  In this paper, we prove a normal criteria for family of meromorphic
functions. As an application of that result, we establish a uniqueness theorem
for entire function concerning a conjecture of R. Bruck. The above uniqueness
theorem is an improvement of a problem studied by L. Z. Yang et. al [14].
However, our method differs the method of L. Z. Yang et. al [14]. We mainly use
normal family theory and combine it with Nevanlinna theory instead of using
only the Nevanlinna theory as in [14].
