Show that there is no such entire function

Suppose the inequality is true. Let $f(z) = \sum_{n=0}^{\infty}a_nz^n.$ Let $g(z) = \exp (\bar z ).$ Note that

$$g(re^{it}) = \sum_{n=0}^{\infty}r^ne^{-int}/n!.$$

Using the orthogonality of the exponentials, we then get

$$(1/2\pi)\int_0^{2\pi}|f(re^{it}) - g(re^{it})|^2\, dt = |a_0-1|^2+\sum_{n=1}^{\infty}|a_n|^2r^{2n} + \sum_{n=1}^{\infty}r^{2n}/(n!)^2 \le 9r^2.$$

Because of the second series, this inequality fails for large $r,$ contradiction.