Question is :
Are there any non-constant entire functions $f$ that satisfy an inequality of the form $|f(z)|\leq A+B\log |z|$ for all $z$ with $|z|\geq 1$, where $A,B$ are positive constants.
Let $z_0\in \mathbb{C}$.
As $f$ is entire and $z_0\in \mathbb{C}$, given any $R>0$ and
the circle $D_R$ at $z_0$ of radius $R$, we have
$$f^{(1)}(z_0)=\frac{1}{2\pi i}\int_{\partial D_R} \frac{f(z)}{(z-z_0)^2}dz.$$
For simplicity, take $z_0=0$. Parametrizing the circle we have $z(\theta)=Re^{i\theta}$ then $dz=iRe^{i\theta}d\theta$.
So $$f^{(1)}(0)=\frac{1}{2\pi i}\int_0^{2\pi} \frac{f(Re^{i\theta})}{R^2e^{i2\theta}}dz.$$ Considering modulus, we have $$\left|f^{(1)}(0)\right| \leq\frac{1}{2\pi }\int_0^{2\pi} \frac{\left|f(Re^{i\theta})\right|}{R}d\theta.$$ Now, we have $|f(z)|\leq A+B \log |z|$ for all $z$ with $|z|\geq 1$. We have $|Re^{i\theta}|=R$. So, for $R>1$ we have $$|f^{(1)}(0)| \leq\frac{1}{2\pi }\int_0^{2\pi} \frac{|f(Re^{i\theta})|}{R}d\theta \leq \frac{1}{2\pi }\int_0^{2\pi} \frac{A+B\log R}{R}d\theta=\frac{A+B\log R}{R}.$$ As $R\rightarrow \infty$ we have $\frac{A+B\log R}{R}\rightarrow 0$. So, $|f^{(1)}(0)|=0$.
Everything till now is fine. Now, i wanted to do this for arbitrary $z_0$ other than $0$. Everything would be same except that this $R\geq 1$ may not work.
We want $R$ such that $\left|z_0+Re^{i\theta}\right|\geq 1$. I do not see how to get this. Using the inequality $$\left|z_0+Re^{i\theta}\right|\geq |z_0|-|Re^{i\theta}|=|z_0|-R$$ I thought considerng $R$ such that $-R\geq 1-|z_0|$ equivalently $R\leq |z_0|-1$ may work.. But then this would restrict $R$ and we can not think of $R$ going to infinity and it may happen that $|z_0|<1$ so $R$ would turn out to be negative which is not acceptable.
Seeing all this I think we only have $\left|f'(0)\right|=0$ and for arbitrary $z_0$ it may not be true. Then i have to give one counterexample $f$.. Could not think of any such.