As far as polynomials are concerned, we know by virtue of fundamental theorem of algebra that a polynomial of degree $n$ has exactly $n$ roots in $\mathbb{C}$ and its derivative then will have $n-1$ roots. Thus we can say that the number of roots of a polynomial of degree $n$ is greater than the number of roots of its derivative.
However, my concern is that can I get similar conclusion for transcendental entire functions ?
Note that for the entire function $e^z$ has no zeros in $\mathbb{C}$.
That conclusion does not hold for transcendental entire functions. As an example, if $h$ is an entire function then $f(z) = e^{h(z)}$ has no zeros, but $f'(z) = h'(z)e^{h(z)}$ can have finitely many (take $h$ as a polynomial) or infinitely many (e.g. $h(z) = \sin(z)$) zeros.
There is also no general estimate in the other direction: $g(z) = 1 + e^z$ has infinitely many zeros, but $g'(z) = e^z$ has none.