In this paper, we deal with the problem of uniqueness and weighted sharing of two meromorphic functions with their first derivatives having the same fixed points with the same multiplicities. The results in this paper improve those given
In this paper, the 3IM+1CM theorem with a general difference polynomial
L
z
,
f
will be established by using new methods and technologies. Note that the obtained result is valid when the sum of the coefficient of
L
z
,
f
is equal to zero or not. Thus, the theorem with the condition that the sum of the coefficient of
L
z
,
f
is equal to zero is also a good extension for recent results. However, it is new for the case that the sum of the coefficient of
L
z
,
f
is not equal to zero. In fact, the main difficulty of proof is also from this case, which causes the traditional theorem invalid. On the other hand, it is more interesting that the nonconstant finite-order meromorphic function
f
can be exactly expressed for the case
f
≡
−
L
z
,
f
. Furthermore, the sharpness of our conditions and the existence of the main result are illustrated by examples. In particular, the main result is also valid for the discrete analytic functions.
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