On weighted compositions preserving the Carathéodory class

Let ${\mathbb{D}}$ denote the unit disk in the complex plane and ${\mathcal{H}}({\mathbb{D}})$ the algebra of all functions analytic in ${\mathbb{D}}$. If $\phi\in{\mathcal{H}}({\mathbb{D}})$ and $\phi({\mathbb{D}})\subset{\mathbb{D}}$, we will say that $\phi$ is an analytic self-map of the disk. If, moreover, such $\phi$ satisfies $\phi(0)=0$, we will refer to it as a Schwarz-type function (as in the classical Schwarz lemma).

Denote by ${\mathcal{P}}$ the Carathéodory class of all $f$ in ${\mathcal{H}}({\mathbb{D}})$ with positive real part and normalized so that $f(0)=1$. An important example of a function in this class is the so-called half-plane mapping $\ell$ given by

This conformal map of the disk onto the right half-plane is extremal in many senses for the class ${\mathcal{P}}$. This is manifested, for example, by the growth theorem for the functions in the class:

The above estimate [11, Section 2.1] is a direct corollary of the Schwarz lemma and the elementary subordination principle since every function $f$ in ${\mathcal{P}}$ is of the form $\ell\circ\omega$, where $\omega$ is some Schwarz-type function. In the particular case when $\omega(z)=\lambda z$ with $|\lambda|=1$, we will use the symbol $\ell_{\lambda}$ to denote the functions $\ell\circ\omega$; that is, $\ell_{\lambda}(z)=(1+\lambda z)/(1-\lambda z)$. In view of the Herglotz representation theorem [8, Chapter 1], the class ${\mathcal{P}}$ equals $\overline{\textrm{co\,}({\mathcal{L}})}$, the closed convex hull of the collection ${\mathcal{L}}=\{\ell_{\lambda}\,\colon\,|\lambda|=1\}$ in the topology of uniform convergence on compact subsets of ${\mathbb{D}}$ (the compact-open topology). Every Schwarz-type function has radial limits almost everywhere on the unit circle ${\mathbb{T}}$ with respect to the normalized arc length measure $dm=d\theta/(2\pi)$ (see [7, Chapter 1]), hence so does every $f$ in ${\mathcal{P}}$.

Whenever $\phi$ is an analytic self-map of the disk, it is immediate that $f\circ\phi\in{\mathcal{P}}$ for every $f\in{\mathcal{P}}$ if and only if $\phi$ is a Schwarz-type function. Thus, it makes sense to consider the composition transformation $C_{\phi}$ on ${\mathcal{P}}$ defined by the formula $C_{\phi}(f)=f\circ\phi$. For the theory of composition operators on Banach or Hilbert spaces of analytic functions, we refer the reader to [5] or [13].

It also seems reasonable to consider the multiplication transformations on ${\mathcal{P}}$, given by $M_{F}(f)=Ff$, where $F\in{\mathcal{H}}({\mathbb{D}})$. While there are many cases of such transformations in spaces of analytic functions, it turns out that such mapping can preserve the class ${\mathcal{P}}$ only in the trivial case when $F\equiv 1$.

One can consider the more general weighted composition transformations $T_{F,\phi}$ on ${\mathcal{P}}$, defined by a composition followed by a multiplication: $T_{F,\phi}(f)=F(f\circ\phi)$, where $F$, $\phi\in{\mathcal{H}}({\mathbb{D}})$ and $\phi({\mathbb{D}})\subset{\mathbb{D}}$, and ask whether such a transformation acts on ${\mathcal{P}}$ or preserves ${\mathcal{P}}$ in the sense that $T_{F,\phi}(f)\in{\mathcal{P}}$ for every $f$ in ${\mathcal{P}}$. This will make sense more often since one of the two symbols $F$ and $\phi$ can compensate for the behavior of the other. Also, the nonlinearity of such transformations in our context makes the analysis non-trivial.

The linear weighted composition operators have been studied in a number of papers in the context of Banach spaces of analytic functions. The earliest references on this type of operators are [9] and [10]; among the numerous more recent references, we mention [1], [4], or [6]. Although the class ${\mathcal{P}}$ has no linear space structure, studying the question of when a weighted composition transformation preserves this class should be of interest as it would indicate the degree of rigidity when trying to produce new functions with positive real part from the given ones by composing and multiplying. Moreover, the knowledge of weighted composition transformations that preserve the class ${\mathcal{P}}$ could even have some applications to variational methods for solving non-linear extremal problems in geometric function theory.

The question considered in the present paper is to characterize the ordered pairs of functions $(F,\phi)$ for which $F(f\circ\phi)\in{\mathcal{P}}$ whenever $f\in{\mathcal{P}}$. We solve this problem completely by exhibiting explicit conditions of geometric and analytic character which are equivalent to this property and are relatively easy to check in practice. The precise statement is formulated as Theorem 2.

Although this result is conclusive, the work does not end here by any means. In some situations the theorem allows us to deduce the following rigidity principle: if $\phi$ is of certain type, then the only $F$ possible in the trivial case: $F\equiv 1$; see Proposition 3. Alternatively, when $F$ is of special type then $\phi$ must be unique; cf. Proposition 4.

Another issue of interest is to construct non-trivial examples and to interpret the statement of Theorem 2 for some specific types of maps $F$ and $\phi$ in terms of their qualitative behavior (image compactly contained in ${\mathbb{D}}$, radial limits, angular derivatives, etc.). Roughly speaking, one could ask how much we can push the conditions for these maps to the limit and when one of the symbols is “good” in some sense, how “bad” can the other one be. Thus, various statements such as propositions 6 and 7 and Theorem 10 seem to require a more intricate analysis and may be as interesting as the characterization itself. Different relevant examples are also included.

We end the paper by showing that every transformation $T_{F,\phi}$ that preserves class ${\mathcal{P}}$ has a unique fixed point whenever $\phi$ is not a rotation and this fixed point is obtained by applying the iterates of $T_{F,\phi}$ to an arbitrary function in ${\mathcal{P}}$, as one would expect. This is the content of Theorem 11. In the case when $\phi$ is a rotation, one can easily describe all fixed points. Even though the transformation is non-linear, here one can adapt the methods from linear functional analysis used in the study of composition operators.

We begin by recording the obvious necessary conditions that must be satisfied by all admissible symbols $F$ and $\phi$, i.e., by those which satisfy the initial assumptions $F$, $\phi\in{\mathcal{H}}({\mathbb{D}})$, $\phi({\mathbb{D}})\subset{\mathbb{D}}$ and make the inclusion $T_{F,\phi}({\mathcal{P}})\subset{\mathcal{P}}$ possible.

$\bullet$ If $F(f\circ\phi)\in{\mathcal{P}}$ for all $f$ in ${\mathcal{P}}$, it is clear that then $F\in{\mathcal{P}}$; this is immediately seen by choosing $f\equiv 1$, which is a function in ${\mathcal{P}}$.

$\bullet$ In addition to this, $\phi$ must be a Schwarz-type function since, after choosing $f(z)=1+z$, another function obviously in ${\mathcal{P}}$, we get

hence $\phi(0)=0$.

Thus, from now on we shall always work assuming these hypotheses: $F\in{\mathcal{P}}$ and $\phi$ is a Schwarz-type function.

It is quite easy to establish the lack of non-trivial pointwise multipliers of ${\mathcal{P}}$.

We now characterize in different ways all admissible pairs of symbols. Note that condition (b) below simply states that it suffices to test the action of $T_{F,\phi}$ on the set ${\mathcal{L}}=\{\ell_{\lambda}\,\colon\,|\lambda|=1\}$ of all rotations of the half-plane function in order to know whether the transformation preserves ${\mathcal{P}}$. Condition (c) gives an effective analytic way of testing if a symbol is admissible or not while (d) provides conditions of geometric type. Each can be useful in its own way.

Throughout the paper, we shall consider the principal branch of the argument function with values in $(-\pi,\pi]$. Note that for any function $f$ in ${\mathcal{P}}$, the function $\arg f$ takes on the values only in $(-\pi/2,\pi/2)$ and is a continuous function in the disk. Moreover, the argument of the product of two such functions $f$ and $g$ in ${\mathcal{P}}$, with values in $(-\pi,\pi)$, is still continuous and the formula

holds throughout ${\mathbb{D}}$. We will use this fact repeatedly.

It should be noted that in the above result the inequalities in conditions (c) and (d) are both invariant under rotations of $\phi$ but not under the rotations in $\omega$ (or under the appropriate changes in $F$).

As is usual, for a bounded analytic function $\phi$ in ${\mathbb{D}}$ we write $\|\phi\|_{\infty}=\sup_{z\in{\mathbb{D}}}|\phi(z)|=\operatorname{ess\,sup}_{\zeta\in{\mathbb{T}}}|\phi(\zeta)|$.

Recall that bounded analytic functions in ${\mathbb{D}}$ have radial limits $\phi(\zeta)=\lim_{r\to 1}\phi(r\zeta)$ for almost every point $\zeta$ on the unit circle ${\mathbb{T}}$ with respect to the normalized Lebesgue arc length measure [7, Chapter 1]. An analytic function $\phi$ in the disk is called inner if $|\phi(z)|\leq 1$ for all $z$ in ${\mathbb{D}}$ (equivalently, $\|\phi\|_{\infty}\leq 1$) and also $|\phi(\zeta)|=1$ almost everywhere on ${\mathbb{T}}$. The following result generalizes our Proposition 1.

Here is the counterpart of this statement with assumptions on $\omega$.

It is easily seen from Theorem 2 that any admissible multiplication symbol $F$ can only carry a very small portion of the boundary of the unit disk to the imaginary axis.

In the context of (linear) weighted composition transformations the case in which $F=\phi^{\prime}$ is often important. However, in our context it should be noted that in this case we only obtain another rigidity situation. Namely, assuming that $T_{\phi^{\prime},\phi}({\mathcal{P}})\subset{\mathcal{P}}$ and choosing $f\equiv 1$ we get $\phi^{\prime}\in{\mathcal{P}}$ hence $\phi^{\prime}(0)=1$. The case of equality in the Schwarz lemma forces $\phi(z)=z$, hence $F=\phi^{\prime}\equiv 1$, so our transformation $T_{\phi^{\prime},\phi}$ reduces to the identity map.

Many non-trivial examples of weighted composition transformations that preserve class ${\mathcal{P}}$ are possible when $\phi({\mathbb{D}})$ is compactly contained in ${\mathbb{D}}$ or $F({\mathbb{D}})$ is contained in a sector, as the following results show.

We now formulate a counterpart of Proposition 6 with similar hypotheses on $\omega$ instead of $\phi$ which follows from our previous result.

Many interesting examples in geometric function theory are obtained from special types of conformal mappings such as the lens maps. In what follows, for $0<\alpha<1$ we will denote by $\lambda_{\alpha}$ the standard lens map given by the formula

It is elementary that $\lambda_{\alpha}$ is a conformal map of the unit disk onto a lens-shaped region $L_{\alpha}$ bounded by two circular arcs (symmetric with respect to the real axis) that intersect at the points $\pm 1$ forming an angle of opening $\pi\alpha$ at each of these points; see [13, p. 27].

The following simple geometric observation will be useful. The half-plane map $\ell$ is bijective between a lens-shaped region $L_{\alpha}$ and an angle with vertex at the origin and maps in a one-to-one fashion the largest disk contained in the lens-shaped region onto a disk tangent to the legs of the angle.

Our next result essentially shows that when the multiplication symbol is obtained by composing the half-plane map with a lens map, the statements of Proposition 6 and Proposition 7 can be unified into a single “if and only if” statement.

Recall that an analytic self-map $\phi$ of ${\mathbb{D}}$ is said to have an angular derivative $\phi^{\prime}(\zeta)$ (in the restricted sense of Carathéodory [3, § 298-299]) at a point $\zeta$ on the unit circle ${\mathbb{T}}$ if it satisfies the following two conditions:

(a) the nontangential limit of $\phi$ at $\zeta$ has modulus one;

(b) $\phi^{\prime}(z)$ has a finite nontangential limit as $z\to\zeta$.

The Julia-Carathéodory theorem (see [5] or [13]) states that $\phi$ has an angular derivative at $\zeta$ if and only if

and, in this case, $|\phi^{\prime}(\zeta)|$ equals the above (unrestricted) lower limit (note that this limit is always strictly positive [13, p. 57]. Otherwise it is understood that $|\phi^{\prime}(\zeta)|=\infty$.

Even though the angular derivative of $\phi$ need not exist anywhere on ${\mathbb{T}}$ as a finite number, the function $|\phi^{\prime}|:{\mathbb{T}}\to[0,\infty]$ is well defined in this extended sense; being lower semicontinuous ([2], Lemma 2.5), it attains its minimum on ${\mathbb{T}}$ (cf. also [5, Proposition 2.46]).

One ought to keep in mind that the function $|\phi^{\prime}|$ on ${\mathbb{T}}$ as above, in general, does not coincide at all with the modulus of the boundary values of $\phi^{\prime}$ (if those exist). The most obvious example is the linear map $\phi(z)=az+b$ onto a disk compactly contained in ${\mathbb{D}}$, which happens precisely when $|a|+|b|<1$. Its usual derivative is constant everywhere, while the angular derivative does not exist at any point on the boundary; in this case, we interpret that $|\phi^{\prime}(\zeta)|=\infty$ for every point $\zeta$ on the unit circle.

The concept of angular derivative is fundamental in the study of compactness of composition operators on Hardy and Bergman spaces, as well as in the iteration of analytic self-maps of the unit disk.

Our next result shows that if $\phi$ possesses even a mildly reasonable boundary behavior at a point on the unit circle then $\omega$ automatically cannot be “too good” at the same point.

There are two ways in which the function $\omega$ can fail to have angular derivative: either it does not have a radial limit of modulus one or it does but the differential quotient fails to have a limit at the point in question. Here is an example of the first kind. It deals with the map $\phi$ such that $\phi({\mathbb{D}})$ has a tangential contact with the unit circle. The price we pay for this is that $\omega$ is a dilated self-map of the disk (hence, in this example $\omega({\mathbb{D}})$ is compactly contained in ${\mathbb{D}}$).

The natural question arises as to whether it is possible to have an example where both $\phi$ and $\omega$ can have radial limits of modulus one at the same point (obviously, without having an angular derivative at the point in question) but the weighted composition $T_{F,\phi}$ still preserves ${\mathcal{P}}$. The following example, illustrated by the figure below, gives an affirmative answer.

Note, however, in relation to this “leaf-shaped” region that our mapping $\omega=\phi$ has boundary contact with the unit circle but does not have angular derivative at $z=1$. The intuitive reason for this is that the corners at $-1$ and $1$ are contained in lens-shaped regions and lens maps do not have angular derivatives. A rigorous proof of this fact can be given by using subordination. Alternatively, one can write the equation of the boundary of $\Omega$ in polar coordinates:

Solving for $r$, one obtains

and by elementary calculus one checks that for a fixed $\varepsilon>0$ the integral

diverges and the conclusion follows by the Tsuji-Warschawski criterion [13, p. 72].