This is Chapter 6, section 4, exercise 7 (page 141) from Functions of One Complex Variable by John B. Conway, Second Edition.
Exercise 6.4.7: Let $G=\{z:\Re (z)>0\}$ and let $f:G\rightarrow\mathbb{C}$ be analytic such that $f(1)=0$ and such that $\limsup\limits_{z\rightarrow w}|f(z)|\leq M$ for $w$ in $\partial G$. Also suppose that for every $\delta, 0<\delta<1,$ there is a constant $P$ such that $$|f(z)|\leq P\exp\left(|z|^{1-\delta}\right)$$ Prove that $$|f(z)|\leq M\left[\dfrac{(1-x)^2+y^2}{(1+x)^2+y^2}\right]^{1/2}$$ Hint: Consider $f(z)\left(\dfrac{1+z}{1-z}\right)$
I want to use Corollary 4.2 in this section which says: Let $a\geq\dfrac{1}{2}$ and put $$G=\left\{z:|\arg z|<\dfrac{\pi}{2a}\right\}$$ Suppose that $f$ is analytic on $G$ and there is a constant $M$ such that $\limsup\limits_{z\rightarrow w}|f(z)|\leq M$ for all $w$ in $\partial G$. If there are positive constants $P$ and $b<a$ such that $$|f(z)|\leq P\exp\left(|z|^b\right)$$ for all $z$ with $|z|$ sufficiently large, then $|f(z)|\leq M$ for all $z$ in $G$.