numbers of finite linear measure, not necessarily the same in each case. For any nonconstant meromorphic function h(z), by S(r, h) we denote any quantity satisfying the relation S(r, h) = o(T (r, h)),
r −! 1,
r 62 E.
Suppose that f and g are two nonconstant meromorphic functions and that a 2 C. We say that f and g share a value a-CM (counting multiplicities) provided that f − a and g − a have the same zeros with the same multiplicities. Similarly, we say that f and g share the value a-IM (ignoring multiplicities) provided that f − a and g − a have the same set of zeros and the multiplicities are not taken into account. Moreover, we say that f and g share 1-CM (resp., IM), if 1/f and 1/g share 0-CM (resp., IM). In the course of investigation of the factorization of meromorphic functions, F. Gross [6] first generalized the idea of value sharing by introducing the concept of unique range set. Prior to going into the details of the paper, we first recall the definition of set sharing. Definition 1.1 [15]. For a nonconstant meromorphic function f and any set S ⇢ C [ {1}, we define Ef (S) =
[
a2S
{(z, p) 2 C ⇥ N | f (z) = a with multiplicity p}
1
Ramakrishna Mission Vivekananda Centenary College, Khardaha, West Bengal, India; e-mail: [email protected]. Corresponding author. 3 Jadavpur University, Kolcata, West Bengal, India; e-mail: [email protected]. 2
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 7, pp. 997–1005, July, 2020. Ukrainian DOI: 10.37863/umzh.v72i7.6022. Original article submitted May 29, 2017; revision submitted December 8, 2017. 1164
0041-5995/20/7207–1164
© 2020
Springer Science+Business Media, LLC
O N THE C ARDINALITY OF U NIQUE R ANGE S ETS WITH W EIGHT O NE
and E f (S) =
[
a2S
�
1165
{(z, 1) 2 C ⇥ N | f (z) = a with multiplicity p}.
� If Ef (S) = Eg (S) resp., E f (S) = E g (S) , then it is said that f and g share the set S counting multiplicities (CM) [resp., ignoring multipliciti
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