Exercise 28, § 2

a) Let $D\subset \mathbf{C}$ be the open unit disk. Let $f$ be a germ of a holomorphic function in a neighbourhood of D. For every $z\in D$, one has

where $k(z, w) = (z+w)/(z-w)$ for $(z, w)\in D\times D$. (Note that

and use the Taylor expansion of $f$ about 0.) b) Let $f$ be a holomorphic function on D such that $\mathscr{R}(f(z))>0$ for every $z\in D$. The measures $\mu_r=\frac{1}{2\pi}\mathscr{R}(f(re^{i\theta}))d\theta$ on $[0,2\pi ]$ converge vaguely when $r\rightarrow 1$ to a bounded positive measure $\nu$, and one has

for every $z\in D$. (Reduce to the case where $f(0) = 1$ and observe that the measures $\mu_r$ are then positive measures of total mass 1.)

c) Let $\mathbf{H}$ be the open set of $\mathbf{C}$ formed by the $z\in \mathbf{C}$ such that $\mathscr{I}(z)>0$. Let $f:\mathbf{H}\rightarrow \mathbf{C}$ be a holomorphic function. The image of $f$ is contained in $\mathbf{H}$ if and only if there exist a bounded positive measure $\nu$ on $\mathbf{R}$ and a real number $a\geqslant 0$ such that

for every $z\in \mathbf{H}$.

d) Let I be a nonempty open interval in $\mathbf{R}$ and let $f: I\rightarrow \mathbf{R}$. Put J = $\mathbf{R}-$ I. The following conditions are equivalent ("Loewner's theorem"):

(i) The function $f$ belongs to Loe(I) ;

(ii) There exists an analytic function $\widetilde{f}$ on $(\mathbf{C}-\mathbf{R})\cup I$ whose restriction to I coincides with $f$ and which satisfies $\mathscr{I}(f(z))>0$ if $\mathscr{I}(z)>0$;

(iii) There exist a bounded measure $\nu$ on J and $(a, b)\in \mathbf{R}_+\times \mathbf{R}$ such that

for every $x\in I$.

(First verify directly that (iii) implies (i) and (ii) ; to prove that (i) implies (iii), treat separately the case where $I =\mathbf{R}$ and reduce the other cases to that where $I =\mathbf{R}^*_+$, to which the preceding exercise will be applied ; finally, to prove that (ii) implies (iii), apply question c) and verify that the measure $\nu$ on $\mathbf{R}$ thereby given is supported on J.)