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Mediterr. J. Math. (2023) 20:183

https://doi.org/10.1007/s00009-023-02381-7

1660-5446/23/030001-22

published online April 8, 2023

c

The Author(s), under exclusive licence to Springer

Nature Switzerland AG 2023

Entire Function Sharing Two Values

with Its kth Derivative with the Glint

of Normal Family

Sujoy Majumder, Jeet Sarkar and Nabadwip Sarkar

Abstract. In the paper, we use the idea of normal family to investi-

gate the uniqueness problem of entire functions that share two values

with their kth derivatives and obtain a result which improves as well

as generalizes the recent result due to Li and Yi (Arch Math (Basel)

87(1):52–59, 2006). Also our result solves an unsolved problem of Zhang

and Yang (Comput Math Appl 60:2153–2160, 2010).

Mathematics Subject Classiﬁcation. 30D35, 30D45.

Keywords. Meromorphic functions, Derivative, Nevanlinna theory, Unique-

ness and normal family.

1. Introduction, Deﬁnitions and Results

Let M(C) be the family of non-constant functions which are meromorphic

in C, whereas E(C) denotes the family of non-constant entire functions. On

the other hand we denote by MT(C)andET(C) the family of transcendental

meromorphic functions and transcendental entire functions respectively. In

the paper for f∈M(C), we shall use the standard notations of Nevanlinna’s

value distribution theory such as T(r, f ), m(r, f ), N(r, f ), N(r, f ),S(r, f ),

... (see, e.g., [3,12]). Throughout the paper we denote by ρ(f) the order of

f∈M(C). Let f∈M(C). A meromorphic function ais said to be a small

function of fif T(r, a)=S(r, f ).

Let f,g ∈M(C)anda∈C.Ifg−a= 0 whenever f−a= 0, we write

f=a⇒g=a.Iff=a⇒g=aand g=a⇒f=a, we then write

f=a⇔g=aand we say that fand gshare QIM. If f−aand g−ahave

the same zeros with the same multiplicities, we write f=ag=aand we

say that fand gshare aCM.

Rubel and Yang [9] considered the uniqueness of an entire function

when it shares two values CM with its ﬁrst derivative. In 1977 they proved

the following well-known theorem.

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