A function \(f\in\mathcal{F}\subset \mathcal{H}\) is called an extreme point of
\(\mathcal{F}\) if \(f=\mu f_{1}+ ( 1-\mu ) f_{2}\) implies \(f_{1}=f_{2}=f\) for all \(f_{1}\) and \(f_{2}\) in \(\mathcal{F}\) and \(0<\mu<1\). We shall use the notation \(E\mathcal{F}\) to denote the set of all extreme points of \(\mathcal{F}\). It is clear that \(E\mathcal{F}\subset\mathcal{F}\).
We say that \(\mathcal{F}\) is locally uniformly bounded if for each r, \(0< r<1\), there is a real constant \(M=M ( r ) \) so that \(\vert f(z)\vert \leq M\) where \(f\in\mathcal{F}\) and \(\vert z\vert \leq r\).
We say that a class \(\mathcal{F}\) is convex if \(\mu f+(1-\mu )g\in \mathcal{F}\) for all f and g in \(\mathcal{F}\) and \(0\leq\mu\leq 1\). Moreover, we define the closed convex hull of \(\mathcal{F}\), denoted by \(\overline{co}\mathcal{F}\), as the intersection of all closed convex subsets of \(\mathcal{H}\) (with respect to the topology of locally uniform convergence) that contain \(\mathcal{F}\).
A real-valued functional \(\mathcal{J}:\mathcal{H}\rightarrow \mathbb{R} \) is called convex on a convex class \(\mathcal{F}\subset \mathcal{H}\) if \(\mathcal{J} ( \mu f+ ( 1-\mu ) g ) \leq\mu \mathcal{J} ( f ) + ( 1-\mu ) \mathcal{J} ( g ) \) for all f and g in \(\mathcal{F}\) and \(0\leq\mu\leq1\).
The Krein-Milman theorem (see [16]) is fundamental in the theory of extreme points. In particular, it implies the following.
Lemma 1
If
\(\mathcal{F}\)
is a non-empty compact subclass of the class
\(\mathcal{H}\), then
\(E\mathcal{F}\)
is non-empty and
\(\overline {co}E\mathcal{F}=\overline{co}\mathcal{F}\).
Lemma 2
[7]
Let
\(\mathcal{F}\)
be a non-empty compact convex subclass of the class
\(\mathcal{H}\)
and
\(\mathcal{J}:\mathcal{H}\rightarrow \mathbb{R}\)
be a real-valued, continuous, and convex functional on
\(\mathcal{F}\). Then
$$ \max \bigl\{ \mathcal{J}(f):f\in\mathcal{F} \bigr\} =\max \bigl\{ \mathcal{J}(f):f\in E\mathcal{F} \bigr\} . $$
Since \(\mathcal{H}\) is a complete metric space, Montel’s theorem [17] implies the following.
Lemma 3
A class
\(\mathcal{F}\subset\mathcal{H}\)
is compact if and only if
\(\mathcal{F}\)
is closed and locally uniformly bounded.
Now, we are ready to state and prove our next theorem.
Theorem 5
The class
\(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\)
is a convex and compact subset of
\(\mathcal{H}\).
Proof
For \({0\leq\mu\leq1}\), let \(f_{1},f_{2}\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\) be defined by (2). Then
$$ \mu f_{1}(z)+(1-\mu)f_{2} ( z ) =z+\sum _{n=2}^{\infty } \bigl\{ \bigl( \mu a_{1,n}+ ( 1- \mu ) a_{2,n} \bigr) z^{n}+\overline{ \bigl( \mu b_{1,n}+ ( 1-\mu ) b_{2,n} \bigr) z^{n}} \bigr\} $$
and
$$\begin{aligned} &\sum_{n=2}^{\infty} \bigl\{ \gamma_{n}\bigl\vert \mu a_{1,n}+ ( 1-\mu ) a_{2,n} \bigr\vert +\delta_{n}\bigl\vert \mu b_{1,n}+ ( 1-\mu ) b_{2,n}z^{n}\bigr\vert \bigr\} \\ &\quad\leq\mu\sum_{n=2}^{\infty} \bigl\{ \gamma_{n}\vert a_{1,n}\vert +\delta_{n}\vert b_{1,n}\vert \bigr\} + ( 1-\mu ) \sum_{n=2}^{\infty} \gamma_{n}\vert a_{2,n}\vert +\delta_{n}\vert b_{2,n}\vert \\ &\quad\leq\mu ( B-A ) + ( 1-\mu ) ( B-A ) =B-A. \end{aligned}$$
Thus, the function \({\phi}=\mu f_{1}+(1-\mu)f_{2}\) belongs to the class \(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\). This means that the class \(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\) is convex.
On the other hand, for \(f\in\mathcal{H}_{\mathcal{T}}^{\lambda }(A,B)\), \(\vert z\vert \leq r\) and \(0< r<1\), we have
$$ \bigl\vert f(z)\bigr\vert \leq r+\sum_{n=2}^{\infty} \bigl( \vert a_{n}\vert +\vert b_{n}\vert \bigr) r^{n}\leq r+\sum_{n=2}^{\infty} \bigl( \gamma_{n}\vert a_{n}\vert +\delta_{n} \vert b_{n}\vert \bigr) \leq r+ ( B-A ) . $$
Therefore, \(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\) is locally uniformly bounded. Let
$$ f_{k}(z)=z+\sum_{n=2}^{\infty}a_{k,n}z^{n}+ \sum_{n=2}^{\infty }\overline{b_{k,n}z^{n}}\quad ( z\in{\mathbb{D}}, k\in\mathbb{N}) $$
and let \(f=h+\overline{g}\) be given by (1). Using Theorem 3 we have
$$ \sum_{n=2}^{\infty} \bigl( \gamma_{n} \vert a_{k,n}\vert +\delta_{n}\vert b_{k,n} \vert \bigr) \leq B-A \quad( k\in \mathbb{N}) . $$
(10)
If we assume that \(f_{k}\rightarrow f\), then we conclude that \(\vert a_{k,n}\vert \rightarrow \vert a_{n}\vert \) and \(\vert b_{k,n}\vert \rightarrow \vert b_{n}\vert \) as \(k\rightarrow\infty \) (\(n\in\mathbb{N}\)). Let \(\{ \sigma_{n} \} \) be the sequence of partial sums of the series \(\sum_{n=2}^{\infty} ( \gamma_{n}\vert a_{n}\vert +\delta_{n}\vert b_{n}\vert ) \). Then \(\{ \sigma _{n} \} \) is a nondecreasing sequence and by (10) it is bounded above by \(B-A\). Thus, it is convergent and
$$ \sum_{n=2}^{\infty} \bigl( \gamma_{n} \vert a_{n}\vert +\delta_{n}\vert b_{n} \vert \bigr) =\lim_{n\rightarrow \infty }\sigma_{n}\leq B-A. $$
Therefore, \(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\), and therefore the class \(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\) is closed. In consequence, by Lemma 3, the class \(\mathcal{H}_{\mathcal{T}}^{\lambda }(A,B)\) is compact subset of \(\mathcal{H}\), which completes the proof. □
Our next theorem is on the extreme points of \(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\).
Theorem 6
Extreme points of the class
\(\mathcal{H}_{\mathcal {T}}^{\lambda } ( A,B ) \)
are the functions
f
of the form (1) where
\(h=h_{n} \)
and
\(g=g_{n} \)
are of the form
$$\begin{aligned} &h_{1}(z) =z,\qquad h_{n}(z)=z-\frac{B-A}{\gamma_{n}}e^{i ( 1-n )\phi }z^{n},\\ &g_{n}(z) = ( -1 ) ^{\lambda}\frac{B-A}{\delta_{n}}e^{i ( n-1 ) \phi } \overline{z}^{n} \quad\bigl(z\in{\mathbb{D}}, n\in \{ 2,3,\ldots \} \bigr). \end{aligned}$$
(11)
Proof
Let \(g_{n}=\mu f_{1}+ ( 1-\mu ) f_{2}\) where \(0<\mu<1\) and \(f_{1},f_{2}\in\mathcal{S}_{\mathcal{T}}^{\lambda} ( A,B ) \) are functions of the form (2). Then, by (5), we have \(\vert b_{1,n}\vert =\vert b_{2,n}\vert =\frac{B-A}{\delta _{n}}\), and therefore \(a_{1,k}=a_{2,k}=0\) for \(k\in \{ 2,3,\ldots \} \) and \(b_{1,k}=b_{2,k}=0\) for \(k\in \{ 2,3,\ldots \} \diagdown \{ n \} \). It follows that \(g_{n}=f_{1}=f_{2}\) and consequently \(g_{n}\in E\mathcal{S}_{\mathcal{T}}^{\ast}(A,B)\). Similarly, we can verify that the functions \(h_{n}\) of the form (11) are the extreme points of the class \(\mathcal{S}_{\mathcal{T}}^{\lambda} ( A,B ) \).
Now, suppose that a function f of the form (1) belongs to the set \(E\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\) and f is not of the form (11). Then there exists \(m\in \{ 2,3,\ldots \} \) such that
$$ 0< \vert a_{m}\vert < \frac{B-A}{{\alpha}_{m}}\quad\mbox{or}\quad 0< \vert b_{m}\vert < \frac{B-A}{{\beta}_{m}}. $$
If \(0<\vert a_{m}\vert <\frac{B-A}{{\alpha}_{m}}\), then putting
$$ \mu=\frac{\vert a_{m}\vert {\alpha}_{m}}{B-A}, \qquad\phi=\frac {1}{1-\mu} ( f-\mu h_{m} ) , $$
we have \(0<\mu<1\), \(h_{m}\neq\phi\), and
$$ f=\mu h_{m}+ ( 1-\mu ) \phi. $$
Thus, \(f\notin E\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\). Similarly, if \(0<\vert b_{m}\vert <\frac{B-A}{\delta_{n}}\), then putting
$$ \mu=\frac{\vert b_{m}\vert {\beta}_{m}}{B-A},\qquad \phi=\frac {1}{1-\mu} ( f-\mu g_{m} ) , $$
we have \(0<\mu<1\), \(g_{m}\neq\phi\), and
$$ f=\mu g_{m}+ ( 1-\mu ) \phi. $$
It follows that \(f\notin E\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\), and so the proof is completed. □
It is clear that if the class \(\mathcal{F}= \{ f_{n}\in\mathcal {H}: n\in\mathbb{N}\} \) is locally uniformly bounded, then
$$ \overline{co}\mathcal{F}= \Biggl\{ \sum_{n=1}^{\infty} \mu _{n}f_{n}: \sum_{n=1}^{\infty} \mu_{n}=1, \mu_{n}\geq0\ ( n\in \mathbb{N}) \Biggr\} . $$
Thus, by Theorem 6, we have the following.
Corollary 4
Let
\(h_{n}\), \(g_{n}\)
be defined by (11). Then
$$ \mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)= \Biggl\{ \sum _{n=1}^{\infty } ( \mu_{n}h_{n}+ \delta_{n}g_{n} ) : \sum_{n=1}^{\infty} ( \mu _{n}+\delta_{n} ) =1, \delta_{1}=0, \mu_{n},\delta_{n}\geq 0\ ( n\in\mathbb{N} ) \Biggr\} . $$
For all fixed values of \(m,n,\lambda\in\mathbb{N}\), \(z\in{\mathbb{D}}\), the following real-valued functionals are continuous and convex on \(\mathcal{H}\):
$$ \mathcal{J} ( f ) =\vert a_{n}\vert , \qquad \mathcal{J} ( f ) =\vert b_{n}\vert , \qquad\mathcal {J} ( f ) =\bigl\vert f ( z ) \bigr\vert , \qquad\mathcal{J} (f ) =\bigl\vert J_{\mathcal{H}}^{\lambda}f ( z ) \bigr\vert \quad ( f\in{ \mathcal{H}} ) . $$
Moreover, for \(\mu>0\), \(0< r<1\), the real-valued functional
$$ \mathcal{J} ( f ) = \biggl( \frac{1}{2\pi} \int_{0}^{2\pi }\bigl\vert f \bigl( re^{i\theta} \bigr) \bigr\vert ^{\mu}d\theta \biggr) ^{1/\mu} \quad( f\in{ \mathcal{H}} ) $$
is continuous on \(\mathcal{H}\). For \(\mu\geq1\), by Minkowski’s inequality it is also convex on \(\mathcal{H}\).
Therefore, by Lemma 2 and Theorem 6, we have the following corollaries.
Corollary 5
Let
\(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\)
be a function of the form (8). Then
$$ \vert a_{n}\vert \leq\frac{B-A}{\gamma_{n}}, \qquad \vert b_{n}\vert \leq\frac{B-A}{\delta_{n}}\quad (n=2,3,\ldots), $$
where
\(\gamma_{n}\), \(\delta_{n}\)
are defined by (6). The result is sharp and the functions
\(h_{n}\), \(g_{n}\)
of the form (11) are the extremal functions.
Corollary 6
Let
\(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\)
and
\(\vert z\vert =r<1\). Then
$$\begin{aligned}& r-\frac{B-A}{2^{\lambda} ( 1+2B-A ) }r^{2} \leq\bigl\vert f(z)\bigr\vert \leq r+ \frac{B-A}{2^{\lambda} ( 1+2B-A ) }r^{2},\\& r-\frac{B-A}{1+2B-A}r^{2} \leq\bigl\vert J_{\mathcal{H}}^{\lambda }f(z) \bigr\vert \leq r+\frac{B-A}{1+2B-A}r^{2} \quad( \lambda =1,2,3,\ldots ) . \end{aligned}$$
The result is sharp and the function
\(h_{2}\)
of the form (11) is the extremal function.
The following covering result follows from Corollary
6.
Corollary 7
If
\(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\)
then
\({\mathbb{D}} ( r ) \subset f ( {\mathbb{D}} ) \)
where
$$ r=1-\frac{B-A}{2^{\lambda} ( 1+2B-A ) }. $$
We also conclude to the following.
Corollary 8
Let
\(0< r<1\)
and
\(\mu\geq1\). If
\(f\in\mathcal{H}_{\mathcal {T}}^{\lambda }(A,B)\)
then
$$\begin{aligned}& \frac{1}{2\pi} \int_{0}^{2\pi}\bigl\vert f\bigl(re^{i\theta} \bigr)\bigr\vert ^{\mu}d\theta\leq\frac{1}{2\pi} \int_{0}^{2\pi}\bigl\vert h_{2} \bigl(re^{i\theta}\bigr)\bigr\vert ^{\mu}d\theta, \\& \frac{1}{2\pi} \int_{0}^{2\pi}\bigl\vert J_{\mathcal {H}}^{\lambda }f(z) \bigr\vert ^{\mu}d\theta\leq\frac{1}{2\pi} \int _{0}^{2\pi }\bigl\vert J_{\mathcal{H}}^{\lambda}h_{2} \bigl(re^{i\theta}\bigr)\bigr\vert ^{\mu }d\theta \quad( \mu=1,2,\ldots ) . \end{aligned}$$