Abstract
In this article, we investigate the uniqueness problem on a transcendental entire function
{f(z)}
with its linear mixed-operators Tf, where T is a linear combination of differential-difference operators
{D^{\nu}_{\eta}:=f^{(\nu)}(z+\eta)}
and shift operators
{E_{\zeta}:=f(z+\zeta\/)}
, where
{\eta,\nu,\zeta}
are constants.
We obtain that if a transcendental entire function
{f(z)}
satisfies
{\lambda(f-\alpha)<\sigma(f\/)<+\infty}
, where
{\alpha(z)}
is an entire function with
{\sigma(\alpha)<1}
, and if f and Tf share one small entire function
{a(z)}
with
{\sigma(a)<\sigma(f\/)}
, then
{\frac{Tf-a(z)}{f(z)-a(z)}=\tau,}
where τ is a non-zero constant.
Furthermore, we obtain the value τ and the expression of f
by imposing additional conditions.
Entire function which shares two distinct finite values IM with two of its difference operators
