On zeros and deficiencies of differences of meromorphic functions

For a transcendental entire function f ( z ) in the complex plane, we study its divided differences G n ( z ) . We partially prove a conjecture posed by Bergweiler and Langley under the additional condition that the lower order of f ( z ) is smaller than 1 2 . Furthermore, we prove that if zero is a deficient value of f ( z ) , then δ ( 0 , G ) < 1 , where G ( z ) = ( f ( z + c ) − f ( z ) ) / f ( z ) .